IB Mathematics (HL)/Functions

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Topic 2: Core - Functions and Equations[edit]

The Axis of Symmetry for the Graph of a Quadratic Function[edit]

The axis of symmetry is

Ex.

The axis of symmetry of the graph is

Solving Quadratics[edit]

Quadratic Equations are in the form or in the form . To be solved the equations either have to be factored or be solved using the quadratic formula :

Ex. Since this cannot be factored, it is possible to use the quadratic formula

Discriminant[edit]

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

 : The equation has 2 real roots

 : The equation has 1 real root

 : The equation has 0 real roots

If the middle number is even in then the discriminant can be calculated as . The properties of this modified equation remain the same

Higher level Functions[edit]

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.