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 or 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.
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:
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.
So, 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 (or turning point) of the parabola is the point (0, 0).
In general:
Example 4
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. So, 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).
