Questions: Follow the nine-step graphing strategy to sketch the graph of the rational function. f(x) = 3 / (x^2 - 16) B. The function has no x-intercept. 6. Find the equations of any vertical asymptotes. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The vertical asymptote(s) is(are) x = -4, x = 4. (Type an equation. Use a comma to separate answers as needed.) B. The function has no vertical asymptotes. 7. Determine whether the graph has a horizontal asymptote or a slant asymptote. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The horizontal asymptote is . (Type an equation) B. The function has no horizontal asymptote. It has a slant asymptote of . (Type an equation. Type your answer in slope-intercept form.) C. The function has no horizontal asymptote and has no slant asymptote.

Follow the nine-step graphing strategy to sketch the graph of the rational function.
f(x) = 3 / (x^2 - 16)
B. The function has no x-intercept.
6. Find the equations of any vertical asymptotes. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The vertical asymptote(s) is(are) x = -4, x = 4. (Type an equation. Use a comma to separate answers as needed.)
B. The function has no vertical asymptotes.
7. Determine whether the graph has a horizontal asymptote or a slant asymptote. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The horizontal asymptote is . (Type an equation)
B. The function has no horizontal asymptote. It has a slant asymptote of . (Type an equation. Type your answer in slope-intercept form.)
C. The function has no horizontal asymptote and has no slant asymptote.

Transcript text: Follow the nine-step graphing strategy to sketch the graph of the rational function. \[ f(x)=\frac{3}{x^{2}-16} \] B. The function has no $x$-intercept. 6. Find the equations of any vertical asymptotes. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The vertical asymptote(s) is(are) $x=-4, x=4$. (Type an equation. Use a comma to separate answers as needed.) B. The function has no vertical asymptotes. 7. Determine whether the graph has a horizontal asymptote or a slant asymptote. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The horizontal asymptote is $\square$ . (Type an equation) B. The function has no horizontal asymptote. It has a slant asymptote of $\square$ ]. (Type an equation. Type your answer in slope-intercept form.) C. The function has no horizontal asymptote and has no slant asymptote.

Solution

Solution Steps

Step 1: Identify the vertical asymptotes

To find the vertical asymptotes, we need to determine where the denominator of the function $ f(x) = \frac{3}{x^2 - 16} $ is equal to zero.

\[ x^2 - 16 = 0 \]

Solving for $ x $:

\[ x^2 = 16 \implies x = \pm 4 \]

Thus, the vertical asymptotes are at $ x = -4 $ and $ x = 4 $.

Step 2: Identify the horizontal asymptote

To find the horizontal asymptote of the function $ f(x) = \frac{3}{x^2 - 16} $, we compare the degrees of the numerator and the denominator. The degree of the numerator is 0 (constant term 3), and the degree of the denominator is 2.

Since the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is:

\[ y = 0 \]

Final Answer

The vertical asymptotes are $ x = -4 $ and $ x = 4 $.
The horizontal asymptote is $ y = 0 $.

{"axisType": 3, "coordSystem": {"xmin": -10, "xmax": 10, "ymin": -1, "ymax": 1}, "commands": ["y = 3/(x**2 - 16)"], "latex_expressions": ["$y = \\frac{3}{x^2 - 16}$"]}

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