# Algebra/Printable version

Algebra

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# Welcome To Algebra!

This Wikibook helps you to learn algebra. If you wish to read this book, click the link above.

# Authors

## Authors

The authors of Algebra include:

# Contributors

## Ongoing merging work

TO DO

To be merged/unsorted

1. In this book
1. Theory (A section pertaining to proofs can potentially can go under Elementary Arithmetic chapter)
2. Iteration
3. Stemplots (Do we want this?)
4. Talk:Algebra/ToInclude
5. Arithmetic/Numerical Axioms (Was orphaned from the book, should keep an eye out for others)
6. Equalities and Inequalities (Was orphaned from the book)
7. Algebra
9. Solving equations
10. Theory of Equations
2. Redirect to another book (why?)
1. Logarithms

ORPHANED PAGES

# What is math, exactly?

Of all the subjects taught in schools throughout the world, mathematics is possibly the one which has collected the image of being most feared and disliked.

So, to start, what is math? What makes it so different from other areas of study, like languages or history? And more importantly, why on Earth do you need to know anything about it?

## Definition

Mathematics is such a wide and broad field of study. To define it would be very hard.

### Study of patterns

Mathematics is a study of patterns—finding patterns and explaining why such patterns exist. Patterns are everywhere: shapes (What is the area of this? What is the volume of that?), counting (How many ways are there to do this? How many are there of that?), and more.

One particularly interesting class of patterns is the patterns of numbers. Whole numbers, for example, look simple but they're not: 1, 2, 3... everyone knows what they are, everyone is familiar with their addition and multiplication. But there are subtle and profound patterns lurking. For example, we can look at what numbers are formed when we repeatedly add 2. 2, 4, 6, 8... We can look at what numbers are formed when we repeatedly add 3: 3, 6, 9, 12... It's easy to see that every number falls into at least one of these sequences, but how many does it fall into, and which ones? For example, 12 is in the sequence 2, 4, 6, 8, 10, 12...; 3, 6, 9, 12...; 4, 8, 12...; 6, 12...; and 12... What numbers fall into only one sequence? These numbers get a special name, prime numbers, because they are only divisible by one and themselves. Finding prime numbers is an extremely tricky question and forms the mechanism for protecting your privacy on the internet. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. How many of them are there? 3 and 5 are 2 apart, so are 5 and 7, and 11 and 13, and 17 and 19. How many pairs are there like this?

There are all sorts of questions to ask about these simple sequences, and some of them lead to very profound statements about the structure of numbers.

But why are there such patterns? Why is it that there has to be an infinite number of primes? It's possible to imagine that this stream of numbers just stops, maybe after billions and billions of primes, maybe there are so many that people will never be able to compute them all—but why is that impossible? Why is it that if you imagine that, then, if you are very imaginative, you will see that it is absurd?

The asking of questions about pure patterns is mathematics, as well as answering why it must be that way.

### Art of making conjectures and theorems

Mathematics is an art of making conjectures and theorems—guesses about patterns. It is about explaining patterns like primes (How many are there?). There are so many conjectures and theorems, especially in geometry.

Conjectures and theorems are important in math. They say what can be done and what cannot. These things are discoveries, made by people who looked at patterns and found rules about them. For example, the numbers formed when we multiply by 2 are called the even numbers. There are special rules about them. For example, if you add any two even numbers together, you get another even number. That is what is called a theorem. It is a discovery about a pattern.

### Study of logic

Mathematics is a study of logic—proving the conjectures. It is about showing why the conjectures and theorems are true.

This is important in geometry. In fact, many mathematicians want to prove many things, even up to today. There are many conjectures about patterns that still aren't proven.

### Conclusion

Now, why would you want to know about it? Simple. Because it is fun and because it is interesting.

## Understanding Our World

To give an exact definition for a subject as broad as mathematics is not easy. It is not just the study of numbers but taking what we know, realizing patterns and organizing it all into a something that we can work with and understand. Throughout the history of math there have been several ways to organize numbers, the most common way now is the decimal system but even today we still use several others. When man was just a nomad there was no need for numbers, the number of people in clan was small and all you had worry about was food and surviving day to day. As we started to settle down, make camps, towns and eventually cities and empires we needed new ways to talk about numbers. How many sheep are in the flock? How many people live in the tribe? How far is it to the next town? There were all things we needed numbers for to be able to communicate and understand what others were saying. There are countless things that we describe using numbers and quantities to understand their meaning.

## Discovery

A large part of Mathematics is discovery. After we defined the numbers and how we were going to measure lengths there was math everywhere just waiting to be unveiled. The area of a rectangle has always been the length times the width(${\displaystyle A=lw}$ ) but until we had numbers to define our length and width it could not be discovered. Volumes, prime numbers and multiples are all parts of mathematics that were just waiting to be discovered. The discoveries now are a lot more complicated to understand but they still exist.

## Invention

Often math research leads us to a point where we can’t go any further without a little invention. Imaginary numbers, which you will work with later in this book, are one example of where we had to create something to make the math work. Imaginary numbers in the real world do not exist but they have to exist to explain some of the things that happen in our physical world. We invented them to make math work in a manner that could explain the patterns we could observe.

## Abstract

Mathematics is where we find some our first abstract thoughts. A number is not the same as a letter; letters each make a noise and when we put them together they make a word that represents a noun or verb or some other part of speech. Every number represents a different quantity and even more confusing is that two 3s do not make 6 but 33. It is easy to see that 3 fingers on your left hand and 4 fingers on your right come together to make 7 and that is why you will often see grade school students adding in this fashion. Even as we become more accustomed to adding, the simple facts, like adding the numbers between 1 and 9, are often memorized instead of picturing 5 of something and 8 more of them in our mind we memorize that 8 + 5 = 13. To convince you just little bit more that numbers are abstract define the letter B using only words in the definition. Now try to define the number 6 without using numbers.

# Who should read this book

This book is intended to be a comprehensive look at the mathematics topic of algebra. That said, it would be well suited for a wide variety of individuals, ranging from students (at any grade level) to adults interested in refreshing or improving their understanding of basic math. It could be used either as a primary text or a reference.

This book will avoid explaining subjects with only rigorous mathematical abstractions whenever possible. Math can be tricky and frustrating enough without it seeming inaccessible. So while every topic will be covered fully and correctly and in many times using proper mathematical terminology, there will always be a backup definition or simple explanation to complete the concept. This allows a wider variety of individuals to learn from this text, from an ambitious 12-year-old to a forgetful college professor. Algebra is applicable to your daily life in addition to academic settings, so an algebra textbook should be accessible to everyone.

## Prerequisites

While this book is meant to be accesible to everyone, it is advisable to get a very good grasp on arithmetic before taking a deep dive into algebra. For this reason, the very first chapter of this book acts as a comprehensive review of all of the prequesites that are necessary to start tackling the many topics the book has to offer. This chapter can be skipped entirely if you are confident enough with your ability to perform basic arithmetic. Nonetheless, a quick glimpse at the chapter, and filling in any gaps in your knowledge wouldn't hurt. In fact, it may help one in the long run.

# Structure

## Book Scope

The book is meant to be a more advanced take on High School level Algebra than the Basic Algebra book, covering topics that are typically found in Algebra 1, Algebra 2, and Trigonometry by merging these curriculums together. Many topics that are not typically discussed in a typical Algebra class will also be covered here.

# Chapter 1

Chapter 1: Elementary Arithmetic

## Introduction

This chapter is meant to be a review of all of the prerequisites that are neccessary for understanding algebra.

# Chapter 2

Chapter 2: An Introduction to Algebra

# Chapter 3

Chapter 3: Solving Equations

# Chapter 4

Chapter 4: Inequalities

# The Coordinate (Cartesian) Plane

## A Quick Review...

What is the Cartesian Plane?

Named for "the father of analytical geometry," 17th century French mathematician René Descartes (Cartesius), uniform regular grid (Cartesian) coordinates is one system used for graphing. Many algebraic expressions lend themselves to graphical analysis. The location of a point's Cartesian plot is found by indexing numerical values (coordinates) along numbered grid lines. The trivial single R number line comprises a one dimensional, single ordinate, system with all locations existing only on that line. This study begins with graphing in two dimensions (see the two diagrams below). Plots and points are located and labeled with the offsets of their 'projections' from two number lines (R2 axes) anywhere on the page and plane.

We choose the values or ordinates for points on the Cartesian plane using two perpendicular numbered axes. By convention the point where the two axes cross is labeled as 0 on each axis, making the ordinate for their intersection a special point called the origin and labeled (0,0). Generally, the horizontal axis is labeled x and the vertical axis y. An ordered pair (x,y) specifies the location of a point P on the plane. If we don't want to talk about ordered pairs as x and y we can refer to the first variable in the ordered pair as the abscissa and the second as the ordinate. The axes cross where the abscissa is 0 and the ordinate is 0. We can generalize about the signs of the abscissa and ordinate in ordered pairs because the two axes form four quadrants when they cross. Moving counter clockwise, starting in the upper right quadrant, the quadrants are labeled I, II, III, and IV. In quadrant I, all x- and y-values are positive, in II, x is negative and y positive, in III, all values are negative, and only x is positive in IV. The ordered pair (0,0) represents the origin where the axes intersect.

We use the axes to define relationships between two sets of numbers for which we substitute variables. A relationship implies that changing the value of one variable determines the value of the other. We call the variable that is changing the independent variable and the variable whose value changes the dependent variable. Generally we let x represent the independent variable, y represent the dependent variable so that we can model the relationship between x and y with mathematical symbols. The set of numbers represented by the independent variable is called the domain. The set of numbers represented by the dependent variable is called the range. A special kind of relationship is called a function. A function is a relationship where any value in the domain maps onto one and only one value in the range.

When we graph the points of a relationship on a Cartesian plane then we can determine if the relationship is a function--all vertical lines of the plane cross our graph once and only once. Functions are useful for modeling cause and effect relationships - where the cause is the independent variable and the effect is the dependent variable.

I Positive Positive
II Negative Positive
III Negative Negative
IV Positive Negative

Note that the only thing unique about the Cartesian plane is that it contains a point that we call "the origin" and have labeled (0,0) If we draw another axis perpendicular to the two dimensional graph through the origin we can graph 3 dimensional objects (R3). Imagine a thin extensible numbered wire through the origin of the two dimensional graph. By using a 3rd coordinate this line (X, Y, Z) we can locate points that are above or below the page in space. This idea can be expanded to higher dimensions (including time) and is the basis for a field of theoretical physics called [[w:String_theory| string theory

## Determining Points

Continuous sets of points can be represented by lines or curves on a graph showing how continuous paired quantities relate to each other. Such numbers or quantities are called variables. A function, relation, or equation can define how, as an input quantity varies, a related output quantity can vary. A graph can then illustrate how these variables are related to each other. The independent variable is one which is sent to the function because a person can control or vary it at will. The dependent variable is the one that comes out of the function. The definition of the function determines the value of the dependent variable depending on the value of the independent variable.

Any equation which has two variables can effectively define a relation between the two variables and the relation can be plotted on a two-dimensional Cartesian coordinate graph. In any particular equation, relation, or function definition, numbers which stay the same, i. e. which are not variables, are often called constants. Even if someone does not know much fancy algebra, a person may start getting an idea of what a function or a two-variable equation or relation looks like in a graph by choosing various numbers for one of the variables from the domain and calculating corresponding numbers of the other variable (in the range) to determine as many ordered pairs as practical and plotting those points on a graph. After enough points are plotted, one may be able to estimate what a continuous relation looks like by connecting the calculated points by a straight or curved line, depending on the situation.

Determine as many points as you feel comfortable to make a plot of this equation on a graph. Then on graph paper draw x and y axes on a grid of squares. Then plot the (x,y) ordered pairs as points on the graph. Finally, when you think you have plotted enough points to get a feel for the shape of the function, connect the points to see what the function graph looks like. If you are not sure in a part of the graph, you can always calculate more points to fill in that place.

## Plotting Points on a Graph

A formula, equation or inequality that is solved for just a single instance of one variable is an implicit relation of, and dependent in, that variable in terms of the remaining and independent variable(s). An explicit relation is one noting which variable(s) is(are) independent. A point solution is a group of values (one for each independent variable) together with a corresponding calculated dependent value. Consensus-notation comma-separates these with the dependent variable's value last in an ordered group (parenthesis are optional iff there's no ambiguity). The independent variable's(s') valid-values solution set(s) constitutes the relation's domain and the dependent's its range. A function is a relation restricted to independent value(s) which generate unique dependent values.

A complete plot depicts all possible point solutions to a relation. 'One' independent variable plus 'the' dependent variable presents a 'two' dimensional map (graph) with 'two' perpendicular (orthogonal) number lines (axes) which cross (intercept) at the origin (O). By consensus the first coordinate relates to offset along the horizontal axis and the second to the vertical (an ordered pair).

This Function Graphing (section) uses a two-dimensional rectangular Cartesian coordinate system, the first coordinate is abscissa and commonly references an x-axis and the second is ordinate for a y-axis. The axes values are numbers and are elements of and closed to R and together constitute R2. The origin point is the number pair 0,0. Positive numbers increment to the right and up and negative numbers decrement to the left and down. The notation x,y is the common form/formula of an ordered-pair point and is plotted on/in the following graph.

To remember the order of coordinates think of a roller coaster--"If you do it RIGHT, you might throw UP"; or if the graph were a street with tall buildings—you have to go along the street (RIGHT/LEFT) before you can enter and go UP the stairs; or you crawl (RIGHT/LEFT) before you walk (Stand UP).

