Depic a function for the graphjust in case you can’t see the numbers are -3, (0,0), 3 ### Understanding Asymptotes and Graphing Rational Functions#### Explanation of the GraphThe 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.

Answered: Depic a function for the graph just in…

Math

Algebra

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)

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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Question

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.

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