Algebra/Chapter 17/Parabolas

Algebra

Geometric Definition

A parabola is the set of points that is the same distance away from a single point called the focus and a line called the directrix. The distance to the line is taken as the perpendicular distance.

One important point on the parabola itself is called the vertex, which is the point which has the smallest distance between both the focus and the directrix. Parabolas are symmetric, and their lines of symmetry pass through the vertex. Also due to this symmetry, only the vertex and one other point are necessary to completely define a parabola.

Algebraic definition

To derive an equation for a parabola, let's suppose (for now) that the directrix is horizontal and therefore has the equation:

$y=a$ where a is some constant.

Also suppose that the focus is given by (x,y) = (b,c). The vertex is then, by definition and inspection, located at:

$(h,k)=\left(b,{\frac {|a+c|}{2}}\right)$

Now let's examine the distance between some point in the x-y plane (x*,y*) and focus and directrix. The distance between the point and the focus is given by the distance formula:

$D_{1}={\sqrt {(x^{*}-b)^{2}+(y^{*}-c)^{2}}}$

Since the directrix is horizontal, the distance between the point and the line is simply $D_{2}=|{y^{*}-a}|$

By the definition of a parabola, both of these distances must be equal, and therefore:

${\sqrt {(x^{*}-b)^{2}+(y^{*}-c)^{2}}}=|y^{*}-a|$

This is the equation of a parabola, but let's make it a little nicer to work with. First square both sides, and notice the absolute values are no longer needed since the square of a number is always positive:

$(x^{*}-b)^{2}+(y^{*}-c)^{2}=(y^{*}-a)^{2}$

FOILing and then simplifying:

$(x^{*}-b)^{2}+y^{*2}-2y^{*}c+c^{2}=y^{*2}-2y^{*}a+a^{2}$

$(x^{*}-b)^{2}-2y^{*}c+c^{2}=-2y^{*}a+a^{2}$

$(x^{*}-b)^{2}+2y^{*}(a-c)+c^{2}-a^{2}=0$. By differences of two squares we arrive at:

$(x^{*}-b)^{2}+2y^{*}(a-c)+(c-a)(c+a)=0$. Factoring:

$(x^{*}-b)^{2}+(c-a)((c+a)-2y^{*})=0$

Now notice that the quantity $a+c$ is twice the y-coordinate of the vertex, and that b is the x-coordinate of the vertex. Therefore:

$(x^{*}-h)^{2}+(c-a)(2k-2y^{*})=0$

Factoring a 2 out of the second term,

$(x^{*}-h)^{2}+2(c-a)(k-y^{*})=0$ or, letting D be the distance between the distance between the vertex and the focus (which is ${\frac {c-a}{2}}$) we arrive at the most useful form of a parabolic equation:

$(x^{*}-h)^{2}=4D(y^{*}-k)$ where D is the distance from the vertex to the focus, and (h,k) is the vertex

When written in this form, the latus rectum has a length of 4D. In addition, the coordinates of the vertex itself are (x,y)=(h,k). Using this information, and the symmetry of the parabola, it is straightforward to graph it.

But, the formula that is written in the Algebra Textbook is usually $x^{2}=4py$ if the parabola is vertical and $y^{2}=4px$ if the parabola is horizontal. And, the focus is (0,p) and the directrix is -p if the parabola is vertical as the focus is (p,0) and the directrix is -p if the parabola is horizontal.

Alternate forms

The standard form of a parabola is:

$y=Ax^{2}+Bx+C$

where A, B, and C are constants. From this form we can deduce that the y-intercept of the parabola is C. It can be shown (and will be shown in a later section) that the x-coordinate of the vertex is ${\frac {-B}{2A}}$.