The axes and plane field extend out as close to infinity (positive and negative) as is necessary for any analysis. In practice graphs are abbreviated to areas of examination. In our example the generic point x,y is assigned to number values (x = 3, y =4) by plotting it at point 3,4. A line and a dotted line show that this position is located by starting at (0,0), moving 3 units in the positive x direction (right), and moving 4 units in the positive y direction (upwards).

## Practice Problems

### Conceptual Questions

Problem 5.1 Determine the quadrant each point is located on.

# Functions

## Function as Box

Functions are another way of describing certain things mathematically. They are often described as a machine in a box open on two ends; you put something in one end, something happens to it in the middle, and something pops out the other end. The function is the machine inside, and it's defined by what it does to whatever you give it.

Let's say the machine has a blade that slices whatever you put into it in two and sends one half out the other end. If you put in a banana, you'd get back half a banana. If you put in an apple, you'd get back half an apple.

A good question to ask about this machine might be what happened to the other half of the piece of fruit? But, since this is algebra the things that go in and come out of functions will be numbers, so we're pretty sure the box won't fill up with numbers and break. Let's define the function to take what you give it and cut it in half, that is, divide it by two. If you put in 2, you'd get back 1. If you put in 57, you'd get back 28.5. The function machine allows us to alter expressions. Functions are typically named with a single letter. We'll call this one h for half. (There's nothing special about the letter we choose—we could just as well called this function f. The letter doesn't have to stand for anything.)

Now we need the notation. To put 2 into the function, we write ${\displaystyle h(2)}$  (read h of 2). We know that

${\displaystyle h(2)=1}$

We can also calculate

${\displaystyle h(57)=28.5}$

Using algebraic notation we can describe what this machine does as:

${\displaystyle h(x)={\frac {x}{2}}}$

Instead of listing all the things we can put in our machine we represent them with a variable ${\displaystyle x}$ . When we write ${\displaystyle h(x)}$  we are saying that we sent ${\displaystyle x}$  through the machine and it was cut in half. Using this form we don't have to count the halves that come out of our machine when we put 57 apples or oranges in our machine. We know that we put 57 or anything into our machine we will only have 28.5 of those things come out the other end. When using algebraic notation to specify a relationship; we have created something called an algebraic function definition. (This example illustrates the difference between mathematics and science and engineering. Since this is an imaginary machine we only need to specify what comes out the other end of the box. In real machines we also need to think about what to do with the halves that don't come out the other end).

## Practice Problems

Use ^ on exponents

For problems 1-6, Use algebra to define the function described.
Example: x) Three-quarters of a number
Answer: ${\displaystyle f(x)=3x/4}$


1 Four-fifths of a number

2 A number added to itself five times

3 A number increased by three times the number

4 A number multiplied by itself seven times, then reduced by the number

5 One number reduced by a different number three times

6 Half a number times three more than another number

Passing a number through the function box is called evaluating the function for a number. For problems 7-22, evaluate the described function for the given numbers.
Example: x) ${\displaystyle f(x)=3x-4}$
a) ${\displaystyle f(0)}$  b) ${\displaystyle f(4)}$
c) ${\displaystyle f(1/3)}$   d) ${\displaystyle f(1/2)}$

Answer: a) ${\displaystyle f(0)=3(0)-4=-4}$    b) ${\displaystyle f(4)=3(4)-4=8}$
c) ${\displaystyle f(1/3)=3(1/3)-4=-3}$  d) ${\displaystyle f(1/2)=3(1/2)-4=-5/2}$


${\displaystyle f(x)=10x}$

7

 ${\displaystyle f(1)=}$

8

 ${\displaystyle f(1/2)=}$

9

 ${\displaystyle f(3.5)=}$

10

 ${\displaystyle f(-1)=}$
${\displaystyle g(x)=3x-1}$

11

 ${\displaystyle g(4)=}$

12

 ${\displaystyle g(2/3)=}$

13

 ${\displaystyle g(1.1)=}$

14

 ${\displaystyle g(-3)=}$
${\displaystyle h(x)=x^{2}}$

15

 ${\displaystyle h(7)=}$

16

 ${\displaystyle h(3/5)=}$

17

 ${\displaystyle h(5.3)=}$

18

 ${\displaystyle h(-6)=}$
${\displaystyle i(x)=1/(x-4)}$

19

 ${\displaystyle i(5)=}$

20

 ${\displaystyle i(2/3)=}$

21

 ${\displaystyle i(7.25)=}$

22

 ${\displaystyle i(0)=}$

## Function as Relation

Functions can also be thought of as a subset of relations. A relation is a connection between numbers in one set and numbers in another.

In other words, each number you put in is associated with each number you get out. The difference is that in a function, every 'input' number is associated with exactly one 'output' number whereas in a relation an 'input' number may be associated with multiple or no 'output' numbers. This is an important fact about functions. Notice that the relation depicted by the diagram above is not a function because it does not meet this requirement, unlike the relation depicted by the second diagram below, which is a function.

All functions are relations. Not all relations are functions.

## Practice Problems

Which of the following definitions are functions and which are relations:

1 ${\displaystyle f(x)=x+2}$

 function relation

2 ${\displaystyle g(x)=x-2}$

 function relation

3 ${\displaystyle h(x)=x*2}$

 function relation

4 ${\displaystyle i(x)=x/2}$

 function relation
Which of the following definitions are functions and which are relations:

5 ${\displaystyle j(x)=x+x}$

 function relation

6 ${\displaystyle k(x)=x-x}$

 function relation

7 ${\displaystyle l(x)=x*x}$

 function relation

8 ${\displaystyle m(x)=x/x}$

 function relation

9 Can you write descriptions of the functions in problems 1-8: What is the difference? (answer on paper)

## Domain and Range

### Domain

The domain of a function is the set of 'input' numbers for which the function is defined. The domain is part of the definition of a function. In the function in the illustration above, the domain is {-1,1,7,1/2}.

The natural domain of an algebraically-defined function is the set of numbers for which the function is defined.

In most algebra formulas, x is usually the variable associated with Domain.

Example

The function ${\displaystyle f(x)={\sqrt {x}}}$  has a domain of ${\displaystyle x\geq 0}$  because the square root function is only defined for positive numbers (assuming that we are dealing with only real numbers).

### Range

The range of a function is the set of results or solutions to the equation for a given input. A true function only has one result for every Domain.

In most algebra formulas, y is usually the variable associated with Range. As such, it can also be expressed f(x), which says that its value is a function of x.

Example

The function ${\displaystyle f(x)=x^{2}}$  has a range of ${\displaystyle y\geq 0}$  because the square of a number is always positive.

### Functions in terms of Domain and Range

In taking both domain and range into account, a function is any mathematical formula that produces one and only one result for each input. Hence, it can be said that in a valid function, Domain (x) and Range (y) have a many to one correspondence so that every given Domain value has one and only one Range value as a result, but not necessarily vice versa. This makes sense since results can repeat, but inputs cannot.

As a result, if x is horizontal and y is vertical, a function in terms of y (e.g. y = mx + b) will produce a set of results such that if intersected by a vertical line at any point on the graph it will only pass through the graph once. An asymptotic function (one with at least one undefined result) would also count as valid since it did not pass through more than one point of the graph. This is called the "vertical line" test.

The terms domain and range can be applied to all relations and not just functions. A relation is a definition where one item in the definition's domain maps to more than one item in the definition's range. We use the terms domain and range to define the difference between a function and a relation.

## A Bit of Function Terminology

When speaking or writing about functions, different terminologies are used to describe how the functions work or what they do.

### f of x Terminology

When we write ${\displaystyle f(x)}$ , we say f of x. Thus, if we have a function defined with the equation ${\displaystyle g(x)={\frac {x+2}{7}}}$  then we say that

g of x equals the sum of x and 2 altogether divided by 7.

or

g of x is x plus 2 all over 7.

When we plug a value (say 5) into the function for x we write ${\displaystyle g(5)={\frac {5+2}{7}}={\frac {7}{7}}=1}$  but we say that

g of 5 equals 1.

Algebraic notation is the easiest way mathematicians have to express relationships defined by arithmetic operators like ${\displaystyle +,-./*,}$  exponents, and roots. Once some of these functions have been defined it is easier to refer to the function name, and to refer to the values of the function as above.

### Function "value of" Terminology

If we have a function defined with the equation ${\displaystyle g(x)={\frac {x+2}{7}}}$  then we say that The value of g at x is the sum of x and 2 altogether divided by 7. This way, ${\displaystyle g(5)={\frac {5+2}{7}}={\frac {7}{7}}=1}$

## Piecewise-defined function

A function whose definition depends on the input.

### Absolute value function

f(x)=|x|
or
${\displaystyle f(x)={\begin{cases}x&{\mbox{if }}x\geq 0\\-x&{\mbox{if }}x<0\end{cases}}}$

One can interpret ${\displaystyle |x|}$  as the undirected distance between x and 0, (which is always non-negative). Going on, ${\displaystyle |x-y|}$  can be interpreted as the distance between the numbers x and y on the number line.

## Even and odd functions

### Even functions

An even function is defined as a function ${\displaystyle f}$  such that ${\displaystyle f(-x)=f(x)}$ .
Geometrically an even function can be defined as a function that exhibits a mirror image symmetry across the y-axis (the vertical line that passes through the origin).

An example of an even function is ${\displaystyle h(x)=x^{2}}$  because ${\displaystyle f(5)=25=f(-5)}$  and because ${\displaystyle f(x)=x^{2}=f(-x)}$  for all real numbers x.

### Odd functions

An odd function is defined as a function ${\displaystyle f}$  such that ${\displaystyle f(-x)=-f(x)}$ .
Geometrically an odd function can be defined as a function that exhibits a 180 degree rotational symmetry about the origin.

An example of an odd function is ${\displaystyle f(x)=x^{3}}$  because for all real numbers x, ${\displaystyle f(x)=x^{3}=-((-x)^{3})=-f(-x)}$  for example ${\displaystyle f(2)=2^{3}=8=-((-2)^{3})=-(-8)=-((-2)^{3})=-f(-2)}$

## Composite function

A composite function ${\displaystyle h}$  can be defined as the composite of the two functions ${\displaystyle f}$  and ${\displaystyle g}$  and denoted as ${\displaystyle h(x)=f(g(x))}$  (read h of x is equal to f of g of x) or ${\displaystyle h(x)=(f\circ g)(x)}$ .

Example:

Let ${\displaystyle f(x)=2x+1}$      ${\displaystyle g(x)=5x-3}$
${\displaystyle h(x)=f(g(x))}$
${\displaystyle h(x)=f(5x-3)}$
${\displaystyle h(x)=2(5x-3)+1}$
${\displaystyle h(x)=10x-6+1}$
∴${\displaystyle h(x)=10x-5}$


Example:

Let ${\displaystyle f(x)=-{\sqrt {16-x}}\,}$      ${\displaystyle g(x)=4x^{2}\,}$
${\displaystyle (f\circ g)(x)=f(4x^{2})\,}$
${\displaystyle (f\circ g)(x)=-{\sqrt {16-(4x^{2})}}\,}$
${\displaystyle (f\circ g)(x)=-{\sqrt {4(4-x^{2})}}\,}$
${\displaystyle (f\circ g)(x)=-{\sqrt {4}}{\sqrt {(4-x^{2})}}\,}$
${\displaystyle (f\circ g)(x)=-2{\sqrt {4-x^{2}}}\,}$
Domain: ${\displaystyle -2\leq x\leq 2}$
Range: ${\displaystyle -4\leq y\leq 0}$


## Inverse function

The function ${\displaystyle g}$  is the inverse of the one-to-one function ${\displaystyle f}$  if and only if the following are true:

${\displaystyle g(f(x))=x\,}$
${\displaystyle f(g(x))=x\,}$

The inverse of function ${\displaystyle f}$  is denoted as ${\displaystyle f^{-1}}$  .

Geometrically ${\displaystyle f^{-1}}$  is the reflection of ${\displaystyle f}$  across the line ${\displaystyle y=x}$ . Conceptually, using the box analogy, a function's inverse box undoes what the function's regular box does.

Example:

${\displaystyle f(x)=2x\,}$
${\displaystyle f^{-1}(x)={\frac {1}{2}}x\,}$

${\displaystyle f(f^{-1}(x))=f({\frac {1}{2}}x)\,}$
${\displaystyle f(f^{-1}(x))=2({\frac {1}{2}}x)\,}$
${\displaystyle f(f^{-1}(x))=x\,}$

${\displaystyle f^{-1}(f(x))=f^{-1}(2x)\,}$
${\displaystyle f^{-1}(f(x))={\frac {1}{2}}(2x)\,}$
${\displaystyle f^{-1}(f(x))=x\,}$


To find the inverse of a function, remember that when we use ${\displaystyle f^{-1}(x)}$  as an input to ${\displaystyle f}$  the result is ${\displaystyle x}$ . So start by writing ${\displaystyle x=f\left(f^{-1}(x)\right)}$  and solve for ${\displaystyle f^{-1}}$

Example:

Suppose:${\displaystyle f(x)=2x+1\,}$
Then ${\displaystyle x=f\left(f^{-1}(x)\right)}$
${\displaystyle x=2f^{-1}(x)+1\,}$
${\displaystyle x-1=2f^{-1}(x)\,}$
${\displaystyle f^{-1}(x)={\frac {x-1}{2}}\,}$


The Domain of an inverse function is exactly the same as the Range of the original function. If the Range of the original function is limited in some way, the inverse of a function will require a restricted domain.

