Chapter 11.7, Problem 10CB

a.

To determine

Graph $y=\frac{1}{x}$ , $y=\frac{1}{x}-4$ and $y=\frac{1}{x}+3$

Answer to Problem 10CB

  

Explanation of Solution

Given function:

  $y=\frac{1}{x}$ , $y=\frac{1}{x-4}$ and $y=\frac{1}{x+3}$

Concept used:

Transformation of functions: For any positive integer a , we have

• $y=f\left(x+a\right)$ is the graph of $y=f\left(x\right)$ shifted a units left.
• $y=f\left(x-a\right)$ is the graph of $y=f\left(x\right)$ shifted a units right.
• $y=f\left(x\right)+a$ is the graph of $y=f\left(x\right)$ shifted a units up.
• $y=f\left(x\right)-a$ is the graph of $y=f\left(x\right)$ shifted a units down.
• $y=af\left(x\right)$ is the graph of $y=f\left(x\right)$ stretches a units in y direction if $a>1$ and compresses a units in y direction if $0<a<1$
• $y=f\left(ax\right)$ is the graph of $y=f\left(x\right)$ stretches a units in x direction if $0<a<1$ and compresses a units in x direction if $a>1$ .
• $y=-f\left(x\right)$ is the graph of $y=f\left(x\right)$ reflected about x axis.
• $y=f\left(-x\right)$ is the graph of $y=f\left(x\right)$ reflected about y axis.

Calculation:

Graph of the given functions is below

  

b.

To determine

Make a Conjecture: How does adding or subtracting a positive number on the right side of $y=\frac{1}{x}$ translate the graph?

Explanation of Solution

Given function:

  $y=\frac{1}{x}$

Concept used:

Transformation of functions: For any positive integer a , we have

• $y=f\left(x+a\right)$ is the graph of $y=f\left(x\right)$ shifted a units left.
• $y=f\left(x-a\right)$ is the graph of $y=f\left(x\right)$ shifted a units right.
• $y=f\left(x\right)+a$ is the graph of $y=f\left(x\right)$ shifted a units up.
• $y=f\left(x\right)-a$ is the graph of $y=f\left(x\right)$ shifted a units down.
• $y=af\left(x\right)$ is the graph of $y=f\left(x\right)$ stretches a units in y direction if $a>1$ and compresses a units in y direction if $0<a<1$
• $y=f\left(ax\right)$ is the graph of $y=f\left(x\right)$ stretches a units in x direction if $0<a<1$ and compresses a units in x direction if $a>1$ .
• $y=-f\left(x\right)$ is the graph of $y=f\left(x\right)$ reflected about x axis.
• $y=f\left(-x\right)$ is the graph of $y=f\left(x\right)$ reflected about y axis.

Calculation:

In adding a positive number on the right side $y=\frac{1}{x}$ of the graph, the translates graph shifts up to the graph of $y=\frac{1}{x}$

If we add 3 on the right side of $y=\frac{1}{x}$ then the graph of $y=\frac{1}{x}+3$ shifts 3 units up of the graph $y=\frac{1}{x}$

In subtracting a positive number on the right side $y=\frac{1}{x}$ of the graph, the translates graph shifts down to the graph of $y=\frac{1}{x}$

If we subtract 4 on the right side of $y=\frac{1}{x}$ then the graph of $y=\frac{1}{x}-4$ shifts 4 units down of the graph $y=\frac{1}{x}$