# 5.3 The Range of a Sinusoidal Function

Handout: FOM 12 5.3 Determine the Range

The range of the graph is  The two tranformations we can make to the values are to

• multiply (or divide)

In general, a sinusoidal graph has equation . It is only the values and that alter the range of the graph.

# Multiply

To draw the graph we consider particular points (easy points), and multiply the coordinate by 3 as follows: Track each point in turn. For example, on the blue line we have the point therefore we plot the new point . The coordinate (that is, ) is multiplied by .

Next we draw a line through our new points: We see that the range of the green curve is .

The amplitude of this curve is .

The sinusoidal axis is not changed, it is still .

To draw the graph again we consider particular points and add 2 to the value as follows: As before, track each point in turn. For example, the point on the blue curve at will become the point . We see that the range of the green curve is .

The sinusoidal axis is the horizontal line .

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

# Try each transformation here:

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 to .

First, lets draw a new sinusoidal axis at  Now let’s find the multiples of 180 on the line 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 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 , which we can see is the same as .

In general, we can say that the range of the sinusoidal function is (when 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