## Parabolas

Quadratic functions have graphs that are parabolas.
The general equation of a quadratic function can be given in two forms:

y = ax 2 + bx + c OR y = a(x − h) 2 + k

In both forms the highest exponent of x is 2.

The vertex, stationary point or turning point gives either the maximum or minimum value of the function.

### Sketching Parabolas

Method 1: Factorisation.

If the equation is in the form y = ax2 + bx + c the following method should be used:

Step (a) Factorise the function.

Step (b) Find the x- and y-intercepts by putting y = 0 and x = 0.

Step (c) Find the axis of symmetry − always midway between the two x-intercepts.

Step (d) Find the coordinates of the vertex or turning point.

 Example Answer Graph Sketch the graph of the function y = x2 − 6x + 8 Step (a) y = x2 − 6x + 8 = (x − 4)(x − 2) Step (b) Put x = 0 into the equation. y = 02 − 6 × 0 + 8y = 8 The y-intercept is 8 Put y = 0 0 = (x − 4)(x − 2)x = 4 or x = 2 The x-intercepts are 4 and 2 Step (c) Axis of symmetry is always midwaybetween the x-intercepts. Axis of symmetry is the line x = 3 Step (d) The vertex occurs at x = 3 When x = 3y = (3 − 4)(3 − 2)y = -1 The vertex is at the point (3, -1) Now sketch the graph :

Method 2: Transformation.

This method finds the coordinates of the vertex.
The y-intercept is found by putting x = 0.

The graph of the basic parabola y = x2 is shown in the diagram. This basic parabola is moved or transformed as follows.

 (a) y = ax 2 The a has the effect of changing theparabola in the y- direction. It affects the steepness of the graph. If a is large, the parabola is steeper. If a is small, the parabola is flatter. If a is negative, the parabola is inverted. (b) y = x 2 + k The k has the effect of moving the parabola along the y-axis by k units. (c) y = (x − h)2 The h has the effect of moving the basic parabola along the x-axis by h units. 