Chapter 11.7, Problem 22PPE

To determine

Identify the asymptotes of the graph of given function. Then graph the function.

Answer to Problem 22PPE

Vertical asymptote $x=3$

Horizontal asymptote $y=-5$

Explanation of Solution

Given function:

  $f\left(x\right)=\frac{-1}{x-3}-5$

Concept used:

• The vertical asymptotes of a function occur at the points where the function is undefined. Vertical asymptote is the vertical line that the graph of rational function follows till infinity but never intersects.

The horizontal asymptote of a rational 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.
• We know that the vertical asymptotes occur at the values of x for which the denominator is zero.

Calculation:

The given Rational function is

  $\begin{array}{c}f\left(x\right)=\frac{-1}{x-3}-5\\ x-3=0\\ x=3\end{array}$

Vertical asymptote $x=3$

Horizontal asymptote $y=-5$

X	Y
1	-4.5
-1	-4.75
2	-4
4	-6
5	-5.5
7	-5.25

• Plot the points from the table.
• Join the points in Quadrant 3 and 4 to make a curve.
• Join the points in Quadrant 4 to make a curve.