Example:

${\displaystyle f(x)={\sqrt {x-1}}\,}$
${\displaystyle x=f\left(f^{-1}(x)\right)}$
${\displaystyle x={\sqrt {f^{-1}(x)-1}}\,}$
${\displaystyle x^{2}={\sqrt {f^{-1}(x)-1}}^{2}\,}$
${\displaystyle x^{2}=f^{-1}(x)-1}$
${\displaystyle f^{-1}(x)=x^{2}+1\,}$
The Range of ${\displaystyle f(x)}$  is ${\displaystyle f(x)\geq 0}$ . So the Domain of ${\displaystyle f^{-1}(x)}$  is ${\displaystyle x\geq 0}$ .


### One-to-one function

A function that for every input there exists an output unique to that input.

Equivalently, we may say that a function ${\displaystyle f}$  is called one-to-one if for all ${\displaystyle x,x'\in A,f(x)=f(x')}$  implies that ${\displaystyle x=x'}$  where A is the domain set of f and both x and x' are members of that set.

Horizontal Line Test
If no horizontal line intersects the graph of a function in more than one place then the function is a one-to-one function.

## Creating Functions

In the previous chapter we reviewed what you've already learned about mathematics: Numbers, Variables, and Relationships. We reviewed the types of numbers, the operations you can perform on numbers, the properties of these operations, and how these properties can allow you to write expressions, or if we know about enough the constraints on the expressions you can write equations and inequalities that define things that are true.

In the section above we've looked at the concept of a function. First we showed how to created equations with a function on one side of the equals operator and an expression on the other. Then we looked at more complicated ways to use function notation.

Once you get used to them functions give you a different way to look at math. When you think about math with numbers you are thinking about just one answer. When you think about math with functions you are looking for relationships and you are building mathematical models.

Give an example of a 3 X 3 square with a diagonal. What is the area of one of the triangles from the diagonal. Apply the Area function: l X w, and then the half function.

# Function Graphing

## Functions have an Independent Variable and a Dependent Variable

When we look at a function such as  ${\displaystyle f(x)={\frac {1}{2}}x,}$   we call the variable that we are changing—in this case  ${\displaystyle x\,}$  --the independent variable. We assign the value of the function to a variable we call the dependent variable. The reason that we say that  ${\displaystyle x\,}$   is independent is because we can pick any value for which the function is defined—in this case real  ${\displaystyle \mathbb {R} }$   is implied—as an input into the function. Once we pick the value of the independent variable the same result will always come out of the function. We say the result is assigned to the dependent variable, since it depends on what value we placed into the function.

Equating  ${\displaystyle y\,}$   with our function  ${\displaystyle y={\frac {1}{2}}x,}$   then  ${\displaystyle 2y=2({\frac {1}{2}}x),}$   then  ${\displaystyle 2y=x,\,}$   then  ${\displaystyle g(y)=2y.\,}$

The independent variable is now  ${\displaystyle y\,}$   and the dependent variable  ${\displaystyle x.\,}$

• Note: this is a very unusual case where the ordered pair  ${\displaystyle (g(y),y)\,}$   is reverse mapped  ${\displaystyle y\mapsto x\,}$   and corresponding reverses (dependent, independent), (range, domain), and now  ${\displaystyle x\,}$   must be singular for each and every  ${\displaystyle y\,}$   corresponding to an horizontal line test of function! It would be less desireable to rotate or swap the positions of axes, the order of coordinate pairs  ${\displaystyle (x,y)\,}$   and (abscissa, ordinate).

Have we used Algebra to change the nature of the function? Let's look at the results for three functions

${\displaystyle f(x)={\frac {1}{2}}x}$      ${\displaystyle g(y)=2y\,}$      ${\displaystyle h(x)=2x\,}$
${\displaystyle x\,}$    ${\displaystyle f(x)\,}$
${\displaystyle 2\,}$      ${\displaystyle 1\,}$
${\displaystyle 0\,}$      ${\displaystyle 0\,}$
${\displaystyle -2\,}$      ${\displaystyle -1\,}$
${\displaystyle y\,}$    ${\displaystyle g(y)\,}$
${\displaystyle 1\,}$      ${\displaystyle 2\,}$
${\displaystyle 0\,}$      ${\displaystyle 0\,}$
${\displaystyle -1\,}$      ${\displaystyle -2\,}$
${\displaystyle x\,}$      ${\displaystyle h(x)\,}$
${\displaystyle 1\,}$      ${\displaystyle 2\,}$
${\displaystyle 0\,}$      ${\displaystyle 0\,}$
${\displaystyle -1\,}$      ${\displaystyle -2\,}$

If we look at the table above we can see that the independent variable for  ${\displaystyle f(x)\,}$   gives the same results as the dependent variable of  ${\displaystyle g(y).\,}$   We can see what this means when we look at the values for  ${\displaystyle h(x).\,}$   The function  ${\displaystyle g(y)\,}$   is the same as the function  ${\displaystyle f(x),\,}$   but when we switch which variable we use as the independent variable between  ${\displaystyle g(y)\,}$   and  ${\displaystyle h(x)\,}$   we see that we have discovered that  ${\displaystyle g(y)\,}$   and  ${\displaystyle h(x)\,}$   are inverse functions.

Let's take a look at how we can draw functions in  ${\displaystyle x\,}$   and  ${\displaystyle y\,}$   and then come back and look at this idea of independent and dependent variables again.

### Explicit and Implicit Functions

Variables like  ${\displaystyle x\,}$   and  ${\displaystyle y\,}$   formulate a 'relation' using simple algebra.  ${\displaystyle f,\,}$    ${\displaystyle g,\,}$   and   ${\displaystyle h\,}$   commonly denote functions. Function notation  ${\displaystyle f(x),\,}$   read "eff of ex", denotes a function with 'explicit' dependence on the independent variable  ${\displaystyle x.\,}$   By assigning variable  ${\displaystyle y\,}$   to  ${\displaystyle f(x),\,}$    ${\displaystyle y=f(x),\,}$    ${\displaystyle y\,}$   is now an 'implicit' function of   ${\displaystyle x\,}$   using equation notation. If  ${\displaystyle f(x)\,}$   is  ${\displaystyle {\frac {x}{2}},}$   then  ${\displaystyle y={\frac {x}{2}}}$    [ ${\displaystyle y(x)\,}$   would denote an 'explicit' function of  ${\displaystyle x\,}$  ]. A relation is also a function when the dependent variable has one and only one value for each and every independent variable value.

## The Cartesian Coordinate System

The Cartesian Coordinate System is a uniform rectangular grid used for plane graph plots. It's named after pioneer of analytic geometry, 17th century French mathematician René Descartes, whose Latinized name was Renatus Cartesius. Recall that each point has a unique location, different from every other point. We know that a line is a collection of points. If we pick a direction of travel for the line that starts at a point then all of the other points can be thought of as either behind our starting point or ahead of it. Finally, a plane can be thought of as a collection of lines that are parallel to each other. We can draw another line that is composed of one point from each of the lines that we chose to fill our plane. If we do this then we can locate the other lines as behind or ahead of the line with the point we chose to start on. Descartes decided to pick a line and call it the  ${\displaystyle x\,}$  -axis, and to then pick a line perpendicular to this line and call it the  ${\displaystyle y\,}$  -axis. He then labeled this intersection point  ${\displaystyle (0,0)\,}$   and origin O. The points to the left (or behind) of this point each represent a negative number that we label as  ${\displaystyle (-x,0).\,}$   The points to the right (or ahead) of this point each represent a positive number that we label as  ${\displaystyle (x,0).\,}$   The points on the  ${\displaystyle y\,}$  -axis that are above  ${\displaystyle 0,0\,}$   are labeled as positive  ${\displaystyle (0,y),\,}$   and the points on the  ${\displaystyle y\,}$  -axis below  ${\displaystyle 0,0\,}$   are labeled as negative  ${\displaystyle (0,-y).\,}$   A point is plotted as a location on the plane using its coordinates from the grid formed by the  ${\displaystyle x\,}$   and  ${\displaystyle y\,}$  -axes. If you draw a line perpendicular to the  ${\displaystyle x\,}$  -axis from a point you pick then that point has the same  ${\displaystyle x\,}$  -coordinate as the point where that line crosses the  ${\displaystyle x\,}$  -axis. If you draw a line perpendicular to the  ${\displaystyle y\,}$  -axis from your point then it has the same  ${\displaystyle y\,}$  -coordinate as the point where that line crosses the  ${\displaystyle y\,}$  -axis. If you need to sharpen your knowledge in this area, this link/section should help: The Coordinate (Cartesian) Plane

An equation and its graph can be referred to as equal. This is true since a graph is a representation of a specific equation. This is because an equation is a group of one or more variables along with one or more numbers and an equal sign (  ${\displaystyle x=1,\,}$   ${\displaystyle y=x+1,\,}$   and  ${\displaystyle y=x^{2}+2x+1\,}$   are all examples of equations). Since variables were introduced as way of representing the many possible numbers that could be plugged into the equation. A graph of an equation is a way of drawing the relationship between the numbers that can be input (the independent variable) and the possible outputs that would be produced. For example, in the equation:  ${\displaystyle y=x+2,\,}$   we could choose to make the  ${\displaystyle x\,}$   the independent variable and the output number would be two more than the input number every time. The graph of this equation would be a picture showing this relationship. On the graph, each  ${\displaystyle y\,}$  -value (the vertical axis) would be two higher than the (horizontal)  ${\displaystyle x\,}$  -value that is plugged in because of the  ${\displaystyle +\,2\,}$   in the equation.

## Linear Equations and Functions

This section shows the different ways we can algebraically write a linear function. We will spend some time looking at a way called the "slope intercept form" that has the equation  ${\displaystyle y=f(x)=mx+b\,.\,}$

Unless a domain for  ${\displaystyle x\,}$   is otherwise stated, the domain for linear functions will be assumed to be all real numbers  ${\displaystyle \mathbb {R} }$   and so the lines in graphs of all linear functions extend infinitely in both directions. Also in linear functions with all real number domains, the range of a linear function may cover the entire set of real numbers for  ${\displaystyle y\,,\,}$   one exception is when the slope  ${\displaystyle m=0\,}$   and the function equals a constant. In such cases, the range is simply the constant. Another would be a squaring function where the range would be non-negative when  ${\displaystyle b=0\,.\,}$

## The y-intercept constant b

It was shown that  ${\displaystyle y=f(x)=mx+b\,}$   has infinite solutions (in the UK,  ${\displaystyle y=mx+c\,;\,}$   also common  ${\displaystyle y=ax+b\,,\,}$   ${\displaystyle y=a_{1}x+a_{0}\,}$   and  ${\displaystyle y=-{\frac {A}{B}}x-{\frac {C}{B}}\,}$  ). Points  ${\displaystyle (x,y)\,}$   will be mapped with independent variable  ${\displaystyle x\,}$   assuming the horizontal axis and  ${\displaystyle y\,}$   vertical on a Cartesian grid. By assigning  ${\displaystyle x\,}$   to a value and evaluating  ${\displaystyle y\,,\,}$   a (single) point coordinate solution is found. When  ${\displaystyle x=0\,,\,}$   then by zero-product property term  ${\displaystyle m\times x=0\,}$  ,  and by additive identity terms  ${\displaystyle 0+b=b=y\,.\,}$   The point  ${\displaystyle (0,b)\,}$   is the unique member of the line (linear equation's solution) where the y-axis is 'intercepted'. More about intercepts link:  The  ${\displaystyle x\,}$   and  ${\displaystyle y\,}$   Intercepts

## What does the m tell us when we have the equation ${\displaystyle y=f(x)=mx+b\,}$  ?

${\displaystyle m\,}$   is a constant called the slope of the line. Slope indicates the steepness of the line.

#### Slope

Two separate points fixed anywhere defines a unique straight line containing the points. Confining this study to plane geometry (${\displaystyle R^{2}}$ ) and fixing coordinates for unique points at  ${\displaystyle (x_{1},y_{1})\,}$   and  ${\displaystyle (x_{2},y_{2})\,}$   a straight line is defined relating two variables in a linear-equation mappable on a graph-plot. When the two points are identical, infinite lines result, even in a single plane. When  ${\displaystyle x_{1}=x_{2}\,}$   then a vertical-line mere relation is defined, not a function. Functions are equation-relations evaluating to singularly unique dependent values. Only when (iff)  ${\displaystyle x_{1}\neq x_{2},\,}$   then is the line containing the points a linear 'function' of  ${\displaystyle x.\,}$

For a linear function, the slope can be determined from any two known points of the line. The slope corresponds to an increment or change in the vertical direction divided by a corresponding increment or change in the horizontal direction between any different points of the straight line.
Let ${\displaystyle \Delta y=\,}$  increment or change in the ${\displaystyle y\,}$  -direction (vertical) and
Let ${\displaystyle \Delta x=\,}$  increment or change in the ${\displaystyle x\,}$  -direction (horizontal).
For two points  ${\displaystyle (x_{1},y_{1})\,}$   and  ${\displaystyle (x_{2},y_{2}),\,}$   the slope of the function line m is given by: ${\displaystyle m={\frac {\Delta y}{\Delta x}}={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}}$

• This formula is called the formula for slope measure but is sometimes referred to as the slope formula.

For a linear function, fixing two unique points of the line or fixing the slope and any one point of the line is enough to determine the line and identify it by an equation. There is an equation form for a linear function called the point-slope form of a line2 which uses the slope  ${\displaystyle m\,}$   and any one point  ${\displaystyle (x_{1},y_{1})\,}$   to determine a valid equation for the function's line: ${\displaystyle y-y_{1}=m(x-x_{1})\,}$
Algebra/Slope

## Other forms of linear equations?

### Intercept Form of a Line

There is one more general form of a linear function we will cover. This is the intercept form of a line, where the constants a and b are such that (a,0) is the x-intercept point and (0,b) is the
y-intercept point.

