Chapter 11.4, Problem 74SR

To determine

To find the asymptotes of the given function −

  $y=\frac{-1}{x+6}+7$

Answer to Problem 74SR

The vertical asymptote of the function $y=\frac{-1}{x+6}+7$ is at $x=-6$ and the Horizontal asymptote is at $y=7$ .

Explanation of Solution

Given: Function: $y=\frac{-1}{x+6}+7$

Formula used:

Asymptote is a straight line that can be horizontal, vertical or obliquous that goes closer and closer to a curve which is the graphic of a given function.

The vertical asymptotes will occur at those values of x for which the denominator is equal to zero.

The horizontal asymptote is the leading coefficient (the coefficient of the largest exponent) of the numerator divided by the leading coefficient of the denominator if the degrees of the numerator and denominator are the same.

Calculation:

Given: Function: $y=\frac{-1}{x+6}+7$

Solving above function, we have:

  $\begin{array}{l}y=\frac{-1+7\left(x+6\right)}{x+6}\\ y=\frac{43+7x}{x+6}\end{array}$

As per the definition of the Vertical asymptotes, we have the denominator as Zero:

  $\begin{array}{l}x+6=0\\ x=-6\end{array}$

Thus, the vertical asymptote of the function $y=\frac{-1}{x+6}+7$ is at $x=-6$ .

As per the definition of the Horizontal asymptotes, we have:

Horizontal asymptote $=$ $\frac{7}{1}$

Horizontal asymptote $=$ $7$

Thus, the Horizontal asymptote of the function $y=\frac{-1}{x+6}+7$ is at $y=7$ .

Conclusion:

Hence, the vertical asymptote of the function $y=\frac{-1}{x+6}+7$ is at $x=-6$ and the Horizontal asymptote is at $y=7$ .