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.

Multiply

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 $y=0$ .

Add

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: