Chapter 3.7, Problem 26E

To determine

To graph: the given equation using parent graph $f\left(x\right)$ , also describe the transformations and find the locations of new asymptotes.

Answer to Problem 26E

  

Location of new vertical asymptote is $x=-3$ .

Location of new horizontal asymptote is $y=-1$ .

Explanation of Solution

Given information:

  $\text{Parent function, }f\left(x\right)=\frac{1}{x}$

  $y=\frac{2}{x+3}-1$

Calculation:

Look at the basic graph function of transformation shown below.

  $y=\frac{1}{x-h}+v$

Here, horizontal and vertical movement is represented by $h$ and $v$ respectively.

The subtraction $(-)$ is down for moving down or moving left. Addition $(+)$ is used form moving up or right. The horizontal movements occur within x-operations $\left(x-h\right)$ .

The equation of transformation shows that $f\left(x\right)$ shifts 3 units to left and then stretched by factor of 2. At last it is moved 1 unit down. Thus, the graph of transformation and parent graph $f\left(x\right)$ is as follows:

  

The red color is of graph $f\left(x\right)$ and blue color is of graph after transformation of $f\left(x\right)$

The new vertical asymptote is at $x=-3$ and horizontal asymptote is $y=-1$ .