Question

graph each function. Be sure to label key points and show at least two cycles. Use the graph to determine the domain and the range of each function.

$y=3cot\left(\frac{1}{2}x\right)-2$

Short answer

The domain of the function is$\left\{x\right|x\ne \mathrm{\pi },\mathrm{k}\mathrm{is}\mathrm{an}\mathrm{integer}\}$

The range of the function is $(-\infty ,\infty )$

The graph of the function $y=3cot\left(\frac{1}{2}x\right)-2$is

Step-by-step solution

Step 1. Given data

The given function is

$y=3cot\left(\frac{1}{2}x\right)-2$

Step 2. Graph of the parent function

Parent function of $y=3cot\left(\frac{1}{2}x\right)-2$is $y=cot\left(x\right)$

Plot the graph of $y=cot\left(x\right)$

Step 3. Vertical stretch

Plot $y=3cot\left(x\right)$by vertical stretching the graph of $y=cot\left(x\right)$by a factor of $3$

Step 4. Horizontal stretch 

Plot $y=3cot\left(\frac{1}{2}x\right)$by Horizontal stretching the graph of $y=3cot\left(x\right)$by a factor of $\frac{1}{2}$

Step 5. Vertical shift

Plot $y=3cot\left(\frac{1}{2}x\right)-2$by vertical shifting the graph of $y=3cot\left(\frac{1}{2}x\right)$to downward by two units

Step 6. Domain and range

The graph of function state that the

The domain of the function $y=3cot\left(\frac{1}{2}x\right)-2$is localid="1646316950947" $\left\{x\right|x\ne \mathrm{\pi },\mathrm{k}\mathrm{is}\mathrm{an}\mathrm{integer}\}$

The range of the function $y=3cot\left(\frac{1}{2}x\right)-2$is$(-\infty ,\infty )$

Step 7. Verification

Plot the graph of $y=3cot\left(\frac{1}{2}x\right)-2$using a graphing utility

The graph matches so our graph is correct.