Chapter 11.7, Problem 3P

a.

To determine

The asymptotes of the graph of $y=\frac{2}{x+3}-4$ .

Answer to Problem 3P

  $x-3\ne 3-x$

Explanation of Solution

Given:

The rational function, $y=\frac{2}{x+3}-4$

Concept used:

Horizontal asymptote of a function is the horizontal line that the function approaches to as the value of x increase and decrease indefinitely.

The horizontal asymptote of a rational function can be found using the rules:

• 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 or oblique asymptote.

Calculation:

In order to find the asymptotes of the given rational function, $y=\frac{2}{x+3}-4$ , observe that the vertical asymptotes occur where the function is undefined, that is the denominator is zero. So in order to find the vertical asymptotes, set the denominator equal to zero and solve for x values.

  $\begin{array}{c}x+3=0\\ x=-3\end{array}$

So, the vertical asymptote occur at $x=-3$ .

Now to find the horizontal asymptote, observe that the degree of the numerator is les than the degree of the denominator, so the horizontal asymptote should occur at $y=0$ . But there is a vertical shift of 4 units downward, so the horizontal asymptote will occur at $y=-4$ .

The graph of the function is shown below,

  

b.

To determine

Is it possible for two different rational functions to have the same vertical and horizontal asymptotes.

Answer to Problem 3P

  $\text{Yes}$

Explanation of Solution

Concept Used:

Horizontal asymptote of a function is the horizontal line that the function approaches to as the value of x increase and decrease indefinitely.

The horizontal asymptote of a rational function can be found using the rules:

• 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 or oblique asymptote.

Calculation:

Yes, two different rational functions can have the same vertical and horizontal asymptotes.

For example, the rational function $y=\frac{5}{2x+6}-4$ have the same vertical and horizontal asymptotes as of the rational function $y=\frac{2}{x+3}-4$ .