Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Rational Function, write the function for the graph given below. All intercepts are integers. Please show work so I can follow! Thank you.

The image displays a graph of a rational function illustrating its behavior across different regions of the x-axis. The graph is drawn in purple on a coordinate plane ranging from -13 to 13 on the x-axis and -30 to 30 on the y-axis. ### Key Features of the Graph: 1. **Vertical Asymptotes:** - There are vertical dashed lines at x = -3 and x = 3, indicating vertical asymptotes. The function approaches infinity near these lines. 2. **Behavior Around Asymptotes:** - As x approaches -3 from the left, the function increases sharply towards positive infinity. - As x approaches -3 from the right, the function decreases sharply towards negative infinity. - As x approaches 3, the behavior is similar, with the function approaching positive infinity from the left and negative infinity from the right. 3. **Intersection with the Y-axis:** - The graph intersects the y-axis at approximately the point (0, 5). 4. **Curve Description:** - The graph displays three distinct sections due to the vertical asymptotes. - For x < -3, the graph rises steeply. - Between -3 and 3, the graph appears as an inverted peak, pointing downwards. - For x > 3, the graph rises again. This graph provides insight into how rational functions behave near their asymptotes and how they are defined in segments between these critical points. Understanding these features is essential for analyzing and graphing rational functions effectively.
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Transcribed Image Text:The image displays a graph of a rational function illustrating its behavior across different regions of the x-axis. The graph is drawn in purple on a coordinate plane ranging from -13 to 13 on the x-axis and -30 to 30 on the y-axis. ### Key Features of the Graph: 1. **Vertical Asymptotes:** - There are vertical dashed lines at x = -3 and x = 3, indicating vertical asymptotes. The function approaches infinity near these lines. 2. **Behavior Around Asymptotes:** - As x approaches -3 from the left, the function increases sharply towards positive infinity. - As x approaches -3 from the right, the function decreases sharply towards negative infinity. - As x approaches 3, the behavior is similar, with the function approaching positive infinity from the left and negative infinity from the right. 3. **Intersection with the Y-axis:** - The graph intersects the y-axis at approximately the point (0, 5). 4. **Curve Description:** - The graph displays three distinct sections due to the vertical asymptotes. - For x < -3, the graph rises steeply. - Between -3 and 3, the graph appears as an inverted peak, pointing downwards. - For x > 3, the graph rises again. This graph provides insight into how rational functions behave near their asymptotes and how they are defined in segments between these critical points. Understanding these features is essential for analyzing and graphing rational functions effectively.
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