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) = ax² + 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 : (-b ± √(b² - 4ac))/2a

Ex. y = x² + 2x - 1 Since this cannot be factored, it is possible to use the quadratic formula x = -1 ± √5

Discriminant

The discriminant of the equation is important in determining whether the equation has 2, 1, 0 roots The equation of the discriminant: b² - 4ac

b² - 4ac > 0 : The equation has 2 real roots

b² - 4ac = 0 : The equation has 1 real root

b² - 4ac < 0 : The equation has 0 real roots

If the middle number is even in ax² + bx + c then the discriminant can be calculated as (b²)/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.