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Determine the intervals over which the given rational function graph is increasing and decreasing. -16 4 8

Determine the intervals over which the given rational function graph is increasing and decreasing. -16 4 8

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.5: Graphs Of Functions
Problem 35E
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### Understanding the Behavior of Rational Function Graph **Objective:** Determine the intervals over which the given rational function graph is increasing and decreasing. **Graph Description:** The image features a graph of a rational function. Specifically, the graph shows a hyperbola. Below is a detailed explanation of the graph: - **Axes:** - The x-axis ranges from approximately -8 to 8. - The y-axis ranges from approximately -16 to 16. - **Graph Characteristics:** - The graph has two branches, one in the first quadrant and one in the third quadrant. - The branches approach the vertical line \(x = 0\) but never touch it, indicating a vertical asymptote. - Similarly, the branches approach the horizontal line \(y = 0\) but never touch it, indicating a horizontal asymptote. **Intervals of Increase and Decrease:** - The function is **decreasing** for \(x < 0\). - As x moves from negative infinity towards 0 (but not reaching 0), the y-values decrease from 0 to negative infinity. - The function is **increasing** for \(x > 0\). - As x moves from 0 to positive infinity, the y-values increase from negative infinity to 0 again. **Note:** - The function does not cross the x- or y-axes. - The points where the function approaches but does not touch the axes represent the asymptotes. Analyzing the intervals of increase and decrease helps in understanding the behavior of the rational function and aids in sketching its graph.
Transcribed Image Text:### Understanding the Behavior of Rational Function Graph **Objective:** Determine the intervals over which the given rational function graph is increasing and decreasing. **Graph Description:** The image features a graph of a rational function. Specifically, the graph shows a hyperbola. Below is a detailed explanation of the graph: - **Axes:** - The x-axis ranges from approximately -8 to 8. - The y-axis ranges from approximately -16 to 16. - **Graph Characteristics:** - The graph has two branches, one in the first quadrant and one in the third quadrant. - The branches approach the vertical line \(x = 0\) but never touch it, indicating a vertical asymptote. - Similarly, the branches approach the horizontal line \(y = 0\) but never touch it, indicating a horizontal asymptote. **Intervals of Increase and Decrease:** - The function is **decreasing** for \(x < 0\). - As x moves from negative infinity towards 0 (but not reaching 0), the y-values decrease from 0 to negative infinity. - The function is **increasing** for \(x > 0\). - As x moves from 0 to positive infinity, the y-values increase from negative infinity to 0 again. **Note:** - The function does not cross the x- or y-axes. - The points where the function approaches but does not touch the axes represent the asymptotes. Analyzing the intervals of increase and decrease helps in understanding the behavior of the rational function and aids in sketching its graph.
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