# IB Mathematics (HL)/Functions

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## 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 $x=-3$ ### Solving Quadratics

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.