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.

Factorising
Transforming

Parabola_Water.jpgSketching 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 + 8
y = 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 = 3
y = (3 − 4)(3 − 2)
y = -1

The vertex is at the point (3, -1)

Now sketch the graphY10_Parabolas_01.gif

:

 

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.Y10_Parabolas_02.gif

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.
Y10_Parabolas_03.gif

(b) y = x 2 + k

The k has the effect of moving the parabola along the y-axis by k units.

 

Y10_Parabolas_04.gif

(c) y = (x − h)2

The h has the effect of moving the basic parabola along the x-axis by h units.

 

Y10_Parabolas_05.gif