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Expert-verified Found in: Page 775 ### Algebra 2

Book edition Middle English Edition
Author(s) Carter
Pages 804 pages
ISBN 9780079039903 # State the vertical shift, amplitude, period and phase shift of the function $y=6\mathrm{cot}\left[\frac{2}{3}\left(\theta -90°\right)\right]+0.75$ and then graph the function.

The vertical shift of $y=6\mathrm{cot}\left[\frac{2}{3}\left(\theta -90°\right)\right]+0.75$ is $0.75$.

The amplitude of $y=6\mathrm{cot}\left[\frac{2}{3}\left(\theta -90°\right)\right]+0.75$ is not defined.

The period of $y=6\mathrm{cot}\left[\frac{2}{3}\left(\theta -90°\right)\right]+0.75$ is $\frac{3\pi }{2}$.

The phase shift of $y=6\mathrm{cot}\left[\frac{2}{3}\left(\theta -90°\right)\right]+0.75$ is $90°$.

See the step by step solution

## Step 1. Write down the given information.

The given function is $y=6\mathrm{cot}\left[\frac{2}{3}\left(\theta -90°\right)\right]+0.75$.

## Step 2. Concept used.

A function of the form:

$y=a\mathrm{sin}b\left(\theta -h\right)+k,y=a\mathrm{cos}b\left(\theta -h\right)+k\text{and}y=a\mathrm{tan}b\left(\theta -h\right)+k$ has vertical shift $\left(k\right)$. And period $\frac{360°}{\left|b\right|}\text{or}\frac{2\pi }{\left|b\right|}$ for sine and cosine functions and a period of $\frac{180°}{\left|b\right|}\text{or}\frac{\pi }{\left|b\right|}$ for tangent function. The phase shift for the functions is $\left(h\right)$.

The amplitude of tangent and cotangent functions is not defined.

## Step 3. Evaluating vertical shift, amplitude, period and phase shift of the given function.

With the help of concept stated above, the vertical shift, amplitude, period and phase shift of the function is evaluated as:

The vertical shift of the function $y=6\mathrm{cot}\left[\frac{2}{3}\left(\theta -90°\right)\right]+0.75$ is $0.75$.

The amplitude of $y=6\mathrm{cot}\left[\frac{2}{3}\left(\theta -90°\right)\right]+0.75$ is not defined.

The period of $y=6\mathrm{cot}\left[\frac{2}{3}\left(\theta -90°\right)\right]+0.75$ is $\frac{\pi }{\left|\frac{2}{3}\right|}=\frac{3\pi }{2}$.

The phase shift of $y=6\mathrm{cot}\left[\frac{2}{3}\left(\theta -90°\right)\right]+0.75$ is $90°$.

## Step 4. Sketch the graph of the function.

The graph of the function $y=6\mathrm{cot}\left[\frac{2}{3}\left(\theta -90°\right)\right]+0.75$ is shown below. ## Step 5. Conclusion.

The vertical shift of the function $y=6\mathrm{cot}\left[\frac{2}{3}\left(\theta -90°\right)\right]+0.75$ is $0.75$.

The amplitude of $y=6\mathrm{cot}\left[\frac{2}{3}\left(\theta -90°\right)\right]+0.75$ is not defined.

The period of $y=6\mathrm{cot}\left[\frac{2}{3}\left(\theta -90°\right)\right]+0.75$ is $\frac{3\pi }{2}$.

The phase shift of $y=6\mathrm{cot}\left[\frac{2}{3}\left(\theta -90°\right)\right]+0.75$ is $90°$. ### Want to see more solutions like these? 