IB Mathematics (HL)/Functions

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.