Your best friend asks you to describe how the graph of

### Translating Sine and Cosine Functions

Just like other functions, sine and cosine curves can be moved to the left, right, up and down. The general equation for a sine and cosine curve is **phase shift**, and

Let's graph

This function will be shifted

Now, let's graph

Because -2 is not written in terms of

Finally, let's find the equation of the sine curve below.

First, we know the amplitude is 1 because the average between 2 and 0 (the maximum and minimum) is 1. Next, we can find the vertical shift. Recall that the maximum is usually 1, in this equation it is 2. That means that the function is shifted up 1 unit

Because

To determine the value of the horizontal shift, you might have to estimate. For example, we estimated that the negative shift was -3 because the maximum value of the parent graph is at

**Examples**

**Example 1**

Earlier, you were asked to tell your friend how the graph of

If you compare

Therefore, the graph of

**In Examples 2 & 3, g****raph the function from [π,3π].**

#### Example 2

Shift the parent graph down one unit.

#### Example 3

Shift the parent graph to the left

#### Example 4

Find the equation of the cosine curve below.

The parent graph is in green. It moves up 3 units and to the right

If you moved the cosine curve backward, then the equation would be \begin{align*}y=\cos \left(x+\frac{5 \pi}{4}\right)+3\end{align*}.

### Review

For questions 1-4, match the equation with its graph.

- \begin{align*}y=\sin \left(x-\frac{\pi}{2}\right)\end{align*}
- \begin{align*}y=\cos \left(x-\frac{\pi}{4}\right)+3\end{align*}
- \begin{align*}y=\cos \left(x+\frac{\pi}{4}\right)-2\end{align*}
- \begin{align*}y=\sin \left(x-\frac{\pi}{4}\right)+2\end{align*}

Which graph above also represents these equations?

- \begin{align*}y=\cos(x-\pi)\end{align*}
- \begin{align*}y=\sin \left(x+\frac{3 \pi}{4}\right)-2\end{align*}
- Write another sine equation for graph A.
**Writing**How many sine (or cosine) equations can be generated for one curve? Why?- Fill in the blanks below.
- \begin{align*}\sin x=\cos(x-\underline{\;\;\;\;\;})\end{align*}
- \begin{align*}\cos x=\sin(x-\underline{\;\;\;\;\;})\end{align*}

For questions 10-15, graph the following equations from \begin{align*}[-2\pi, 2\pi]\end{align*}.

- \begin{align*}y=\sin \left(x+\frac{\pi}{4}\right)\end{align*}
- \begin{align*}y=1+\cos x\end{align*}
- \begin{align*}y=\cos(x+\pi)-2\end{align*}
- \begin{align*}y=\sin(x+3)-4\end{align*}
- \begin{align*}y=\sin \left(x-\frac{\pi}{6}\right)\end{align*}
- \begin{align*}y=\cos(x-1)-3\end{align*}
**Critical Thinking**Is there a difference between \begin{align*}y=\sin x +1\end{align*} and \begin{align*}y=\sin(x+1)\end{align*}? Explain your answer.

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 14.2.