# IB Mathematics (HL)/Functions

## Topic 2: Core - Functions and Equations

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

$f(x) = a(x-p)^2 + q$

The axis of symmetry is $x = p$

Ex. $y = 2(x+3)^2 + 4$

The axis of symmetry of the graph is $x = -3$

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

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

#### Discriminant

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

$b^2 - 4ac > 0$ : The equation has 2 real roots

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

$b^2 - 4ac < 0$ : The equation has 0 real roots

If the middle number is even in $ax^2 + bx + c$ then the discriminant can be calculated as $\frac{b^2}{4} - ac$. 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.