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 $x=-3$

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.