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Select all properties that are true of the given rational function graph. 20 -20 2 There are no maximum or minimum values. The graph has a horizontal asymptote at y = -1. The graph has a vertical asymptote at x = 2. The graph has odd symmetry.

Select all properties that are true of the given rational function graph. 20 -20 2 There are no maximum or minimum values. The graph has a horizontal asymptote at y = -1. The graph has a vertical asymptote at x = 2. The graph has odd symmetry.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.5: Rational Functions
Problem 54E
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### Properties of the Rational Function Graph **Graph Description:** The graph represents a rational function that consists of two distinct curves. One curve exists in the upper right and upper left quadrants, and another exists in the lower left and lower right quadrants. The horizontal and vertical axes intersect at the origin (0,0). **Key Features:** 1. **Horizontal Asymptote:** Present at \( y = 0 \). This indicates the function approaches zero as \(x\) moves towards positive or negative infinity. 2. **Vertical Asymptote:** Present at \( x = 0 \). This indicates that the function approaches negative or positive infinity as \(x\) approaches zero. 3. **Symmetry:** The graph appears to be symmetric with respect to the origin, indicating odd symmetry. **Properties to Evaluate:** - [ ] There are no maximum or minimum values. - [ ] The graph has a horizontal asymptote at \( y = -1 \). - [ ] The graph has a vertical asymptote at \( x = 2 \). - [x] The graph has odd symmetry. ### Explanation of Graph Behavior **Horizontal Asymptote:** - The function approaches \( y = 0 \) as the values of \( x \) go to infinity or negative infinity. **Vertical Asymptote:** - The function moves towards positive or negative infinity as \( x \) nears zero. **Symmetry:** - The function is symmetric with respect to the origin. This type of symmetry is called odd symmetry, indicating \( f(-x) = -f(x) \). Therefore, the correct properties of the graph are selected accordingly.
Transcribed Image Text:### Properties of the Rational Function Graph **Graph Description:** The graph represents a rational function that consists of two distinct curves. One curve exists in the upper right and upper left quadrants, and another exists in the lower left and lower right quadrants. The horizontal and vertical axes intersect at the origin (0,0). **Key Features:** 1. **Horizontal Asymptote:** Present at \( y = 0 \). This indicates the function approaches zero as \(x\) moves towards positive or negative infinity. 2. **Vertical Asymptote:** Present at \( x = 0 \). This indicates that the function approaches negative or positive infinity as \(x\) approaches zero. 3. **Symmetry:** The graph appears to be symmetric with respect to the origin, indicating odd symmetry. **Properties to Evaluate:** - [ ] There are no maximum or minimum values. - [ ] The graph has a horizontal asymptote at \( y = -1 \). - [ ] The graph has a vertical asymptote at \( x = 2 \). - [x] The graph has odd symmetry. ### Explanation of Graph Behavior **Horizontal Asymptote:** - The function approaches \( y = 0 \) as the values of \( x \) go to infinity or negative infinity. **Vertical Asymptote:** - The function moves towards positive or negative infinity as \( x \) nears zero. **Symmetry:** - The function is symmetric with respect to the origin. This type of symmetry is called odd symmetry, indicating \( f(-x) = -f(x) \). Therefore, the correct properties of the graph are selected accordingly.
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