### Understanding Rational Functions#### Learning ObjectiveBy the end of this lesson, you will be able to write an equation for the rational function graphed below.#### Graph ExplanationThe graph presented shows a rational function, illustrated on a Cartesian plane with the x-axis ranging from -7 to 7 and the y-axis ranging from -5 to 5.##### Features of the Graph:1. **Vertical Asymptotes**: - The graph has three vertical asymptotes, indicated by red dashed lines, located at: - $ x = -3 $ - $ x = 1 $ - $ x = 4 $ These lines represent the values of x where the function is undefined and the graph tends to infinity.2. **Horizontal Behavior**: - As $ x $ approaches $ \pm \infty $, the graph approaches the x-axis, suggesting a horizontal asymptote at $ y = 0 $.3. **Curve Behavior**: - The curve exhibits typical behavior seen with rational functions. The segments between asymptotes show the graph approaching infinity near the vertical lines and leveling off away from them.##### Graph:(Graph of a rational function with vertical asymptotes at $ x = -3 $, $ x = 1 $, and $ x = 4 $, and a horizontal asymptote at $ y = 0 $). #### TaskUsing the above information and observing the behavior of the rational function in the graph, please write the equation of the function in the form:\[ y = \frac{P(x)}{Q(x)} \]where $ P(x) $ and $ Q(x) $ are polynomials.#### Equation Input Box**y =** [Insert your equation here]#### Additional Resources- For more information on rational functions and their behaviors, [click here](#).- To practice more problems on this topic, please visit our [practice section](#).--- Submit your answer in the input box provided above and check your work against the solution hints available on the next page.---**Keywords:** Rational Function, Vertical Asymptotes, Polynomial, Horizontal Asymptote

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Answered: Write an equation for the function…

Math

Calculus

Write an equation for the function graphed below 5+ 4 2 1 -7 -6 -5 -4 -3 -2 -1 1 2 -1 3 4 5 6 7 -2 -3 + -4 -5+

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Concept explainers

Vertices Of An Ellipse

In the study of the conic section of the geometry, An ellipse is a plane curve surrounding two focal points, or foci where the set of all locus of points in the plane has their sum of the distances to the two foci is constant. On the ellipse curve, it is the intersection point of the main axis. In an ellipse graph, the center of the axis is the midpoint of the line segment connecting the foci. The major axis is the line section that runs through the ellipse's foci, while the minor axis runs through the middle and is perpendicular to the major axis.. The ellipse has two endpoints on the major axis which are known as the vertices (in plural), vertex in the singular. The equation of the major axis is 2a, the equation of the minor is 2b and the distance between the major axis and minor axis is 2c. The semi-major axis has a length of a and the semi-minor axis has a length of b. The ellipse's vertices are determined by the elliptical orbit's equation and graph.

Question

### Understanding Rational Functions

#### Learning Objective
By the end of this lesson, you will be able to write an equation for the rational function graphed below.

#### Graph Explanation
The graph presented shows a rational function, illustrated on a Cartesian plane with the x-axis ranging from -7 to 7 and the y-axis ranging from -5 to 5.

##### Features of the Graph:
1. **Vertical Asymptotes**:
- The graph has three vertical asymptotes, indicated by red dashed lines, located at:
- $ x = -3 $
- $ x = 1 $
- $ x = 4 $
These lines represent the values of x where the function is undefined and the graph tends to infinity.

2. **Horizontal Behavior**:
- As $ x $ approaches $ \pm \infty $, the graph approaches the x-axis, suggesting a horizontal asymptote at $ y = 0 $.

3. **Curve Behavior**:
- The curve exhibits typical behavior seen with rational functions. The segments between asymptotes show the graph approaching infinity near the vertical lines and leveling off away from them.

##### Graph:
(Graph of a rational function with vertical asymptotes at $ x = -3 $, $ x = 1 $, and $ x = 4 $, and a horizontal asymptote at $ y = 0 $).

#### Task
Using the above information and observing the behavior of the rational function in the graph, please write the equation of the function in the form:
\[ y = \frac{P(x)}{Q(x)} \]
where $ P(x) $ and $ Q(x) $ are polynomials.

#### Equation Input Box
**y =** [Insert your equation here]

#### Additional Resources
- For more information on rational functions and their behaviors, [click here](#).
- To practice more problems on this topic, please visit our [practice section](#).

---

Submit your answer in the input box provided above and check your work against the solution hints available on the next page.

---
**Keywords:** Rational Function, Vertical Asymptotes, Polynomial, Horizontal Asymptote

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