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.
