Q. 39 - Precalculus Enhanced with Graphing Utilities

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.

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

Step-by-Step Solution

Answer

The domain of the function is$\left\{x\right|x\ne k3\mathrm{\pi }, \mathrm{k} \mathrm{is} \mathrm{an} \mathrm{integer}\}$

The range of the function is $(-\infty ,-3]\cup [1,\infty )$

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

Step 1. Given data

The given function is

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

Step 2. Graph of the parent function

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

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

Step 3. Vertical stretch

Plot $y= 2 csc\left(x\right)$ by vertical stretching the graph of $y= csc\left(x\right)$ by a factor of $2$

Step 4. Horizontal stretch

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

Step 5. Vertical shift

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

Step 6. Domain and range

The graph of function state that the

The domain of the function is $\left\{x\right|x\ne 3k\mathrm{\pi }\}$

The range of the function is$(-\infty ,-3]\cup [1,\infty )$

Step 7. Verification

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

The graph matches so our graph is correct.