Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012

6.3 The quadratic function

Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012

The quadratic function

The quadratic function takes the form f(x) = ax2 + bx + c where a ≠ 0.

The graph of a quadratic function has a characteristic shape.

Whether the graph is or depends on the value of a as shown

below.

−2 0 2 4 6 8 10 12

3

5

7

9

x

y

a > 0 a < 0

Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012

The quadratic function

When the quadratic function is graphed, there are several key

properties that need to be identified. These are as follows.

−2 2 4 6 8 10

−2

2

4

x

y

axis of symmetry

y-intercept

x-intercepts

vertex

Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012

The quadratic function

At the y-intercept, the x-coordinate is 0.

Therefore the value x = 0 can be substituted into the equation y = ax2 + bx + c.

This gives y = c.

−2 2 4 6 8 10

−2

2

4

x

y

y-intercept

Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012

The quadratic function

At the x-intercept, the y-coordinate is 0.

Therefore the value y = 0 can be substituted into the equation

y = ax2 + bx + c.

This gives ax2 + bx + c = 0.

−2 2 4 6 8 10

−2

2

4

x

y

x-intercepts

Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012

The quadratic function

ax2 + bx + c = 0 can sometimes be solved by factorizing to give a(x + p)(x + q) = 0 .

If it cannot be factorized, x can be found by using the quadraticformula

a

acbbx

2

42

Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012

The quadratic function

The axis of symmetry for the quadratic y = ax2 + bx + c is foundusing the equation

Because the graph is symmetrical, the axis of symmetry also passes

through the point that is halfway between the two x-intercepts.

−2 2 4 6 8 10

−2

2

4

x

y

axis of symmetry

x-intercepts

a

bx

2

Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012

The quadratic function

The vertex lies on the axis of symmetry, therefore its x-coordinate isalso found by the formula

To find the y-coordinate of the vertex, substitute the x-value foundabove into the equation of the quadratic y = ax2 + bx + c.

−2 2 4 6 8 10

−2

2

4

x

y

vertex

axis of symmetry

a

bx

2