Questions: Follow the steps for graphing a rational function to graph the function R(x) = x^2 / (x^2 + x - 20). (Type an equation. Use integers or fractions for any numbers in the equation.) B. The function has two horizontal asymptotes. The top asymptote is , and the bottom asymptote is . (Type equations. Use integers or fractions for any numbers in the equations.) C. There is no horizontal asymptote. Determine the oblique asymptote(s), if one exists. Select the correct choice below and, if necessary, fill in the answer box(es) within your choice. A. The function has one oblique asymptote, (Type an equation. Use integers or fractions for any numbers in the equation.) B. The function has two oblique asymptotes. The oblique asymptote with a negative slope is , and the oblique asymptote with a positive slope is (Type equations. Use integers or fractions for any numbers in the equations.) C. There is no oblique asymptote. Determine the points, if any, at which the graph of R intersects the horizontal or oblique asymptote, if one exists. Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The graph of R intersects the horizontal or oblique asymptote at . (Simplify your answer. Type an ordered pair. Use a comma to separate answers as needed.) B. There is no point at which the graph of R intersects the horizontal or oblique asymptote. C. There is no horizontal or oblique asymptote.

Follow the steps for graphing a rational function to graph the function R(x) = x^2 / (x^2 + x - 20). (Type an equation. Use integers or fractions for any numbers in the equation.) B. The function has two horizontal asymptotes. The top asymptote is , and the bottom asymptote is . (Type equations. Use integers or fractions for any numbers in the equations.) C. There is no horizontal asymptote. Determine the oblique asymptote(s), if one exists. Select the correct choice below and, if necessary, fill in the answer box(es) within your choice. A. The function has one oblique asymptote, (Type an equation. Use integers or fractions for any numbers in the equation.) B. The function has two oblique asymptotes. The oblique asymptote with a negative slope is , and the oblique asymptote with a positive slope is (Type equations. Use integers or fractions for any numbers in the equations.) C. There is no oblique asymptote. Determine the points, if any, at which the graph of R intersects the horizontal or oblique asymptote, if one exists. Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The graph of R intersects the horizontal or oblique asymptote at . (Simplify your answer. Type an ordered pair. Use a comma to separate answers as needed.) B. There is no point at which the graph of R intersects the horizontal or oblique asymptote. C. There is no horizontal or oblique asymptote.
Transcript text: Follow the steps for graphing a rational function to graph the function $R(x)=\frac{x^{2}}{x^{2}+x-20}$. (Type an equation. Use integers or fractions for any numbers in the equation.) B. The function has two horizontal asymptotes. The top asymptote is $\square$ , and the bottom asymptote is $\square$ . (Type equations. Use integers or fractions for any numbers in the equations.) C. There is no horizontal asymptote. Determine the oblique asymptote(s), if one exists. Select the correct choice below and, if necessary, fill in the answer box(es) within your choice. A. The function has one oblique asymptote, $\square$ (Type an equation. Use integers or fractions for any numbers in the equation.) B. The function has two oblique asymptotes. The oblique asymptote with a negative slope is $\square$ , and the oblique asymptote with a positive slope is $\square$ (Type equations. Use integers or fractions for any numbers in the equations.) C. There is no oblique asymptote. Determine the points, if any, at which the graph of R intersects the horizontal or oblique asymptote, if one exists. Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The graph of R intersects the horizontal or oblique asymptote at $\square$ . (Simplify your answer. Type an ordered pair. Use a comma to separate answers as needed.) B. There is no point at which the graph of R intersects the horizontal or oblique asymptote. C. There is no horizontal or oblique asymptote.
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Solution

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Solution Steps

Step 1: Identify Horizontal Asymptotes

The horizontal asymptote of a rational function \( R(x) = \frac{P(x)}{Q(x)} \) is determined by the degrees of the polynomials \( P(x) \) and \( Q(x) \). If the degree of \( P(x) \) is equal to the degree of \( Q(x) \), the horizontal asymptote is \( y = \frac{a}{b} \), where \( a \) and \( b \) are the leading coefficients of \( P(x) \) and \( Q(x) \), respectively.

For \( R(x) = \frac{x^2}{x^2 + x - 20} \), both the numerator and the denominator have a degree of 2. The leading coefficients are both 1.

Step 2: Calculate Horizontal Asymptote

Since the degrees are equal, the horizontal asymptote is: \[ y = \frac{1}{1} = 1 \]

Step 3: Determine Oblique Asymptotes

Oblique asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. In this case, both the numerator and the denominator have the same degree, so there is no oblique asymptote.

Final Answer

  • The function has one horizontal asymptote: \( y = 1 \).
  • There is no oblique asymptote.

{"axisType": 3, "coordSystem": {"xmin": -10, "xmax": 10, "ymin": -10, "ymax": 10}, "commands": ["y = x^2/(x^2 + x - 20)"], "latex_expressions": ["$R(x) = \\frac{x^2}{x^2 + x - 20}$"]}

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