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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, equation of midline, amplitude and period of the function $y=\mathrm{cos}\left(\theta \right)-5$ and then graph the function.

The vertical shift of $y=\mathrm{cos}\left(\theta \right)-5$ is $-5$.

The equation of midline of $y=\mathrm{cos}\left(\theta \right)-5$ is $y=-5$.

The amplitude of $y=\mathrm{cos}\left(\theta \right)-5$ is 1.

The period of $y=\mathrm{cos}\left(\theta \right)-5$ is $2\pi$.

See the step by step solution

## Step 1. Write down the given information.

The given function is $y=\mathrm{cos}\left(\theta \right)-5$.

## 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, a period of $\frac{360°}{\left|b\right|}\text{or}\frac{2\pi }{\left|b\right|}$ for sine, cosecant, secant and cosine functions and a period of $\frac{180°}{\left|b\right|}\text{or}\frac{\pi }{\left|b\right|}$ for tangent and cotangent function. The phase shift for the functions is $\left(h\right)$.

The amplitude of secant, cosecant, tangent and cotangent functions is not defined.

The equation for the midline is written as, $y=k$. A midline is a new reference line when the parent graph is stretched vertically up or down and then the graph oscillates about new reference line called the midline.

## Step 3. Calculation.

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

The vertical shift of $y=\mathrm{cos}\left(\theta \right)-5$ is $-5$.

The equation of midline of $y=\mathrm{cos}\left(\theta \right)-5$ is $y=-5$.

The amplitude of $y=\mathrm{cos}\left(\theta \right)-5$ is $\left|1\right|=1$.

The period of $y=\mathrm{cos}\left(\theta \right)-5$ is $\frac{2\pi }{\left|1\right|}=2\pi$.

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

The graph of the function $y=\mathrm{cos}\left(\theta \right)-5$ is shown below. ## Step 5. Conclusion.

The vertical shift of $y=\mathrm{cos}\left(\theta \right)-5$ is $-5$.

The equation of midline of $y=\mathrm{cos}\left(\theta \right)-5$ is $y=-5$.

The amplitude of $y=\mathrm{cos}\left(\theta \right)-5$ is $\left|1\right|=1$.

The period of $y=\mathrm{cos}\left(\theta \right)-5$ is $\frac{2\pi }{\left|1\right|}=2\pi$. ### Want to see more solutions like these? 