5.3 The Range of a Sinusoidal Function

Handout: FOM 12 5.3 Determine the Range

The range of the graph y=\sin (x^{\circ}) is -1\leq y \leq 1

The two tranformations we can make to the y values are to

  • multiply (or divide)
  • add (or subtract

In general, a sinusoidal graph has equation y=a\sin(b(x-c))^{\circ}+d. It is only the values a and d that alter the range of the graph.


To draw the graph y=3\sin (x^{\circ}) we consider particular points (easy points), and multiply the y coordinate by 3 as follows:

Track each point in turn. For example, on the blue line we have the point (90,1) therefore we plot the new point (90,3). The y coordinate (that is, \sin(90^{\circ})) is multiplied by 3.

Next we draw a line through our new points:

We see that the range of the green curve is -3 \leq y \leq 3.

The amplitude of this curve is 3.

The sinusoidal axis is not changed, it is still y=0.


To draw the graph y=\sin(x^{\circ})+2 again we consider particular points and add 2 to the y value as follows:

As before, track each point in turn. For example, the point on the blue curve at (30,0.5) will become the point (30,2.5).

We see that the range of the green curve is 1 \leq y \leq 3.

The sinusoidal axis is the horizontal line y=2.

The amplitude of the curve is not changed, it is still 1.

Try each transformation here:

Multiply and add

To do both operations, we should multiply first then add. However, in practice it is easier to draw a new sinusoidal axis, and plot the correct amplitude from there.

For example, transform y=\sin(x^{\circ}) to y=2\sin(x^{\circ})+3.

First, lets draw a new sinusoidal axis at y=3

Now let’s find the multiples of 180 on the line y=3 to plot our new ‘zeros’:

Now let’s track the multiples of 90, and plot our new max and min but remembering that the amplitude of y=3\sin (x^{\circ})+2 is 3, so we plot 3 above and below the sinusoidal axis:

Finally, we can draw our curve and erase the sinusoidal axis:

The range of our new graph is 1 \leq y \leq 5, which we can see is the same as 3-2 \leq y \leq 3+2.

In general, we can say that the range of the sinusoidal function y=a\sin(b(x-c))^{\circ}+d is

    \[d-a \leq y \leq d+a\]

(when a is positive, otherwise the inequality is reversed).

Try both transformations together here:

Practice: Determine the range

CA1 Test out: Determine the range accuracy quiz

Practice: Match the graph

CA2 Test out: Match the graph