Year 10 Interactive Maths - Second Edition

Quadratic Functions

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 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).

Key Terms

quadratic function, quadratic graph, parabola, symmetry, axis of symmetry, turning point, vertex

Study Another Topic in Chapter 13: Quadratic Graphs

[ Quadratic Functions ] [ Graphs of y = ax² + c, a > 0 ] [ Graphs of y = a(x - b)², a > 0 ] [ Graphs of y = a(x - b)² + c, a > 0 ] [ Quadratic Graphs by Transformations ] [ Translation ] [ Sketching Parabolic Graphs ] [ Problem Solving I ] [ Quadratic Inequalities ] [ Simultaneous Equations - Linear/Quadratic ] [ Problem Solving II ] [ Sketching Quadratic Inequalities ] [ Intersection of Half-Planes ] [ Problem Solving Unit ] [ Projects ] [ Symbols ] [ Index ]