${\displaystyle {\frac {x}{a}}+{\frac {y}{b}}=1\qquad }$     where a ≠ 0 and b ≠ 0

Neither constant a nor b can equal 0 because division by 0 is not allowed. The intercept form of a line cannot be applied when the linear function has the simplified form y = m x because the
y-intercept ordinate cannot equal 0.

Multiplying the intercept form of a line by the constants a and b will give

${\displaystyle bx+ay=ab\ }$

which then becomes equivalent to the general linear equation form A x + B y + C where A = b, B = a, and C = ab. We now see that neither A nor B can be 0, therefore the intercept form cannot represent horizontal or vertical lines. Multiplying the intercept form of a line by just b gives

${\displaystyle {\frac {b}{a}}x+y=b}$

if we subtract

${\displaystyle {\frac {b}{a}}x}$

we get:

${\displaystyle y=b-{\frac {b}{a}}x}$

which can, in turn, be rearranged to:

${\displaystyle y=-{\frac {b}{a}}x+b}$

which becomes equivalent to the slope-intercept form where the slope m = -b/a.

Example: A graphed line crosses the x-axis at -3 and crosses the y-axis at -6. What equation can represent this line? What is the slope?

Solution: intercept form: ${\displaystyle \qquad {\frac {x}{-3}}+{\frac {y}{-6}}=1}$

Multiplying by -6 gives ${\displaystyle {\frac {-6}{-3}}x+y=-6}$

${\displaystyle y=-2x-6\ }$

so we see the slope m = -2.

Graph of y = - 2x – 6 showing intercepts.

The line can also be written as ${\displaystyle -6x-3y=(-3)(-6)\ }$

${\displaystyle 6x+3y=-18\ }$

Example: Can the equation

${\displaystyle {\frac {x}{2}}+{\frac {y}{4}}=0}$

be transformed into an intercept form of a line, (x/a) + (y/b) =1, to find the intercepts?

Solution: No, no amount of valid mathematical manipulation can transform it into the intercept form. Instead multiplying by 4, then subtracting 2x gives

${\displaystyle y=-2x\ }$

which is of the form y = m x where m = -2. The line intersects the axes at (0,0). Since the intercepts are both 0, the general intercept form of a line cannot be used.

Example: Find the slope and function of the line connecting the points (2,1) and (4,4).

Solution: When calculating the slope of a straight line from two points with the preceding formula, it does not matter which is point 1 and which is point 2. Let's set (x1,y1) as (2,1) and (x2,y2) as (4,4). Then using the two-point formula for the slope m:

${\displaystyle m={\frac {4-1}{4-2}}={\frac {3}{2}}}$

Using the point-slope form:

One substitutes the coordinates for either point into the point-slope form as x1 and y1. For simplicity, we will use x1=2 and y1=1.

${\displaystyle y-1={\frac {3}{2}}(x-2)}$
${\displaystyle y-1={\frac {3}{2}}x-3\,}$
${\displaystyle \ y={\frac {3}{2}}x-2}$

Using the slope-intercept form:

Alternatively, one can solve for b, the y-intercept ordinate, in the general form of a linear function of one variable, y = m x + b.

${\displaystyle b=y-mx\ }$

Knowing the slope m, take any known point on the line and substitute the point coordinates and m into this form of a linear function and calculate b. In this example, (x1,y1) is used.

${\displaystyle b=y_{1}-mx_{1}\,}$
${\displaystyle b=1-{\frac {3}{2}}\cdot 2=1-3=-2}$

Now the constants m and b are both known and the function is written as

${\displaystyle y={\frac {3}{2}}x-2}$       or alternatively as      ${\displaystyle f(x)={\frac {3}{2}}x-2}$

__________end of example__________

For another explanation of slope look here:

Example: Graph the equation 5x + 2y = 10 and calculate the slope.

Solution: This fits the general form of a linear equation, so finding two different points are enough to determine the line. To find the x-intercept, set y = 0 and solve for x.

${\displaystyle 5x+2\cdot 0=10=5x\ }$
${\displaystyle x=10/5=2\ }$

so the x-intercept point is (2,0). To find the y-intercept, set x = 0 and solve for y.

${\displaystyle 5\cdot 0+2y=10=2y\ }$
${\displaystyle y=10/2=5\ }$

so the y-intercept point is (0,5). Drawing a line through (2,0) and (0,5) would produce the following graph.

Graph of 5x + 2y = 10 showing intercepts

To determine the slope m from the two points, one can set (x1,y1) as (2,0) and (x2,y2) as (0,5), or vice versa and calculate as follows:

${\displaystyle m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}={\frac {5-0}{0-2}}=-{\frac {5}{2}}=-2.5}$

__________end of example__________

## Summary of General Equation Forms of a Line

The most general form applicable to all lines on a two-dimensional Cartesian graph is

${\displaystyle Ax+By=C\ }$

with three constants, A, B, and C. These constants are not unique to the line because multiplying the whole equation by a constant factor gives a new set of valid constants for the same line. When B = 0, the rest of the equation represents a vertical line, which is not a function. If B ≠ 0, then the line is a function. Such a linear function can be represented by the slope-intercept form which has two constants.

slope-intercept form:

${\displaystyle y=mx+b\ }$

The two constants, m and b, used together are unique to the line. In other words, a certain line can have only one pair of values for m and b in this form.

The point-slope form given here

${\displaystyle y-y_{1}=m(x-x_{1})\,}$

uses three constants; m is unique for a given line; x1 and y1 are not unique and can be from any point on the line. The point-slope cannot represent a vertical line.

The intercept form of a line, given here,

${\displaystyle {\frac {x}{a}}+{\frac {y}{b}}=1}$     a ≠ 0 and b ≠ 0

uses two unique constants which are the x and y intercepts, but cannot be made to represent horizontal or vertical lines or lines crossing through (0,0). It is the least applicable of the general forms in this summary.

Of the last three general forms of a linear function, the slope-intercept form is the most useful because it uses only constants unique to a given line and can represent any linear function. All of the problems in this book and in mathematics in general can be solved without using the point-slope form or the intercept form unless they are specifically called for in a problem. Generally, problems involving linear functions can be solved using the slope-intercept form
(y = m x + b) and the formula for slope.

## Discontinuity in Otherwise Linear Equations

Let variable y be dependent upon a function of independent variable x

${\displaystyle y=f(x)\,,}$   also  ${\displaystyle y(x)=f\,.}$

y is also the function f, and x is also the argument ( ). Let y be the expressed quotient function

${\displaystyle y={\frac {2x^{2}-5x+3}{x-1}}.}$

The graph of y's solution plots a continuous straight line set of points except for the point where x would be 1. Evaluation of the denominator with ${\displaystyle x=1}$  results in division by zero, an undefined condition not a member element of R and outside algebraic closure. y has a discontinuity (break) and no solution at point 1,-1. It becomes important to treat each side of a break separately in advanced studies.

y's otherwise linear form can be expressed by an equation removed of its discontinuity. Factor ${\displaystyle x-1}$  from the ${\displaystyle 2x^{2}-5x+3}$  numerator (use synthetic division).

${\displaystyle y={\frac {2x^{2}-5x+3}{x-1}}.}$
${\displaystyle y={\frac {(2x-3)(x-1)}{(x-1)}}.}$
${\displaystyle y=2x-3\,,\;x\neq 1}$   (for all x except 1) .

Reducing its (x-1) multiplicative inverse factors (reciprocals) to multiplicative identity (unity) leaves the ${\displaystyle (2x-3)}$  factor (with implied universal-factor 1/1). Limiting this simpler function's domain; 'all ${\displaystyle x}$  except ${\displaystyle x=1}$ , where x is undefined' or simply 'and x ≠ 1' (implying 'and R2 '); equates it to the original function. This expression is a linear function of x, with slope m = 2 and a y-intercept ordinate of -3. The expression ${\displaystyle 2x-3}$  evaluates to -1 at x = 1, but function y is undefined (division by zero) at that point. There is a discontinuity for function y at x = 1. Practically the function has a sort of one-point hole (a skip), shown on the graph as a small hollow circle around that point. Lines, rays and line segments (and arcs, chords and curves) are shown discontinuous by dashed or dotted lines.

Note: non-linear equations may also be discontinuous—see the subsequent graph plot of the reciprocal function y = 1/x, in which y is discontinuous at x = 0 not just for a point, but over a 'double' asymptotic extremum pole along the y-axis. As x is evaluated at smaller magnitudes (both – and +) closer to zero, y approaches no definition in both the – and + mappings of the function.

Example: What would the graph of the following function look like?

${\displaystyle y={\frac {x^{2}-4}{x+2}}.}$

Solution:

${\displaystyle y={\frac {x^{2}-4}{x+2}}={\frac {(x+2)(x-2)}{(x+2)}}.}$

Reduce the reciprocal (x + 2) factors to unity. This makes y = x – 2 for all x except x = -2, where there is a discontinuity. The line y = x – 2 would have a slope m = 1 and a
y-intercept ordinate of -2. So for the final answer , we graph a line with a slope of 1 and a y-intercept of -2, and we show a discontinuity at x = -2, where y would otherwise have been equal to -4.

Example: Write a function which would be graphed as a line the same as y = 2 x – 3 except with two discontinuities, one at x = 0 and another at x = 1.

Solution: The function must have a denominator with the factors

denominator = (x – 0)(x – 1) = x (x – 1) .

to have 'zeros' at the two x values. The function's numerator also gets the factors preserving an overall factor of unity, the expressions are multiplied out:

${\displaystyle y={\frac {(2x-3)x(x-1)}{x(x-1)}}={\frac {2x^{3}-5x^{2}+3x}{x^{2}-x}}.}$

__________end of example__________

# Linear Equations and Functions

## What are Linear Equations?

In the functions section we talked about how a function is like a box that takes an independent input value and uses a rule defined mathematically to create a unique output value. The value for the output is dependent on the value that is put in the box. We call the values that are going into the box the independent values or the domain. We call the values coming out of the box the dependent values or the range.

Unless we specify differently, on Cartesian graphs the domain is the real numbers. In the Cartesian Plane section we saw how running different values through a function to identify the points on the Cartesian plane by picking the first (x) value of the point the domain and the second (y) value from the range. To restate this: by convention the two variables for a function on the Cartesian Plane are x for the domain, the independent variable, and y for the range, the dependent variable. The variable y is the same as writing f(x). Mathematicians recognize this equivalence but generally prefer to write y because its shorter. Because a function definition has an input and an output it must also contain an equal sign. The section in this book solving equations showed the various operations we can perform on both sides of an equal sign and still maintain the notion of equivalence. In this section we plug different values into the independent variable and solve to find the associated dependent variable. For instance if we start with the equation:

${\displaystyle y=x}$
We can add a -x to both sides to get the equation
${\displaystyle y-x=x-x}$
which we then simplify to
${\displaystyle y-x=0}$
Or we can add a -y to both sides to get the equation
${\displaystyle y-y=x-y}$
which we then simplify to
${\displaystyle 0=x-y}$
Since ${\displaystyle y-x=0\equiv 0=x-y}$  then:
${\displaystyle y-x=0=x-y}$
Using the transitive property
${\displaystyle y-x=x-y}$
adding x + y to both sides gives us
${\displaystyle (y-x)+(y+x)=(x-y)+(y+x)}$
using the associative property we change this to
${\displaystyle (y-y)+(x+x)=(x-x)+(y+y)}$
which simplifies to
${\displaystyle 2x=2y}$
And divide both sides by 2
${\displaystyle 2x/2=2y/2}$
To simply reverse the order and show
${\displaystyle x=y}$

We have not really proved anything mathematically above, but these operations allow us to manipulate equations to get the dependent variable by itself on one side of the equals sign. Then we can plug numbers into the independent variable to discover the function values for those numbers. Then we can draw these values as points on the Cartesian plane and get a feel for what the function would look like if we could see all the points defined by the function at once.

## ${\displaystyle y=0x}$

Equations of the form y = C2 are linear functions of the general form y = m x + b where slope m = 0 and the constant C2 is the y-intercept b (in the general form). The graph of this zero-slope function is a straight horizontal line, intercepting the y-axis at C2, including zero and extends infinitely in the positive and negative directions for all R values of x (see the following diagram).

The domain for such functions is R covering all real numbers (unless otherwise specified), but the range is just the set { c }. The equation y = 0 is the x-axis.

## ${\displaystyle 0y=x}$

Equation x = C1, x is one single value C1 and y, being unrestricted, is every R number. The graph of x = C1 is a straight vertical line where x = C1, covering all positive, negative and zero values of y (see the following diagram).

Its domain is set { C1 } and range is set R (unless otherwise specified). x = C1 is technically not a function (there is more than one value of y for each value of x), but it's a relation. Vertical lines have no slope (m = divide by zero, undefined, plus and minus infinity). These are the only types of linear equations of the general form shown previously which are not linear functions. The equation x = 0 is exactly the y-axis. Lines with steepness approaching vertical have very large-magnitude slopes but are still functions.

## CONTINUE

We are going to start by looking at simple functions called linear equations. When none of the instances of x and y in the algebraic expression defining the function rule have exponents then all the instances of x and y can be combined into just two occurrences. the graph of the expression can be represented as a straight line. The equation that expresses the function is considered a linear equation with two variables. The following equation is a simple example of such a linear equation:

${\displaystyle y-x=2\,}$

Since y is the dependent variable it is standing in for the function. We can re-write the expression as f(x) - x = 2. If we add an x to both sides the equality property holds and we get the expression f(x) - x + x = x + 2. Simplifying we get f(x) = x + 2. In the following table we'll pick 3 values for x, and then calculate the dependent (y) values from f(x).

x value y value (abscissa) Coordinates
(x,y)
-1 1 (-1,1)
0 2 (0,2)
1 3 (1,3)

where x and y are variables to be plotted in a two-dimensional Cartesian coordinate graph as shown here:

This function is equivalent to the previous example of a linear equation, y - x = 2. The arrows at each end of the line indicate that the line extends infinitely in both directions. All linear functions of a single input variable have or can be algebraically arranged to have the general form:

${\displaystyle y=f(x)=mx+b\,}$

where x and y are variables, f(x) is the function of x, m is a constant called the slope of the line, and b is a constant which is the ordinate of the y-intercept (i. e. the value of y where the function line crosses the y-axis). The slope indicates the steepness of the line. In the previous example where y = x + 2, the slope m = 1 and the y-intercept ordinate b = 2. The y = mx+b form of a linear function is called the slope-intercept form.

