Chapter 8.3, Problem 39HP

a.

To determine

To compare and contrast the graph of the pair of equation.

Answer to Problem 39HP

The first graph has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$ .

The second graph is translated $7\text{ units}$ up and has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=7$ .

Explanation of Solution

Given information:

The given function is:

  $y=\frac{1}{x}\text{ and }y-7=\frac{1}{x}$

For the function $y=\frac{1}{x}$ :

This represents a transformation of the graph of $f\left(x\right)=\frac{1}{x}$ .

  $a=1$ : The graph is expanded.

  $h=0$ : There is an asymptote at $x=0$ .

  $k=0$ : There is an asymptote at $y=0$ .

For the function $y-7=\frac{1}{x}\text{ or }y=\frac{1}{x}+7$ :

This represents a transformation of the graph of $f\left(x\right)=\frac{1}{x}$ .

  $a=1$ : The graph is expanded.

  $h=0$ : There is an asymptote at $x=0$ .

  $k=7$ : The graph is translated $7\text{ units}$ up. There is an asymptote at $y=7$ .

The first graph has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$ .

The second graph is translated $7\text{ units}$ up and has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=7$ .

b.

To determine

To compare and contrast the graph of the pair of equation.

Answer to Problem 39HP

Both graphs have a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$ .

The second graph is stretched by a factor of $4$ .

Explanation of Solution

Given information:

The given function is:

  $y=\frac{1}{x}\text{ and }y=4\left(\frac{1}{x}\right)$

For the function $y=\frac{1}{x}$ :

This represents a transformation of the graph of $f\left(x\right)=\frac{1}{x}$ .

  $a=1$ : The graph is expanded.

  $h=0$ : There is an asymptote at $x=0$ .

  $k=0$ : There is an asymptote at $y=0$ .

For the function $y=4\left(\frac{1}{x}\right)\text{ or }y=\frac{4}{x}$ :

This represents a transformation of the graph of $f\left(x\right)=\frac{1}{x}$ .

  $a=4$ : The graph is expanded.

  $h=0$ : There is an asymptote at $x=0$ .

  $k=0$ : There is an asymptote at $y=0$ .

Both graphs have a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$ .

The second graph is stretched by a factor of $4$ .

c.

To determine

To compare and contrast the graph of the pair of equation.

Answer to Problem 39HP

The first graph has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$ .

The second graph is translated $7\text{ units}$ up and has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=7$ .

Explanation of Solution

Given information:

The given function is:

  $y=\frac{1}{x}\text{ and }y=\frac{1}{x+5}$

For the function $y=\frac{1}{x}$ :

This represents a transformation of the graph of $f\left(x\right)=\frac{1}{x}$ .

  $a=1$ : The graph is expanded.

  $h=0$ : There is an asymptote at $x=0$ .

  $k=0$ : There is an asymptote at $y=0$ .

For the function $y=\frac{1}{x+5}$ :

This represents a transformation of the graph of $f\left(x\right)=\frac{1}{x}$ .

  $a=1$ : The graph is expanded.

  $h=-5$ : The graph is translated $5\text{ units}$ left. There is an asymptote at $x=-5$ .

  $k=0$ : There is an asymptote at $y=0$ .

The first graph has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$ .

The second graph is translated $5\text{ units}$ left and has a vertical asymptote at $x=-5$ and a horizontal asymptote at $y=7$ .

d.

To determine

To graph the function.

Explanation of Solution

Given information:

The given function is:

  $y-7=4\left(\frac{1}{x+5}\right)$

Graph:

Sketch the graph using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y=}}$ button.

Step 3: Enter the expression $y-7=4\left(\frac{1}{x+5}\right)$ which is required to graph.

Step 4: Press GRAPH button to graph the function.

The graph is obtained as: