Year 9 Interactive Maths - Second Edition

## Parabolic Graphs

A relation is a quadratic function if the highest power of the pronumeral in the relation is two.

## Graphs of y = ax², a > 0

#### Example 1

##### Solution:

When we plot these points and join them with a smooth curve, we obtain the quadratic graph shown above.  The curve is called a parabola.  It has many applications in science and engineering.

For example, the path followed by a projectile and the shape of the reflector in a car’s headlamps or searchlights.

Looking at the graph and the shape of the curve, you could imagine that a mirror is placed along
the y-axis:  the left-hand side and right hand side of the curve are mirror images of each other.
This property is called symmetry.  We say that the graph is symmetrical about the y-axis, and
the y-axis is called the axis of symmetry.  So, the axis of symmetry has equation x = 0 in the example.

The parabola opens upwards.  The minimum value of y is zero and it occurs when x = 0.  The
point (0, 0) is called the turning point or vertex of the parabola.

###### In general:

In the example above, a = 1.

#### Example 2

##### Solution:

When we plot these points and join them with a smooth curve, we obtain the graph shown above.
###### Note:

The graph is a parabola which opens upwards.  The minimum value of  y is 0 and it occurs
when x = 0.  The point (0, 0) is called the vertex of the parabola.  The graph is symmetrical
about x = 0, i.e. the y-axis.

## Graphs of y = ax², a < 0

#### Example 3

##### Solution:

When we plot these points and join them with a smooth curve, we obtain the graph shown above.
###### Note:

The graph is a parabola which opens downwards.  Clearly, the graph is symmetrical about the y-axis.  Therefore, the equation of the axis of symmetry is x = 0.
The maximum value of  y is 0 and it occurs when x = 0.
The vertex of the parabola is the point (0, 0).

###### In general:

In the example above, a = 1.

#### Example 4

##### Solution:

When we plot these points and join them with a smooth curve, we obtain the graph shown above.