Unless a domain for x is otherwise stated, the domain for linear functions will be assumed to be all real numbers and so the lines in graphs of all linear functions extend infinitely in both directions. Also in linear functions with all real number domains, the range of a linear function will cover the entire set of real numbers for y, unless the slope m = 0 and the function equals a constant. In such cases, the range is simply the constant.

Conversely, if a function has the general form y = mx + b or if it can be arranged to have that form, the function is linear. A linear equation with two variables has or can be algebraically rearranged to have the general form1:

${\displaystyle Ax+By=C\,}$

where x and y represent the linear variables, and the letters A, B, and C can represent any real constants, either positive or negative. Conversely, an equation with two variables x and y having that general form, or being able to be arranged in that form, would be linear as long as A and B are not both equal to 0. In the preceding equation, capital letters are to avoid confusion with other constants in this chapter and for consistency with Reference 1.

If one divides the preceding equation by B (when B is not 0) and solves for y, the following form can be obtained:

${\displaystyle y=(-A/B)x+(C/B)\,}$

If one equates -A/B to the slope m and C/B to the y-intercept ordinate b, it can be seen that the general form for a linear equation and the slope-intercept form for a linear function are practically interconvertible except for the fact that, in a linear function, the B constant in the linear equation form cannot equal 0.

# Intercepts

## Intercepts

To find where the equation of a line crosses the X or Y axis, you don't need much information. In this section we will look at how to find the where the line crosses the axes using the standard form for the linear equation. After we look at how slope works we will see we can convert between the various types of linear equations into the standard form.

## X and Y axis intercepts

An axis intercept point is a point where the graph of a function, relation, or equation intersects the X or Y axes. This section is about finding out how a particular set of functions: linear functions cross the axes.

We know that the domains of most lines are infinite because they are defined at every value of X. The exception is lines that are defined as ${\displaystyle X=c}$  where c is a number we choose when we write the function. By definition this line is only defined for one value of X. Since the domain maps onto more than one value for the range this is actually a relationship and not a function. We've seen the graph of this relationships is a vertical line that passes through the point (c,Y). In the picture below we show that when c=0 then our line is the same as the Y axis. When ${\displaystyle c\neq 0}$  (as in the drawing where X = 3) then the line can never intercept the Y axis.

 To do:

Lines with the equation X=C intersect the X axis once and the Y axis 0 or infinite times.

We can also restrict the range of a function by simply writing ${\displaystyle Y=c}$  where c is again any number we choose. The graph of this line is a horizontal line that passes through the point (X,c). When Y=o this line is the same as the X axis. when ${\displaystyle c\neq 0}$  (as in the drawing where Y = 3)then the line can never intercept the X axis.

 To do:

Lines with the equation Y=C intersect the Y axis once and the X axis 0 or infinite times.

When looking at Cartesian graphs and linear equations we run into a mathematical axiom: "Two points determine a line.". We will see how this axiom affects the slope-intercept definition of a line ${\displaystyle y=f(x)=mx+b}$  in the next section. When two lines intersect they intersect at a point. If a line is not horizontal or perpendicular it will have to intersect the X and Y axes once, but only once.

In this book we are going to accept the statement "At most one line can be drawn through any point not on a given line parallel to the given line in a plane." There is a branch of mathematics called "non-euclidean geometry" that was founded a little more than 160 years ago. Even if you are not interested in mathematics it is worth looking at this Wikipedia article on geometry to get a feel for how formalizing geometry with algebraic methods and then moving beyond them has changed civilization. If you continue in a career requiring advanced mathematics such as Engineering or Physics you might want to follow your interests to see the effect of non-euclidean geometry in your career.

We've seen that for the equation Y=mX + b the Y intercept will always be at b because that is where X=0.

Using Algebra we can subtract b from both sides: Y - b = mX

and multiply by ${\displaystyle {\frac {1}{m}}}$

${\displaystyle {\frac {Y}{m}}-{\frac {b}{m}}=X}$

we can see that the X intercept is going to be ${\displaystyle -{\frac {b}{m}}}$ .

An axis intercept may simply refer to the number value on the axis where the intersection occurs. For brevity we may say the line has an X intercept of 1 and a Y intercept of 2. After graphing just a few lines you will be able to tell this line points down and runs through quadrants II, I, and IV. With a little more practice you will be able to know that the equation for the line is Y=-2x + 2. We will see that by specifying the two points we are actually implying the slope of the line. There is an exception to this rule. If we say a line crosses the axes at 0 we know that the line will pass through 2 quadrants instead of 3, but we won't know which quadrants or how steep the line is. When we look at slope in the next section we will see why the equations above specify a point and a slope.

When you are trying to graph a linear equation finding the axes intercepts is often the easiest way to go about doing it. To find the x-intercept, set y = 0 and solve for x. To find the y-intercept, set x = 0 and solve for y. For most examples the intercepts are different points, and a line can be drawn through the two intercepts. If both intercepts are (0,0), then another point must be determined to graph the line. If the equations is in the form x = c or y = c, the horizontal or vertical lines are very simple to plot.

 To do:Need example graphs showing lines

 To do:Need problems

## Example

• ${\displaystyle Y=5\times x+2}$

Original equation

• ${\displaystyle Y=5\times 0+2}$

Substitute zero for x

• ${\displaystyle Y=2}$

Solution

Therefore, the Y-Intercept of Y = 5x + 2 is 2.

This works for any form of equation.

# Slope

## Slope

Slope is the measure of how much a line moves up or down related to how much it moves left to right.

In this image, the slope of the line is ${\displaystyle {\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}}$ .

Parallel lines are those that have the same slope and do not touch. Examples include latitude lines.

### Slope

Algebra/Slope

Slope is the change in the vertical distance of a line on a coordinate plane over the change in horizontal difference. In other words, it is the “rise” over the “run” or the steepness of a line. Slope is usually represented by the symbol ${\displaystyle m}$  like in the equation ${\displaystyle y=mx+b}$ , m the coefficient of x represents the slope of the line.

Slope is computed by measuring the change in vertical distance divided by the change in horizontal difference, i.e.:

${\displaystyle \ m={\frac {\Delta y}{\Delta x}}={\frac {rise}{run}}}$

The Greek uppercase letter ${\displaystyle \Delta }$  represents change, in this case change in the y-coordinates divided by the change in x-coordinates.

Positive Slope/ Negative Slope

If a line goes up from left to right, then the slope has to be positive. For example, a slope of ¾ would have a “rise” of 3, or go up 3; and a “run” of 4, or go right 4. Both numbers in the slope are either negative or positive in order to have a positive slope.

If a line goes down from left to right, then the slope has to be negative. For example, a slope of -3/4 would have a “rise” of -3, or go down 3; and a “run” of 4, or go right 4. Only one number in the slope can be negative for a line to have a negative slope.

Other Types of Slope

There are two special circumstances, no slope and slope of zero. A horizontal line has a slope of 0 and a vertical line has an undefined slope.

Horizontal lines have the form: ${\displaystyle \ y=a}$  ; where a is a constant, i.e. ${\displaystyle a\in R}$
Vertical lines have the form: ${\displaystyle \ x=a}$  ; where as is a constant, i.e. ${\displaystyle a\in R}$

## Determining Slope

To determine the slope you need some information. This can include two (or more) coordinates, a parallel slope and a coordinate, a perpendicular slope and a coordinate, or the y-intercept and slope.

For completely horizontal lines, the difference in y coordinates between any two points is 0, so the slope m = 0, indicating no steepness in the line at all. If the line extends between right-upper (+,+) and left-lower ( -, -) directions, then the slope is positive. As the slope increases, the line becomes steeper until the line is almost vertical when the slope is very large. When the slope m = 1, the line is diagonal with an angle halfway between the x and y axes. If the line extends between left-upper (-,+) and right-lower (+, -) directions, then the slope is negative. As the slope changes from 0 to very negative numbers, the steepness in the opposite direction increases. Compare the slope ( m ) values in the following graph of functions y = 1 (where
m = 0), y = (1/2) x + 1, y = x + 1, y = 2 x, y = -(1/2) x + 1, y = -x + 1, and y = -2 x + 1. For all two-variable linear equations that can be converted to linear functions, the same calculation applies to slopes for those lines.

## Finding It

For the most part finding slope when given information is a simple matter. Simply take the slope equation y=mx+b and replace the variable with whatever information you know, and solve.

### Two Coordinates

To find the slope with two coordinates, you must first find the slope. Use the standard equation ${\displaystyle {\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}}$ . Put that into the equation as m, and replace x and y with x and y from one of the coordinates. Solve for b. Put that into the equation and you're done.

Example: (1,4) (4,8)

${\displaystyle m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}}$

${\displaystyle m={\frac {4-1}{4-8}}}$

${\displaystyle m={\frac {3}{-4}}}$

Plug that right in.

${\displaystyle y={\frac {3}{-4}}x+b}$

${\displaystyle 4={\frac {3}{-4}}(1)+b}$

${\displaystyle 4={\frac {3}{-4}}+b}$

${\displaystyle 4-{\frac {3}{-4}}=b}$

${\displaystyle {\frac {-19}{4}}=b}$

Put that in the equation and you're done.

${\displaystyle y={\frac {3}{-4}}x+{\frac {-19}{4}}}$

### Parallel Lines

If you have to find the slope of a line(Let's say AB) which is parallel to line(Let's say XY) then using the coordinates of line XY you can find the coordinates of slope of line AB As, Slope of line of a line parallel to another line is equal,i.e. Slope of AB = Slope of XY

Example:-

AB and XY are PARALLEL Lines.Find the slope of AB.

Let line XY have the coordinates:-

${\displaystyle X(2,4)=(x1,y1)}$  and ${\displaystyle Y(3,6)=(x2,y2)}$

Slope of XY

${\displaystyle m=(y2-y1)/(x2-x1)}$

${\displaystyle =(6-4)/(3-2)}$

${\displaystyle =2/1}$

${\displaystyle m=2}$

Therefore,

Slope of AB = Slope of XY

⇒Slope of AB = 2

Thus,Solved

# Standard Form and Solving Slope

## Standard Form

Standard form is another way to write slope-intercept form (as opposed to y=mx+b). It is written as Ax+By=C where A, B, C are all integers. You can also change slope-intercept form to standard form like this: Y=-3/2x+3. Next, you isolate the y-intercept(in this case it is 3) like this: Add 3/2x to each side of the equation to get this: 3/2x+y=3. You can not have a fraction in standard form so you solve this. 2(3/2x+y)=3(2). To get: 3x+2y= 6. Now you have a standard form equation! However, there are some rules for standard form. A, B, C are integers (positive or negative whole numbers) No fractions nor decimals in standard form. "Ax" term is positive. If these are not followed, it is not standard form. i.e. -1/3x+1/4y=4 is NOT standard form.

## Solving Slope

### Solving Standard Form

Slope intercept equations (y=mx+b) are the easiest to graph. So if you encounter an equation in standard form that you are required to graph, you must convert it to slope intercept form. To do this you must take the equation and solve for Y.

Example:

${\displaystyle 9x+7y=-3}$

${\displaystyle 9x-9x+7y=-3-9x}$

${\displaystyle 7y=-3-9x}$

${\displaystyle {\frac {7y}{7}}={\frac {-3-9x}{7}}}$

${\displaystyle y={\frac {-3}{7}}-{\frac {9}{7}}x}$

That is technically slope intercept form, but if you want to make it true (y=mx+b) simply follow the rule of negatives (a - b = a + -b):

${\displaystyle y={\frac {-9}{7}}x+{\frac {-3}{7}}}$

Solving an equation in slope-intercept form- how?

If you come upon an equation in slope-intercept form and require it to be in standard form, simply solve for m (c).

Example:

${\displaystyle y=10x+9}$

${\displaystyle y-10x=9+10x-10x}$

${\displaystyle y-10x=9}$

Standard form cannot have fractions. If they are in a fraction, you must multiply each side to get rid of it.

Example:

${\displaystyle {\frac {9}{10}}x+9y=5}$

${\displaystyle 10[{\frac {9}{10}}x+9y]=10[5]}$

${\displaystyle 9x+90y=50}$

# Graphing Inequalities

## Inequalities in 2 variables

Linear inequalities in 2 variables are typically in the form of ${\displaystyle y , where m is the slope of the line and b is the y-intercept.

Graphing an inequality is easy. First, graph the inequality as if it were an equation. If the sign is ≤ or ≥, graph a normal line. If it is > or <, then use a dotted or dashed line. Then, shade either above or below the line, depending on if y is greater or less than mx + b.

# Other Types of Graphs

## Sample Graphs of Various Functions and Relations

y = |x|

y = x2

y = 1/x

y =x      y = -x      y = |x|      y=-|x|      y=x2      y=-x2      y=x3      y = 1/x      y=-1/(x-1)

y=x! other functions and relations in other sections

inequalities parabolas y=(10 or e) to the x y = log x

polynomials

cubics and squares

what this means is that the graphs of y = x^N(even) and y = x^N (odd) will always look in certain ways.

