Questions: Follow the steps for graphing a rational function to graph the function R(x)=(x^2+x-12)/(x^2-4). A. The graph has x-intercept(s) -4,3 and y-intercept 3 . (Type integers or simplified fractions. Use a comma to separate answers as needed. Type each answer only once.) B. The graph has x-intercept(s) and no y-intercept. (Type an integer or a simplified fraction. Use a comma to separate answers as needed. Type each answer only once.) C. The graph has y-intercept and no x-intercept. (Type an integer or a simplified fraction.) D. The graph has neither x-intercepts nor y-intercepts. Determine the behavior of the graph of R at any x-intercepts. Select the correct choice and, if necessary, fill in the answer box(es) to complete your choice. A. The graph will cross the x-axis at x= and touch the x-axis at x= . (Type integers or simplified fractions. B. The graph will touch the x-axis at x= . (Type an integer or a simplified fraction. Use a comma to separate answers as needed. Type each answer only once.) C. The graph will cross the x-axis at x= . (Type an integer or a simplified fraction. Use a comma to separate answers as needed. Type each answer only once.) D. There is no x-intercept.

Follow the steps for graphing a rational function to graph the function R(x)=(x^2+x-12)/(x^2-4). A. The graph has x-intercept(s) -4,3 and y-intercept 3 . (Type integers or simplified fractions. Use a comma to separate answers as needed. Type each answer only once.) B. The graph has x-intercept(s) and no y-intercept. (Type an integer or a simplified fraction. Use a comma to separate answers as needed. Type each answer only once.) C. The graph has y-intercept and no x-intercept. (Type an integer or a simplified fraction.) D. The graph has neither x-intercepts nor y-intercepts. Determine the behavior of the graph of R at any x-intercepts. Select the correct choice and, if necessary, fill in the answer box(es) to complete your choice. A. The graph will cross the x-axis at x= and touch the x-axis at x= . (Type integers or simplified fractions. B. The graph will touch the x-axis at x= . (Type an integer or a simplified fraction. Use a comma to separate answers as needed. Type each answer only once.) C. The graph will cross the x-axis at x= . (Type an integer or a simplified fraction. Use a comma to separate answers as needed. Type each answer only once.) D. There is no x-intercept.
Transcript text: Follow the steps for graphing a rational function to graph the function $R(x)=\frac{x^{2}+x-12}{x^{2}-4}$. A. The graph has $x$-intercept(s) $-4,3$ and $y$-intercept 3 . (Type integers or simplified fractions. Use a comma to separate answers as needed. Type each answer only once.) B. The graph has $x$-intercept(s) $\square$ and no $y$-intercept. (Type an integer or a simplified fraction. Use a comma to separate answers as needed. Type each answer only once.) C. The graph has $y$-intercept $\square$ and no x-intercept. (Type an integer or a simplified fraction.) D. The graph has neither $x$-intercepts nor $y$-intercepts. Determine the behavior of the graph of $R$ at any $x$-intercepts. Select the correct choice and, if necessary, fill in the answer box(es) to complete your choice. A. The graph will cross the $x$-axis at $x=$ $\square$ and touch the $x$-axis at $x=$ $\square$ . (Type integers or simplified fractions. B. The graph will touch the $x$-axis at $x=$ $\square$ . (Type an integer or a simplified fraction. Use a comma to separate answers as needed. Type each answer only once.) C. The graph will cross the $x$-axis at $x=$ $\square$ . (Type an integer or a simplified fraction. Use a comma to separate answers as needed. Type each answer only once.) D. There is no x-intercept.
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Solution

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Solution Steps

Step 1: Find the $y$-intercept

The $y$-intercept is found by evaluating $R(0) = \frac{-12}{-4} = 3$.

Step 2: Find the $x$-intercepts

The $x$-intercepts are found by solving $P(x) = 0$. The solutions are $x = [-4, 3]$.

Step 3: Determine vertical asymptotes

Vertical asymptotes occur where $Q(x) = 0$. The solutions are $x = [-2, 2]$.

Step 4: Analyze the behavior around vertical asymptotes

This step involves determining the sign of $R(x)$ just to the left and right of each vertical asymptote to understand how the graph approaches the asymptote. This analysis is qualitative and requires further investigation.

Step 5: Determine horizontal or oblique asymptotes

If the degree of $P(x)$ is less than, equal to, or greater than the degree of $Q(x)$, the horizontal asymptote can be determined. In this case, the horizontal asymptote is $y = 1$.

Final Answer:

The graph of the rational function involves identifying the $y$-intercept, $x$-intercepts, vertical asymptotes, and horizontal or oblique asymptotes. The behavior around the asymptotes and intercepts is crucial for sketching the graph accurately.

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