8.1 Graphs of the sine and cosine functions  (Page 6/13)

Begin by comparing the equation to the general form and use the steps outlined in [link] .

• Step 1. The function is already written in general form.
• Step 2. Since $A=-2,$ the amplitude is $\left|A\right|=2.$
• Step 3. $\left|B\right|=\frac{\pi }{2},$ so the period is $P=\frac{2\pi }{\left|B\right|}=\frac{2\pi }{\frac{\pi }{2}}=2\pi \cdot \frac{2}{\pi }=4.$ The period is 4.
• Step 4. $C=-\pi ,$ so we calculate the phase shift as $\frac{C}{B}=\frac{-\pi ,}{\frac{\pi }{2}}=-\pi \cdot \frac{2}{\pi }=-2.$ The phase shift is $-2.$
• Step 5. $D=3,$ so the midline is $y=3,\hspace{0.17em}$ and the vertical shift is up 3.

Since $A$ is negative, the graph of the cosine function has been reflected about the x -axis.

[link] shows one cycle of the graph of the function.

Recall that, for a point on a circle of radius r , the y -coordinate of the point is $y=r\sin (x),$ so in this case, we get the equation $y(x)=3\sin (x).$ The constant 3 causes a vertical stretch of the y -values of the function by a factor of 3, which we can see in the graph in [link] .

Sketching the height, we note that it will start 1 ft above the ground, then increase up to 7 ft above the ground, and continue to oscillate 3 ft above and below the center value of 4 ft, as shown in [link] .

Although we could use a transformation of either the sine or cosine function, we start by looking for characteristics that would make one function easier to use than the other. Let’s use a cosine function because it starts at the highest or lowest value, while a sine function    starts at the middle value. A standard cosine starts at the highest value, and this graph starts at the lowest value, so we need to incorporate a vertical reflection.

Second, we see that the graph oscillates 3 above and below the center, while a basic cosine has an amplitude of 1, so this graph has been vertically stretched by 3, as in the last example.

Finally, to move the center of the circle up to a height of 4, the graph has been vertically shifted up by 4. Putting these transformations together, we find that