Second Graphing section: translations symmetries +/- inversions inverse relations ellipse circle square roots inequalities of these

Other functions and relations

Symmetry about: x-axis y-axis y=x y=-x

Translation (shift) in x and y directions

asymptotes

inverse functions (to be originally introduced in Functions, graphing aspects covered here)

circles

ellipses

inequalities in non-linear relations

stretching relations about x or y axes

Newton's method

Given 0=x^a y^b + x^c y^d etc, can deduce asymptotes/intersects from smallest polygon containing points (a,b) (c,d) etc

References:

1. ELEMENTARY GEOMETRY for College Students, 2nd Edition, by Daniel Alexander and

Geralyn Koeberlein, Houghton Mifflin Company, Boston, MA 1999.

2. ALGEBRA AND TRIGONOMETRY with Analytic Geometry, Ninth Edition, by Earl Swokowski

and Jeffery Cole, Brooks/Cole Publishing Company 1997.

### Pie Chart

Pie charts are best used to compare parts to the whole by percentages. By measuring the number of degrees that a piece of the pie chart is, one can find the percentage it represents.

${\displaystyle {\mbox{degrees}}={\mbox{percent}}\cdot {\frac {360}{100}}}$

which simplifies to degrees * 18 / 5

### Bar Chart

Bar charts are best for plotting the change in something over a period of time. It is nearly the same as a line chart, except that the points are not connected, and instead extend to the bottom of the chart.

# Systems of Equations

## Systems of Simultaneous Equations

In a previous chapter, solving for a single unknown in one equation was already covered. However, there are situations when more than one unknown variable is present in more than one equation. When in a given problem, more than one algebraic equation is true at a time, it is said there is a system of simultaneous equations which are all true together at once. Such sets of multiple equations may help solve for more than one unknown variable in a problem, since having more than one unknown in one equation is typically not enough information to "solve" any of the unknowns.

An unknown quantity is something that needs algebraic information in order to solve it. An equation involving the unknown is typically a piece of information which may provide the information to "solve" the unknown, i. e. to determine a specific number value (or limited number of discrete values) that the unknown is (or can be) equal to. Some equations provide little or no information and so do little or nothing to narrow down the possibilities for solutions of the unknowns. Other equations make it impossible to satisfy an unknown with any real number, so the solution set for the unknown is an empty set. Many other useful equations make it possible to solve an unknown with one or just a few discrete solutions. Similar statements can be made for systems of simultaneous equations, especially regarding the relationships between them.

## Linear Simultaneous Equations with Two Variables

In the previous module, linear equations with two variables were discussed. A single linear equation having two unknown variables is practically insufficient to solve or even narrow down the solutions for the two variables, although it does establish a relationship between them. The relationship is shown graphically as a line. Another linear equation with the same two variables may be enough to narrow down the solution to the two equations to one value for the first variable and one value for the second variable, i. e. to solve the system of two simultaneous linear equations. Let's see how two linear equations with the same two unknowns might be related to each other. Since we said it was given that both equations were linear, the graphs of both equations would be lines in the same two-dimensional coordinate plane (for a system with two variables). The lines could be related to each other in the following three ways:

1. The graphs of both equations could coincide giving the same line. This means that the two equations are providing the same information about how the variables are related to each other. The two equations are basically the same, perhaps just different versions or forms of each other. Either one could be mathematically manipulated to produce the other one. Both lines would have the same slope and the same y-intercept. Such equations are considered dependent on each other. Since no new information is provided, the addition of the second equation does not solve the problem by narrowing the solution set down to one solution.

Example: Dependent linear equations

${\displaystyle 6x-3y=12\ }$
${\displaystyle y=2x-4\ }$

The above two equations provide the same information and result is the same graph, i. e. lines which coincide as shown in the following image.

Let's see how these equations can be mathematically manipulated to show they are basically the same.

Divide both sides of the first equation ${\displaystyle 6x-3y=12\ }$  by 3 to give

${\displaystyle 2x-y=4\ }$
Now add y to both sides
${\displaystyle 2x=4+y\ }$
Now subtract 4 from both sides
${\displaystyle y=2x-4\ }$

This is the same as the second equation in the example. This is the slope-intercept form of the equation, from which a slope and a y-intercept unique to the line can be compared with any other equations in the slope-intercept form.

2. The graphs of two lines could be parallel although not the same. The two lines do not intersect each other at any point. This means there is no solution which satisfies both equations simultaneously, i. e. at the same time. The solution set for this system of simultaneous linear equations is the empty set. Such equations are considered inconsistent with each other and actually give contradictory information if it is claimed they are both true at the same time in the same problem. The parallel lines have equal slopes but different y-intercepts.

Sets of equations which have at least one common point which might provide a solution set are consistent with each other. For example, the dependent equations mentioned previously are consistent with each other.

Example: Inconsistent linear equations

${\displaystyle 3x-2y=-2\ }$
${\displaystyle 3x-2y=2\ }$

To compare slopes and y-intercepts for these two linear equations, we place them in the slope-intercept forms. Subtract 3x from both sides of both equations.

${\displaystyle 3x-2y=-2\qquad \qquad 3x-2y=2}$  ${\displaystyle -2y=-3x-2\qquad \qquad -2y=-3x+2}$

Divide both sides of both equations by -2 and simplify to get slope-intercept forms for comparison.

${\displaystyle (-2y)/(-2)=(-3x-2)/(-2)\qquad (-2y)/(-2)=(-3x+2)/(-2)}$
${\displaystyle y={\frac {3}{2}}x+1\qquad \qquad \qquad \qquad \qquad y={\frac {3}{2}}x-1}$

Now, both slopes are equal at 3/2, but the y-intercepts at 1 and -1 are different.
The lines are parallel. The graphs are shown here:

3. If the two lines are not the same and are not parallel, then they would intersect at one point because they are graphed in the same two-dimensional coordinate plane. The one point of intersection is the ordered pair of numbers which is the solution to the system of two linear equations and two unknowns. The two equations provide enough information to solve the problem and further equations are not needed. Such equations intersecting at a point providing a solution to the problem are considered independent of each other. The lines have different slopes but may or may not have the same y-intercept. Because such equations provide at least one solution point, they are consistent with each other.

Example: Consistent independent linear equations

${\displaystyle y=3x-5\ }$
${\displaystyle y=-x-1\ }$

Both of these equations are given in the slope-intercept, so it is easy to compare slopes and y-intercepts. For these two linear functions, both slopes are different and both y-intercepts are different. This means the lines are neither dependent nor inconsistent, so on a two-dimensional graph they must intersect at some point. In fact, the graph shows the lines intersecting at (1,-2), which is the ordered pair solution to this system of independent simultaneous equations. Visual inspection of a graph cannot be relied on to give perfectly accurate coordinates every time, so either the point is tested with both equations or one of the following two methods is used to determine accurate coordinates for the intersection point.

## Solving Linear Simultaneous Equations

Two ways to solve a system of linear equations are presented here, the addition method and the substitution method. Examples will show how two independent linear simultaneous equations with two unknown variables could be solved for both unknown variables using these methods.

The elimination by addition method is often simply called the addition method. Using the addition method, one of the equations is added (or subtracted) to the other equation(s), usually after multiplying the entire equation by a constant, in order to eliminate one of the unknowns. If the equations are independent, then the resulting equation(s) should be one(s) which will have one less unknown. For an original system of two equations and two unknowns, the resulting equation with one less unknown would have one unknown left which could easily be solved for. For systems with more than two equations and two unknowns, the process of elimination by addition continues until an equation with one unknown results. This unknown could then be solved for and the solved value then substituted into the other equations resulting in a system with one less unknown. The elimination by addition process is repeated until all of the unknowns are solved.

If a system has two equations which are dependent, then the addition of the equations could or would eliminate both unknowns at once. If the equations are parallel lines which are inconsistent, then a contradictory equation could result. The addition method is useful for solving systems of simultaneous linear equations, particularly if the equations are given in the form Ax + By = C, where x and y are the two unknown variables and A, B, and C are constants.

Example: Solve the following system of two equations for unknowns x and y using the addition method:

${\displaystyle x+2y=4\ }$
${\displaystyle 3x-y=5\ }$

Solution: We can either multiply the first equation by -3 and add the result to the second equation to eliminate x, or we can multiply the second equation by 2 and add the result to the first equation to eliminate y. Let's multiply [both sides of ] the second equation by 2.

${\displaystyle 2\cdot (3x-y)=2\cdot 5}$
${\displaystyle 2\cdot 3x-2\cdot y=10}$
${\displaystyle 6x\ -2y=10\ }$

Now we add this resulting equation to the first equation; i. e. each of the two sides of the equations are added together to give a combined equation as shown here:

${\displaystyle x\ +\ 2y=4\ }$
${\displaystyle +\ (6x-2y=10)\ }$
_____________________
${\displaystyle 7x+0\cdot y=14\ }$

This means that we add x + 2y and 6x - 2y to get 7x + 0·y and we add 4 and 10 to get 14.

This eliminates y from the combined equation to give an equation in x only:

${\displaystyle 7x=14\ }$
Now we solve for x:
${\displaystyle x=14/7=2\ }$

Now that we have x, we can substitute the value for x into either of the original two equations and then solve for y. Let's pick the first equation for the substitution into x.

${\displaystyle 2+2y=4\ }$
Solving for y:
${\displaystyle 2y=4-2=2\ }$
${\displaystyle y=2/2=1\ }$

So the solution set consists of the ordered pair ( 2,1) which is the point of intersection for the two linear functions as shown here:

### Elimination by Substitution Method

The elimination by substitution method is often simply called the substitution method. With the substitution method, one of the equations is solved for one of the unknowns in terms of the other unknown(s). Then that expression for the first unknown is substituted into the other equation(s) to eliminate it such that the equation(s) then have only the other unknown(s) left. If the equations are independent, then the resulting equation(s) should be one(s) which will have one less unknown. For an original system of two equations and two unknowns, the resulting equation with one less unknown would have one unknown left which could easily be solved for. For systems with more than two equations and two unknowns, the process of elimination by substitution is repeated until an equation with one unknown results. This unknown could then be solved for and the solved value then substituted into the other equation(s), resulting in a system with one less unknown. The process of elimination by substitution continues until all of the unknowns are solved.

If a system has two equations which are dependent, then applying the substitution method would either eliminate two unknowns at once or result in an equation which do not yield single values for the remaining unknown(s). If the equations are parallel lines which are inconsistent, then a contradictory equation could result.

Example: Solve the following system of two equations for unknowns x and y using the substitution method:

${\displaystyle x-y=-1\ }$
${\displaystyle x+2y=-4\ }$

Solution: We can start by solving for either x or y in terms of the other unknown in either one of the equations. Let's start by solving for x in terms of y in the first equation.

${\displaystyle x=y-1\ }$

Next, we substitute this expression for x into the other equation in order to eliminate x from the equation.

${\displaystyle (y-1)+2y=-4\ }$
${\displaystyle 3y-1=-4\ }$

We have eliminated x and now we have an equation in terms of y only. We now solve for y in this equation.

${\displaystyle 3y=-4+1=-3\ }$
${\displaystyle y=-3/3=-1\ }$

We have found the solution for y to be -1. We substitute this value for y into the expression for x in terms of y we determined from the first equation earlier.

${\displaystyle x=y-1=-1-1\ }$

Finally, we calculate the value of x.

${\displaystyle x=-2\ }$

So the solution set consists of the ordered pair (-2,-1) which is the point of intersection for the two linear functions as shown here:

## Slopes of Parallel and Perpendicular Lines

• In a two-dimensional Cartesian coordinate plane, linear functions which are dependent or whose graphs are parallel lines will have the same slope. CONVERSELY, linear functions having equal slopes are either dependent or have graphs that are parallel lines in a two-dimensional Cartesian coordinate plane. Of course, vertical parallel lines of the general form x=c are not functions and have no defined slopes.

This paragraph restates itself. It should be reworded.

• In a two-dimensional Cartesian coordinate plane, two lines that are perpendicular to each other will form right angles (90° angles) with each other at the point where they intersect. When the slopes of two linear functions whose graphs are lines that are perpendicular are multiplied together, the product of the two slopes equals -1. Conversely, if multiplying the slopes of two linear functions gives a product equal to -1, then their graphs are perpendicular lines on a two-dimensional Cartesian coordinate plane.

In other words, if two perpendicular lines have slopes m1 and m2, then
${\displaystyle m_{1}m_{2}=-1\ }$  .
If a pair of perpendicular lines consists of a horizontal line (of the form y = c) and a vertical line (of the form x = c), then the preceding rule does not apply. A vertical line has no slope and the slope of a horizontal line = 0.

Example: Find the slope-intercept form of a [new] line which intersects y = (1/2)x - 3 at (4,-1) and is perpendicular to it.

Solution: First, find slope of the new line from slope of the given line. Let m = slope of the new line.

${\displaystyle \left({\frac {1}{2}}\right)m=-1}$

${\displaystyle 2\cdot \left({\frac {1}{2}}\right)m=2\cdot (-1)}$

${\displaystyle m=-2\ }$

The slope-intercept form of the new line will be:

${\displaystyle y=-2x+b\ }$

where b is the y-intercept of the new line. Next, solve for y-intercept of new line using the intersecting point (4,-1) and the new slope of -2. Substitute x = 4 and y = -1 into the preceding equation and solve for b.

${\displaystyle -1=-2\cdot 4+b\ }$

${\displaystyle -1=-8+b\ }$

${\displaystyle b=-1+8=7\ }$

Finally, the slope-intercept form of the new perpendicular line is :

${\displaystyle y=-2x+7\ }$  .

Graph showing perpendicular lines in above example.

## Solving Systems of Simultaneous Equations Involving Equations Of Degree 2

The substitution method should be used for efficiency when solving nonlinear simultaneous equations, unless other methods such as the graphing method provide clear and simple solutions quickly (when they would be faster than substitution).

Example: Solve the system of simultaneous equations.

${\displaystyle y^{2}+(2x+3)^{2}=10\ }$
${\displaystyle 2x+y=1\ }$

With the second equation, make a given term (here, 2x should be used) the subject.

${\displaystyle 2x=1-y\ }$

Substitute the third equation into the first, and through factorization of the resulting, simplified quadratic with one variable the solutions can be found.

${\displaystyle y^{2}+((1-y)+3)^{2}=10\ }$
${\displaystyle y^{2}+(4-y)^{2}=10\ }$
${\displaystyle 2y^{2}-8y+16=10\ }$
${\displaystyle y^{2}-4y+3=0\ }$
${\displaystyle (y-1)(y-3)=0\ }$

Hence we know ${\displaystyle y=1\ }$  or ${\displaystyle y=3\ }$

Then, we calculate that the two possibilities are: ${\displaystyle y=1\ }$ , ${\displaystyle x=0\ }$  or; ${\displaystyle y=3\ }$ , ${\displaystyle x=-1\ }$

## Solving Systems of Simultaneous Equations Using a Graphing Calculator

TI-83 (Plus) and TI-84 Plus:

1. Press "Y="
2. Enter both equations, solved for Y
3. Press "GRAPH"
4. If all intersection points are not visible, press "ZOOM" then 0 or select "0: ZoomFit"
5. Press "2nd" then "TRACE"
6. Press 5 or select "5: intersect"
7. Move the cursor to one of the intersection points. (There may be only one) Each of these points represents one solution to the system.
8. Press "ENTER" three times
9. The coordinates of the intersection are shown at the bottom of the screen. Repeat steps 5-8 for other solutions.

TI-89 (Titanium):

via Graphing:

1. Press the green "diamond key", located directly beneath the "2nd" (blue) button.
2. Follow steps 2-5 as listed above. To access "Y=" and "GRAPH", press the green "diamond key" , then press F1 (it activates the tertiary function, "Y=") and F3 ( "GRAPH"). To access "ZOOM" and "TRACE", press F2 and F3 (diamond function activated), respectively. For "ZoomFit", press F2, then "ALPHA" (white), then "=" (for A).
3. To locate the point of intersection, manually use the directional keypad (arrow keys), or press F5 for "Math", then 5 for "Intersection". (The second option is more difficult to use, however; manual searching and zooming is recommended.)
4. The coordinates are displayed on the bottom of the screen. Repeat steps 2 and 3 until all desired solutions have been found. For new or additional equations, return to the "Y=" as described above.

via Simultaneous Equation Solver:

Note:This is a default App on the TI-89 Titanium. If you are using the TI-89 or no longer have the Solver, visit the Texas Instruments site for a free download.

1. On the APPS screen, select "Simultaneous Equation Solver" and press enter. Press "3" when the next screen appears.
2. Enter the number of equations you wish to solve and the corresponding number of solutions.
3. The two equations are represented simultaneously in a 2 x 3 matrix (assuming that you are solving two equations and searching for two solutions. The size of the matrix depends on the number of equations you wanted to solve). In the corresponding boxes, enter the coefficients/constants of your equations, pressing "ENTER" every time you submit a value. (Remember that all equations must be converted into standard form - Ax + By = C - first!)
4. Once all values have been entered, press F5 to solve.

# Graphing Systems of Inequalities

## Example: Highway and City Gas Mileage

 The gas mileage sticker on cars gives two numbers: one for city driving, and one for highway driving. If a sticker says the car gets 25 mpg in the city and 32 mpg on the highway how far can you drive? The abbreviation mpg stands for miles per gallon. The sticker on our car predicts that we get between 25 and 32 mpg, but when we drive our car we estimate how far we are going and how much gas we have in our tank. When we graph we need to change our inequality to a function where x is the number of gallons of gas in our tank. 25x < f(x) < 32x. The picture on the left has two lines one for y=25x and one for y = 32x. The yellow portion of the picture represents how far we may be able to drive when we have x gallons of gas. The vertical line at 10 shows us that we can drive between 250 and 320 miles on 10 gallons of gas. The difference between these two numbers is 70. If we drew a line at 1 gallon of gas we would see that we could drive between 25 and 32 miles, and the difference would be 7. What this graph shows us is the predicted range of miles we can drive on a given amount of gas. The more gas we have, the more likely our actual mileage is going to be different from what the sticker on our car predicted. If our actual mileage falls outside of this range than we may want to take our car to a mechanic to make sure everything is running correctly on our car. If your mileage is too high the odometer on your car may be broken. If your mileage is too low your car might need a tune up.

# Polynomials

With practice the concept of slope for linear functions becomes intuitive. It makes sense that the line that fits the equation ${\displaystyle y=2x}$  has a steeper ascent then the line that fits the equation ${\displaystyle y=1/2x}$ . You only have to move horizontally one unit to change your vertical direction two for the former when you graph ${\displaystyle y=2x}$ . How many blocks do you need to move horizontally to change your vertical direction by one for the line ${\displaystyle y=1/2x}$ ?

 When we express concepts like ${\displaystyle y=x^{2}}$  the abstract behavior of what is being represented becomes a little harder to see.


A monomial of one variable, let's say x, is an algabraic expression of the form

${\displaystyle cx^{m}\ }$

where

• ${\displaystyle c}$  is a constant, and
• ${\displaystyle m}$  is a non-negative integer (e.g., 0, 1, 2, 3, ...).

The integer ${\displaystyle m}$  is called the degree of the monomial.

The idea of a monomial of degree zero appears a bit mystical since it always represents one, except when the value of the variable is set equal to zero when the result is undefined. This idea allows us preserve the value of the constant in the monomial. We know that ${\displaystyle cx^{0}}$  is always equal to ${\displaystyle c}$  because even though we have 0 x's (somethings) we still have a c. When x = 0 things are difficult because the value we started with, 0, represents nothing.

For a monomial of power 1 we are multiplying C by one instance of our variable. When ${\displaystyle x=0}$  we get ${\displaystyle c*0=0}$ . When ${\displaystyle x\neq 0}$  we are multiplying c by 1 x. If x is less than 1 then c gets smaller, if x is more than 1 c gets bigger. When x is between 0 and -1 c gets smaller slower, when x is less than -1 c gets smaller faster.

A monomial with power two is one that "squares" the value of x. The reference to square is because using the multiplication operation once allows us to measure area. If you have something that is one unit on each side this is called a square unit. If you divide both sides of your square unit in half, you get 4 quarter units. We represent this with math by doing the multiplication ${\displaystyle 1/2*1/2=1/4}$  Squaring something is a non-intuitive operation until you become comfortable with the graph of the function. We can see this with the story of the mathematician who was offered a reward by his king. The mathematician said he wanted a single grain of wheat, squared every day for 30 days. For the first seven days the king's servants delivered 1, 2, 4, 16, 256, 65,536 grains of wheat to the mathematician. On the seventh day the value was 4,294,967,296 (4 gig in computer terms)... Sometimes the story ends with the king re-negotiating, sometimes the story ends with the king executing the mathematician to preserve his kingdom, and sometimes the king is astute enough not to take the deal.

A monomial with power three is one that "cubes" the value of x. This is because we use the operation x*x*x to measure the volume that a given area of x*x takes up. If you have a cube that is 1 unit on each side and cut each side in half you will find that you have created 8 cubes. If the mathematician had asked to have the single grain of wheat cubed than the servants would have delivered 1, 8, 512, ${\displaystyle 134\times 10^{6}}$ , ${\displaystyle 242\times 10^{22}}$  grains of wheat and the kings deal would have needed to be re-negotiated two days earlier.

## Polynomials

A polynomial of one variable, x, is an algebraic expression that is a sum of one or more monomials. The degree of the polynomial is the highest degree of the monomials in the sum. An polynomial ${\displaystyle P(x)}$  can generically be expressed in the form

${\displaystyle P(x)=a_{n}x^{n}+...+a_{i}x^{i}+...a_{2}x^{2}+a_{1}x+a_{0}\ }$

or

${\displaystyle \ P(x)=\sum _{i=0}^{n}a_{i}x^{i}}$

The constants ai are called the coefficients of the polynomial.

Each of the individual monomials in the above sum, whose coefficient ai ≠ 0, is called a term of the polynomial. When i = 0, xi = 1 and the corresponding term simply equals the constant ai. Also when i = 1, the corresponding term equals ai x.

A polynomial having two terms is called a binomial. A polynomial having three terms is called a trinomial.

## Polynomial Equations

We refer to all functions with one independent variable as ${\displaystyle P(x)}$ . Each instance of ${\displaystyle P(x)}$  can be represented by an equation (either a monomial or a polynomial) which may have one or more places where the dependent variable is equal to zero. These places are called roots and they represent the number(s) whose value(s) for x make the function ${\displaystyle P(x)=0}$  true. These roots are called the zeroes of the polynomial (singular is zero). A polynomial of degree 1, will always look like a line when you graph it, and always has 1 real zero. A polynomial of degree 2, a quadratic function, can have 0, 1, or 2 real zeroes. A polynomial of degree 3 (a cubic function) can have 1 or 3 real zeroes. A polynomial of degree 4 can have 0, 2, or 4 real zeroes. Complex (unreal) zeroes, when present, always come in pairs. In general, a polynomial of degree n, where n is odd, can have from 1 to n real zeroes. A polynomial of degree n, where n is even, can have from 0 to n real zeroes.

When we graph polynomials each zero is a place where the polynomial crosses the x axis. A polynomial of degree one can be generically written as ${\displaystyle P(x)=Mx+C}$  where M and C can be any real number. We will see that quadratic functions are curves. The curve can bend before it ever touches the X axis in which case it has no zeroes, It can bend just as it touches the X axis, in which case it can have just one zero, or it can open up above or below the X axis in which case it will have two zeroes. If you think about this you will see that polynomials with an odd degree (1,3,5, ...) have to be positive and negative, so they have to cross the X axis at least once. Polynomials with an even degree (2,4,6,....) might always be positive or negative and never have a zero.

Normally we represent a function in the form ${\displaystyle P(x)=y}$ , but when we are looking for the roots of the function we want y to be equal to zero so we solve for the equation of ${\displaystyle P(x)}$  where ${\displaystyle P(x)=0}$

Order Name Number of bumps Where found
1 linear no bumps - straight line straight line equations
2 quadratic one bump equations involving area and

vibrations

3 cubic two bumps equations involving volumes
4 quartic three bumps some physics equations (melting ice)
n (5+) n-1 bumps very rare

## Solving Polynomial Equations

Some polynomial equations can be solved by factoring, and all equations of degrees 1-4 can be solved completely by formulae. Above degree 4, there are no formulae for solving completely, and you must rely on numerical analysis or factoring. This means that for polynomials of degree greater than 4 it is often impossible to find exact solutions.

### Rational roots of polynomial equations

Often we are interested in the rational roots of polynomials. A root is much like a factor of a number. For instance all even numbers have a factor of two. This means you can write the even numbers as two times another number. That is the numbers 2, 4, 6, 8 ... can be written as 2*1, 2*2, 2*3, 2*4 ... . This fact is helpful when you have a fraction of two even numbers. Given a fraction of two even numbers called N and M ${\displaystyle N/M}$  you could reduce the fraction by re-writing it as ${\displaystyle 2*n/2*m}$ . By keeping fractions in lowest terms it's easier to know when you can add or subtract them without looking for a common denominator.

### Multiplying polynomials together

When we multiply polynomials together we rely heavily on the distributive property.

For instance when we multiply 67 by 5 we can divide the equation into (60 + 7)*5 = (300 + 35) = 335. Additionally we can apply the commutative property to multiply multidigit numbers. 67*25 = (60 + 7)(20 + 5) = ((60 + 7)*20) + ((60 + 7) *5) = (60*20) + (7*20) + (60*5) + (7*5) = 1200 + 140 + 300 + 35 = 1675. These properties are the foundation for the different forms of the mechanical calculating tool the abacus.

When multiplying polynomials together we do similar operations. We use the commutative property to divide the multiplier into its component parts and multiply the multiplicant by each of these parts. For instance to multiply ${\displaystyle x^{2}+x}$  by ${\displaystyle x+1}$  we first write the multiplicand and multiplier in terms of powers of x. This gives us ${\displaystyle x^{2}+x+0x^{0}}$  and ${\displaystyle x+1x^{0}}$  The terms raised to the zero power represent constant integer terms in our equations. Next we apply the commutative property to rewrite the equations as ${\displaystyle [(x^{2}+x+0x^{0})*1x^{1}]+[(x^{2}+x+0x^{0})*1x^{0}]}$ . We simplify these equations to be ${\displaystyle [x^{3}+x^{2}+0x^{1}]+[x^{2}+x+0x^{0}]}$  (notice how our integer term drops out). Finally we combine like terms to get the answer x^3 + 2x^2 + x +0x^0. Let's repeat that in the more familiar columnar format of multiplication:

         1x^2 + 1x^1 + 0x^0
*               1x^1 + 1x^0
--------------------------
1x^2 + 1x^1 + 0x^0
+ 1x^3 + 1x^2 + 0x^1
--------------------------
= 1x^3 + 2x^2 + 1x^1 + 0x^0 = x^3 + 2x^2 + x


By breaking a polynomial into its r

If we have a polynomial P(x)

${\displaystyle P(x)=a_{n}x^{n}+...+a_{i}x^{i}+...a_{2}x^{2}+a_{1}x+a_{0}\ }$

The only possible rational roots (roots of the form p/q) are in the form

${\displaystyle {\frac {p}{q}}={\frac {{\mbox{factor}}\ {\mbox{of}}\ a_{0}}{{\mbox{factor}}\ {\mbox{of}}\ a_{n}}}}$

## Binomials

A binomial is a sum or difference of two monomials. These can also be called polynomials, but to specify, these are binomials.

