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Depic a function for the graph just in case you can’t see the numbers are  -3, (0,0), 3

Depic a function for the graph just in case you can’t see the numbers are  -3, (0,0), 3

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Depic a function for the graph

just in case you can’t see the numbers are 

-3, (0,0), 3

### Understanding Asymptotes and Graphing Rational Functions #### Explanation of the Graph The graph depicted is of a rational function, illustrating its behavior around vertical and horizontal asymptotes. #### Key Features of the Graph: 1. **Axes and Scale:** - The x-axis (horizontal axis) and y-axis (vertical axis) are clearly marked. - Significant points are labeled along the axes: on the x-axis at -3, 0, and 3; and on the y-axis at 1. 2. **Vertical Asymptotes:** - The graph has vertical dashed lines at \( x = -1 \) and \( x = 1 \). These lines represent the vertical asymptotes of the function. - As \( x \) approaches -1 and 1 from both the left and right, the function's value increases or decreases without bound, indicating it approaches infinity or negative infinity. 3. **Function Behavior and Horizontal Asymptote:** - For large values of \( |x| \) (both positive and negative), the function approaches a value near the x-axis, indicating that the x-axis (\( y = 0 \)) is a horizontal asymptote. - The function appears to have significant changes in its direction as it approaches the vertical asymptotes, particularly near \( x = -1 \) and \( x = 1 \). #### Interpretation: 1. **Behavior Near Vertical Asymptotes:** - As the function approaches \( x = -1\) from the left, it decreases towards negative infinity. - As it passes \( x = -1\) from the right, it increases towards positive infinity. - Similarly, as it approaches \( x = 1\) from the left, it increases towards positive infinity, and from the right, it decreases towards negative infinity. 2. **Horizontal Asymptote:** - The graph indicates that as \( x \) moves away from the origin (both positive and negative directions), the function value approaches zero. This means the function has a horizontal asymptote at \( y = 0 \). Understanding the characteristics and behaviors illustrated by this graph is crucial for comprehending the nature of rational functions and their asymptotes. This foundational concept is fundamental in further exploring more complex mathematical functions and their graphical representations.
Transcribed Image Text:### Understanding Asymptotes and Graphing Rational Functions #### Explanation of the Graph The graph depicted is of a rational function, illustrating its behavior around vertical and horizontal asymptotes. #### Key Features of the Graph: 1. **Axes and Scale:** - The x-axis (horizontal axis) and y-axis (vertical axis) are clearly marked. - Significant points are labeled along the axes: on the x-axis at -3, 0, and 3; and on the y-axis at 1. 2. **Vertical Asymptotes:** - The graph has vertical dashed lines at \( x = -1 \) and \( x = 1 \). These lines represent the vertical asymptotes of the function. - As \( x \) approaches -1 and 1 from both the left and right, the function's value increases or decreases without bound, indicating it approaches infinity or negative infinity. 3. **Function Behavior and Horizontal Asymptote:** - For large values of \( |x| \) (both positive and negative), the function approaches a value near the x-axis, indicating that the x-axis (\( y = 0 \)) is a horizontal asymptote. - The function appears to have significant changes in its direction as it approaches the vertical asymptotes, particularly near \( x = -1 \) and \( x = 1 \). #### Interpretation: 1. **Behavior Near Vertical Asymptotes:** - As the function approaches \( x = -1\) from the left, it decreases towards negative infinity. - As it passes \( x = -1\) from the right, it increases towards positive infinity. - Similarly, as it approaches \( x = 1\) from the left, it increases towards positive infinity, and from the right, it decreases towards negative infinity. 2. **Horizontal Asymptote:** - The graph indicates that as \( x \) moves away from the origin (both positive and negative directions), the function value approaches zero. This means the function has a horizontal asymptote at \( y = 0 \). Understanding the characteristics and behaviors illustrated by this graph is crucial for comprehending the nature of rational functions and their asymptotes. This foundational concept is fundamental in further exploring more complex mathematical functions and their graphical representations.
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