Your mission, should you choose to accept it, as Agent Trigonometry is to the find the domain and the range of the function \begin{align*}y=\frac{1}{2}\sin (x+2)-3\end{align*}.

### Guidance

Here you'll combine the previous two concepts and change the amplitude, the horizontal shifts, vertical shifts, and reflections.

#### Example A

Graph \begin{align*}y=4 \sin \left(x-\frac{\pi}{4}\right)\end{align*}. Find the domain and range.

**Solution:** First, stretch the curve so that the amplitude is 4, making the maximums and minimums 4 and -4. Then, shift the curve \begin{align*}\frac{\pi}{4}\end{align*} units to the right.

As for the domain, it is all real numbers because the sine curve is periodic and infinite. The range will be from the maximum to the minimum; \begin{align*}y\in [-4,4]\end{align*}.

#### Example B

Graph \begin{align*}y=-2\cos (x-1)+1\end{align*}. Find the domain and range.

**Solution:** The -2 indicates the cosine curve is flipped and stretched so that the amplitude is 2. Then, move the curve up one unit and to the right one unit.

The domain is all real numbers and the range is \begin{align*}y\in [-1,3]\end{align*}.

#### Example C

Find the equation of the sine curve below.

**Solution:** First, let’s find the amplitude. The range is from 1 to -5, which is a total distance of 6. Divided by 2, we find that the amplitude is 3. Halfway between 1 and -5 is \begin{align*}\frac{1+(-5)}{5}=-2\end{align*}, so that is our vertical shift. Lastly, we need to find the horizontal shift. The easiest way to do this is to superimpose the curve \begin{align*}y=3\sin (x)-2\end{align*} over this curve and determine the movement from one maximum to the closest maximum of this curve.

Subtracting \begin{align*}\frac{\pi}{2}\end{align*} and \begin{align*}\frac{\pi}{6}\end{align*}, we have:

\begin{align*}\frac{\pi}{2}-\frac{\pi}{6}=\frac{3 \pi}{6}-\frac{\pi}{6}=\frac{2 \pi}{6}=\frac{\pi}{3}\end{align*}

Making the equation \begin{align*}y=3 \sin \left(x+\frac{\pi}{3}\right)-2\end{align*}.

**Concept Problem Revisit**

The \begin{align*}\frac{1}{2}\end{align*} indicates the sine curve is smooshed so that the amplitude is \begin{align*}\frac{1}{2}\end{align*}. Then, move the curve down two units and to the left three units.

The domain is all real numbers and the range is \begin{align*}y\in [-\frac{5}{2}, -\frac{3}{2}]\end{align*}.

### Guided Practice

Graph the following functions. State the domain and range. Show two full periods.

1. \begin{align*}y=-2\sin \left(x-\frac{\pi}{2}\right)\end{align*}

2. \begin{align*}y=\frac{1}{3}\cos (x+1)-2\end{align*}

3. Write one sine equation and one cosine equation for the curve below.

#### Answers

1. The domain is all real numbers and the range is \begin{align*}y\in [-2,2]\end{align*}.

2. The domain is all real numbers and the range is \begin{align*}y \in \left [-2\frac{1}{3},-1\frac{2}{3} \right ]\end{align*}.

3. The amplitude and vertical shift is the same, whether the equation is a sine or cosine curve. The vertical shift is -2 because that is the number that is halfway between the maximum and minimum. The difference between the maximum and minimum is 1, so the amplitude is half of that, or \begin{align*}\frac{1}{2}\end{align*}. As a sine curve, the function is \begin{align*}y=-2+\frac{1}{2}\sin x\end{align*}. As a cosine curve, there will be a shift of \begin{align*}\frac{\pi}{2}, y=\frac{1}{2} \cos \left(x-\frac{\pi}{2}\right)-2\end{align*}.

### Explore More

Determine if the following statements are true or false.

- To change a cosine curve into a sine curve, shift the curve \begin{align*}\frac{\pi}{2}\end{align*} units.
- For any given sine or cosine graph, there are infinitely many possible equations that can be written to represent the curve.
- The amplitude is the same as the maximum value of the sine or cosine curve.
- The horizontal shift is always in terms of \begin{align*}\pi\end{align*}.
- The domain of any sine or cosine function is always all real numbers.

Graph the following sine or cosine functions such that \begin{align*}x \in [-2 \pi, 2 \pi]\end{align*}. State the domain and range.

- \begin{align*}y=\sin \left(x+\frac{\pi}{4}\right)+1\end{align*}
- \begin{align*}y=2-3\cos x\end{align*}
- \begin{align*}y=\frac{3}{4} \sin \left(x-\frac{2 \pi}{3}\right)\end{align*}
- \begin{align*}y=-5 \sin(x-3)-2\end{align*}
- \begin{align*}y=2 \cos \left(x+\frac{5 \pi}{6}\right)-1.5\end{align*}
- \begin{align*}y=-2.8 \cos(x-8)+4\end{align*}

Use the graph below to answer questions 12-15.

- Write a sine equation for the function where the amplitude is positive.
- Write a cosine equation for the function where the amplitude is positive.
- How often does a sine or cosine curve repeat itself? How can you use this to help you write different equations for the same graph?
- Write a second sine and cosine equation with different horizontal shifts.

Use the graph below to answer questions 16-20.

- Write a sine equation for the function where the amplitude is positive.
- Write a cosine equation for the function where the amplitude is positive.
- Write a sine equation for the function where the amplitude is
*negative*. - Write a cosine equation for the function where the amplitude is
*negative*. - Describe the similarities and differences between the four equations from questions 16-19.

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 14.3.