2x + 2

2y - 7

### How to factor

To factor binomials, find the greatest common factor between the terms and factor.

#### Example

4x + 2

The greatest common factor between these terms is 2 because both of the terms can be divided by it and the coefficient and constant is still an integer. The example factored would become:

2(2x+1)

# Formulas

Computing factors of polynomials requires knowledge of different formulas and some experience to find out which formula to be applied. Below, we give some important formulas:

${\displaystyle {x^{2}-y^{2}=(x+y)(x-y)}}$

${\displaystyle {x^{2}+2xy+y^{2}=(x+y)^{2}}}$

${\displaystyle {x^{2}-2xy+y^{2}=(x-y)^{2}}}$

${\displaystyle {x^{3}-y^{3}=(x-y)(x^{2}+xy+y^{2})}}$

${\displaystyle {x^{3}+y^{3}=(x+y)(x^{2}-xy+y^{2})}}$

${\displaystyle {a^{3}+b^{3}+c^{3}-3abc=(a+b+c)(a^{2}+b^{2}+c^{2}-ab-ac-bc)}}$

## Examples

:${\displaystyle {a^{2}+ab}}$
:${\displaystyle {a(a+b)}}$


${\displaystyle {x^{2}+3x+2}}$
${\displaystyle {x^{2}+y^{2}=2xy}}$
${\displaystyle {x(x+1)+2(x+1)}}$
${\displaystyle {(x+1)(x+2)}}$

:${\displaystyle {a^{2}-4x^{2}}}$
:${\displaystyle {(a^{2}-2ax)+(2ax-4x^{2})}}$
:${\displaystyle {a(a-2x)+2x(a-2x)}}$
:${\displaystyle {(a-2x)(a+2x)}}$


${\displaystyle {x^{3}+8y^{3}}}$
${\displaystyle {(x^{3}-2x^{2}y+4xy^{2})+(2x^{2}y-4xy^{2}+8y^{3})}}$
${\displaystyle {x(x^{2}-2xy+4y^{2})+2y(x^{2}-2xy+4y^{2})}}$
${\displaystyle {(x^{2}-2xy+4y^{2})(x+2y)}}$
${\displaystyle {(x+2y)(x^{2}-2xy+4y^{2})}}$

:${\displaystyle {x^{3}+2x^{2}-13x+10}}$
:${\displaystyle {(x^{3}+4x^{2}-5x)-(2x^{2}+8x-10)}}$
:${\displaystyle {x(x^{2}+4x-5)-2(x^{2}+4x-5)}}$
:${\displaystyle {(x^{2}+4x-5)(x-2)}}$
:${\displaystyle {(x-2)(x^{2}+4x-5)}}$
:${\displaystyle {(x-2)[(x^{2}-x)+(5x-5)]}}$
:${\displaystyle {(x-2)[x(x-1)+5(x-1)]}}$
:${\displaystyle {(x-2)(x-1)(x+5)}}$


${\displaystyle {3x^{4}-3x^{3}-2x^{2}-x-1}}$
${\displaystyle {(3x^{4}-3x^{3}-3x^{2})+(x^{2}-x-1)}}$
${\displaystyle {3x^{2}(x^{2}-x-1)+1(x^{2}-x-1)}}$
${\displaystyle {(x^{2}-x-1)(3x^{2}+1)}}$

:${\displaystyle {x^{4}+4}}$
:${\displaystyle {(x^{4}+4x^{2}+4)-4x^{2}}}$
:${\displaystyle {(x^{2}+2)^{2}-(2x)^{2}}}$
:${\displaystyle {(x^{2}+2x+2)(x^{2}-2x+2)}}$


(name thorem)

write out the coefficients and if the end is equal to zero, than it is a root

example: ${\displaystyle {9x^{2}-6x+9}}$

${\displaystyle {4r^{2}-12+9^{2}}}$

# Possible Factors

To factor we must first look for possible factors. Possible factors are any number that might be a factor. Once we have a possible factor then we divide that number into the number we are factoring. If they divide evenly then we have a factor! The factor is the possible factor we found and the result of the division problem. Here is an example. Let's say the number we are factoring is 20. 2 is the possible factor. 20 / 2 = 10. They divide evenly which means we have a factor. The factors are 2 (the possible factor), and 10 (the result of the division problem). Now that we have a factor we start over with a new possible factor and find all of the factors.

## Examples

Factor 12

First find all the possible factors

The possible factors are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12

Next we will try them one by one

12/1 = 12 (1 and 12 are factors)

12/2 = 6 (2 and 6 are factors)

12/3 = 4 (3 and 4 are factors)

12/4 = 3 (we already have the factors 3 and 4)

Once we get a factor we already have then we know we have all the factors.

So the factors for 12 are 1, 2, 3, 4, 6, and 12.

Factor 54

First find all the possible factors

The possible factors are {1, 2, 3 ... 52, 53, 54}

Do not worry this is not as much work as it seems!

54/1 = 54 (1 and 54 are factors)

54/2 = 27 (2 and 27 are factors)

54/3 = 18 (3 and 18 are factors)

54/4 = 13r2 (4 is not a factor)

54/5 = 10r4 (5 is not a factor)

54/6 = 9 (6 and 9 are factors)

54/7 = 7r5 (7 is not a factor)

54/8 = 6r6 (8 is not a factor)

54/9 = 6 (we already have the factors 9 and 6)

So the factors for 54 are 1, 2, 3, 6, 9, 18, 27, and 54

Factor 180

First find all the possible factors

The possible factors are {1, 2, 3 ... 178, 179, 180}

Do not worry this is not as much work as it seems!

180/1 = 180 (1 and 180 are factors)

180/2 = 90 (2 and 90 are factors)

180/3 = 60 (3 and 60 are factors)

180/4 = 45 (4 and 45 are factors)

180/5 = 36 (5 and 36 are factors)

180/6 = 30 (6 and 30 are factors)

180/7 = 25r5 (7 is not a factor)

180/8 = 22r4 (8 is not a factor)

180/9 = 20 (9 and 20 are factors)

180/10 = 18 (10 and 18 are factors)

180/11 = 16r4 (11 is not a factor)

180/12 = 15 (12 and 15 are factors)

180/13 = 13r11 (13 is not a factor)

180/14 = 12r12 (14 is not a factor)

180/15 = 12 (we already have the factors 15 and 12)

So the factors for 180 are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, and 180.

# Dividing polynomials

The process of factoring will require dividing polynomials. This form of division is not too different from long division method, and is known as synthetic division.

Consider the polynomial x3 - 21x2 + 143x - 315. In this case, determining factors may require trial and error (until you learn of alternate techniques), and when you do, you will need to divide the polynomial with the discovered factor.

In this example, we will divide by (x-5). The full division starts like this:

1x -5 | 1x^3  -21x^2 +143x - 315


As with long division, you need to find the number used for subtraction and place it on top - in this case, you need to make sure the left-most term becomes zero. Next, multiply the newly added top-most term with the left hand side to get the amount to subtract, and perform the subtraction.

                1x^2
1x -5 | 1x^3  -21x^2 +143x - 315
1x^3   -5x^2
------------
-16x^2 +143x - 315


Repeat until the division is complete:

                1x^2  -16x + 63
1x -5 | 1x^3  -21x^2 +143x - 315
1x^3   -5x^2
------------
-16x^2 +143x - 315
-16x^2 + 80x
-------------
63x - 315
63x - 315
---------
0


(If there is a remainder at this point, place it as the numerator over the term being factored out.)

Some people may find writing the x3 and other variables to be bulky - if writing on pen and paper, they can be omitted as part of shorthand.

           1  -16 + 63
1 -5 | 1 -21 +143 - 315
1  -5
-----
-16 +143 - 315
-16 + 80
---------
63 - 315
63 - 315
--------
0


In this case, factoring is straight forward since you can easily determine the number to use for the next step in division.

# Completing the Square

## Derivation

The purpose of "completing the square" is to either factor a prime quadratic equation or to more easily graph a parabola. The procedure to follow is as follows for a quadratic equation ${\displaystyle y=ax^{2}+bx+c}$ :

1. Divide everything by a, so that the number in front of ${\displaystyle x^{2}}$  is a perfect square (1):

${\displaystyle {\frac {y}{a}}=x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}}$

2. Now we want to focus on the term in front of the x. Add the quantity ${\displaystyle \left({\frac {b}{2a}}\right)^{2}}$  to both sides:

${\displaystyle {\frac {y}{a}}+\left({\frac {b}{2a}}\right)^{2}=x^{2}+{\frac {b}{a}}x+\left({\frac {b}{2a}}\right)^{2}+{\frac {c}{a}}}$

3. Now notice that on the right, the first three terms factor into a perfect square:

${\displaystyle x^{2}+{\frac {b}{a}}x+\left({\frac {b}{2a}}\right)^{2}=\left(x+{\frac {b}{2a}}\right)^{2}}$

Multiply this back out to convince yourself that this works.

4. Therefore the completed square form of the quadratic is:

${\displaystyle {\frac {y}{a}}+\left({\frac {b}{2a}}\right)^{2}=\left(x+{\frac {b}{2a}}\right)^{2}+{\frac {c}{a}}}$  or, multiplying through by a,

${\displaystyle y=a\left(x+{\frac {b}{2a}}\right)^{2}+c-{\frac {b^{2}}{4a}}}$

## Explanation of Derivation

1. Divide everything by a, so that the number in front of ${\displaystyle x^{2}}$  is a perfect square (1):

${\displaystyle x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}={a}}$

Think of this as expressing your final result in terms of 1 square x. If your initial equation is

2. Now we want to focus on the term in front of the x. Add the quantity ${\displaystyle \left({\frac {b}{2a}}\right)^{2}}$  to both sides:

${\displaystyle {\frac {y}{a}}+\left({\frac {b}{2a}}\right)^{2}=x^{2}+{\frac {b}{a}}x+\left({\frac {b}{2a}}\right)^{2}+{\frac {c}{a}}}$

3. Now notice that on the right, the first three terms factor into a perfect square:

${\displaystyle x^{2}+{\frac {b}{a}}x+\left({\frac {b}{2a}}\right)^{2}=\left(x+{\frac {b}{2a}}\right)^{2}}$

Multiply this back out to convince yourself that this works.

4. Therefore the completed square form of the quadratic is:

${\displaystyle {\frac {y}{a}}+\left({\frac {b}{2a}}\right)^{2}=\left(x+{\frac {b}{2a}}\right)^{2}+{\frac {c}{a}}}$  or, multiplying through by a,

## Example

The best way to learn to complete a square is through an example. Suppose you want to solve the following equation for x.

 2x2 + 24x + 23 = 0 Does not factor easily, so we complete the square. x2 + 12x + 23/2 = 0 Make coefficient of x2 a 1, by dividing all terms by 2. x2 + 12x = - 23/2 Add – 23/2 to both sides. x2 + 12x + 36 = - 23/2 + 36 Take half of 12 (coefficient of x), and square it. Add to both sides. (x + 6)2 = 49/2 Factor. Now we can take square roots to easily solve this form of the equation. √(x + 6)2 = √49/√2 Take the square root. x + 6 = 7/√2 Simplify. x = -6 + (7√2)/2 Rationalize the denominator.

# Square Root is Positive

## Square Root Is Not Negative

We often say that ${\displaystyle {\sqrt {x}}}$  is the positive number, which when squared is equal to ${\displaystyle x}$ . If ${\displaystyle x>0}$  this is perfectly correct, but if ${\displaystyle x=0}$  then ${\displaystyle {\sqrt {x}}={\sqrt {0}}=0}$  which is not positive. So to be technically correct (which is part of the fun of math) we should say that ${\displaystyle {\sqrt {x}}}$  is the non-negative number, which when squared is equal to ${\displaystyle x}$ .

If it is needed to express that a square root may be both positive and negative, you will see ${\displaystyle \pm {\sqrt {x}}}$ .

## Derivation

The solutions to the general-form quadratic function ${\displaystyle ax^{2}+bx+c=0}$  can be given by a simple equation called the quadratic equation. To solve this equation, recall the completed square form of the quadratic equation derived in the previous section:

${\displaystyle y=a\left(x+{\frac {b}{2a}}\right)^{2}+c-{\frac {b^{2}}{4a}}}$

In this case, ${\displaystyle y=0}$  since we're looking for the root of this function. To solve, first subtract c and divide by a:

${\displaystyle \left(x+{\frac {b}{2a}}\right)^{2}={\frac {b^{2}}{4a^{2}}}-{\frac {c}{a}}}$

Take the (plus and minus) square root of both sides to obtain:

${\displaystyle x+{\frac {b}{2a}}=\pm {\sqrt {{\frac {b^{2}}{4a^{2}}}-{\frac {c}{a}}}}}$

Subtracting ${\displaystyle {\frac {b}{2a}}}$  from both sides:

${\displaystyle x=-{\frac {b}{2a}}\pm {\sqrt {{\frac {b^{2}}{4a^{2}}}-{\frac {c}{a}}}}}$

This is the solution but it's in an inconvenient form. Let's rationalize the denominator of the square root:

${\displaystyle {\sqrt {{\frac {b^{2}}{4a^{2}}}-{\frac {c}{a}}}}={\sqrt {\frac {b^{2}-4ac}{4a^{2}}}}={\frac {\sqrt {b^{2}-4ac}}{2|a|}}=\pm {\frac {\sqrt {b^{2}-4ac}}{2a}}}$