Chapter 11.7, Problem 28PPE

To determine

Describe how the graph of given function is a translation of the graph of $f\left(x\right)=\frac{7}{x}$ ?

Answer to Problem 28PPE

The graph of the rational function is

 

Explanation of Solution

Given information:

The rational function $y=\frac{7}{x+3}-2$

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.

The horizontal asymptote of a ration function is given by:

• If the degree of the polynomial in numerator is less than the degree of polynomial in denominator, then the horizontal asymptote is the horizontal axis or y=0.

• If the degree of the polynomial in numerator is equal to the degree of polynomial in denominator, then the horizontal asymptote is the ratio of the leading coefficients of numerator and denominator.

• If the degree of the polynomial in numerator is greater than the degree of polynomial in denominator, then there is no horizontal asymptote. In this case there occurs a slant asymptote.

Calculation:

In subtracting a positive number on the right side $f\left(x\right)=\frac{7}{x}$ of the graph, the translates graph shifts down to the graph of $f\left(x\right)=\frac{7}{x}$

In adding a positive number in the denominator $f\left(x\right)=\frac{7}{x}$ of the graph, the translates graph shifts the left side to the graph of $f\left(x\right)=\frac{7}{x}$

The graph of $y=\frac{7}{x+3}-2$ is the graph of $f\left(x\right)=\frac{7}{x}$ shifted 2 unit down and shifts 3 unit left together. So this will shift the corresponding point to 2 unit down and 3 unit left.

The coordinate $\left(1,7\right)$ shifts to coordinate $\left(-2,5\right)$ shown in graph below.