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]

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[edit]

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[edit]

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[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.