Sketch the graph of the following function. Indicate where the function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur.
\[
f(x)=\frac{-3}{x-7}
\]
Determine the horizontal asymptote(s). Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
A. The function has two horizontal asymptotes. The top asymptote is and the bottom asymptote is (Type equations.)
B. The function has one horizontal asymptote, $y=0$. (Type an equation.)
c. The function has no horizontal asymptotes.
Determine the slant asymptote(s). Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The slant asymptote(s) is(are)
(Type an equation. Use a comma to separate answers as needed.)
B. The function has no slant asymptotes.
On what interval(s) is $\mathrm{f}$ concave up and on what interval(s) is $\mathrm{f}$ concave down? Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
A. The function is concave up on and concave down on (Simplify your answers. Type your answers in interval notation. Use a comma tir separate answers as needed.)
B. The function is concave up on and is never concave down. (Simplify your answer. Type your answer in interval notation. Use a comma to separate answers as needed.)
C. The function is concave down on $\square$ and is never concave up. (Simplify your answer. Type your answer in interval notation. Use a comma to separate answers as needed.)
Answer
Finally, we find any intercepts. The function intercepts the y-axis at \(y=-\frac{3}{7}\) and does not intercept the x-axis.
Step 1 ：First, we find the vertical asymptote. The vertical asymptote occurs at the value of \(x\) where the denominator is 0. In this case, the denominator is \(x-7\), so the vertical asymptote is at \(x=7\).
Step 2 ：Next, we determine the horizontal asymptote. Since the degree of \(x\) in the numerator is 0 and the degree of \(x\) in the denominator is 1, the function approaches the horizontal asymptote \(y=0\) as \(x\) approaches infinity or negative infinity.
Step 3 ：Then, we find the slant asymptote. Since the degree of the denominator is greater than the degree of the numerator, there is no slant asymptote.
Step 4 ：Next, we determine where the function is increasing or decreasing. Since the function is a rational function with a negative coefficient, it is decreasing for all \(x\) not equal to 7.
Step 5 ：We then find any relative extrema. Since the function is always decreasing, there are no relative extrema.
Step 6 ：We then determine where the function is concave up or concave down. Since the second derivative of the function is 0 for all \(x\), the function is neither concave up nor concave down.
Step 7 ：We then find any points of inflection. Since the function is neither concave up nor concave down, there are no points of inflection.
Step 8 ：Finally, we find any intercepts. The function intercepts the y-axis at \(y=-\frac{3}{7}\) and does not intercept the x-axis.
