# IB Mathematics (HL)/Functions

## Topic 2: Core - Functions and Equations

### The Axis of Symmetry for the Graph of a Quadratic Function

${\displaystyle f(x)=a(x-p)^{2}+q}$

The axis of symmetry is ${\displaystyle x=p}$

Ex. ${\displaystyle y=2(x+3)^{2}+4}$

The axis of symmetry of the graph is ${\displaystyle x=-3}$

Quadratic Equations are in the form ${\displaystyle f(x)=ax^{2}+bx+c}$  or in the form ${\displaystyle a(x-p)^{2}+q}$ . To be solved the equations either have to be factored or be solved using the quadratic formula : ${\displaystyle {\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}}$

Ex. ${\displaystyle y=x^{2}+2x-1}$  Since this cannot be factored, it is possible to use the quadratic formula ${\displaystyle x=-1\pm {\sqrt {5}}}$

#### Discriminant

The discriminant of the equation is important in determining whether the equation has 2, 1, 0 roots The equation of the discriminant: ${\displaystyle b^{2}-4ac}$

${\displaystyle b^{2}-4ac>0}$  : The equation has 2 real roots

${\displaystyle b^{2}-4ac=0}$  : The equation has 1 real root

${\displaystyle b^{2}-4ac<0}$  : The equation has 0 real roots

If the middle number is even in ${\displaystyle ax^{2}+bx+c}$  then the discriminant can be calculated as ${\displaystyle {\frac {b^{2}}{4}}-ac}$ . The properties of this modified equation remain the same

### Higher level Functions

These functions have a degree of two or higher and as a result have more than 2 roots. An example of a higher polynomial function is y = x3 − 2x. This is a cubic equation, with three roots. To find these roots just factor the equation. In this case, it becomes, x(x2−2). From here you can factor using the difference of squares (a2−b2). Thus the equation then becomes, y=x(x+√2)(x−√2). The roots of the equation then become 0,±√2.