Chapter 3.7, Problem 1CFU

a.

To determine

the asymptotes of the function by graphing $g\left(x\right)$ .

Answer to Problem 1CFU

Vertical asymptotes at $x=2$

Horizontal asymptotes at $y=-6$

Explanation of Solution

Given information:

  $f\left(x\right)=\frac{1}{x}$

Let the function after translating $f\left(x\right)$ to 2 units to right and 6 units down be $g\left(x\right)$ .

Calculation:

The graph of $f\left(x\right)$ is as follows:

  

In order to obtain graph of $g\left(x\right)$ , the above graph should be shifted to 2 units to right side and 6 units down. The resultant graph will be as follows:

  

Normally, the vertical asymptotes and horizontal asymptotes would be at $x=0$ and $y=0$ respectively, but shifting the graph to 2 unit right moves the vertical asymptotes to $x=2$ and horizontal asymptotes to $y=-6$ . Thus, the asymptotes of $g\left(x\right)$ is $x=2$ and $y=-6$ .

b.

To determine

To find: an equation of the translated graph.

Answer to Problem 1CFU

  $g\left(x\right)=\frac{1}{x-2}-6$

Explanation of Solution

Given information:

  $f\left(x\right)=\frac{1}{x}$

The function after translating $f\left(x\right)$ to 2 units to right and 6 units down is $g\left(x\right)$ .

Calculation:

Whenever the function moves horizontally, the x-operation get change and while moving vertically it stand alone. The base function will be as follows:

Since the function shifts 2 units right, so 2 units will be subtracted from $x$ , and for the translation of 6 units downwards will be made by subtracting the 6 units from the given function. Thus, the equation for $g\left(x\right)$ will be as follows:

  $g\left(x\right)=\frac{1}{x-2}-6$