### Understanding the End Behavior of Polynomial Functions#### IntroductionWhen studying the behavior of polynomial functions, particularly as they extend towards positive and negative infinity, it is essential to correctly identify their end behavior. This concept will aid you in anticipating the general direction and shape of the polynomial graph at extreme values of $ x $.#### End Behavior AnalysisConsider the graph of the function $ f(x) $ depicted below:![Graph](image-link) The associated graph is a polynomial function characterized by its ‘W’-shaped curve. Next, let's discuss the behavior of $ f(x) $ as $ x $ approaches positive and negative infinity.#### Visual RepresentationThe graph showcases:- $ x $-axis ranging from $ -4 $ to $ +4 $- $ y $-axis ranging from $ -4 $ to $ +4 $The function appears to continue increasing indefinitely as $ x $ moves away from the origin in both positive and negative directions.#### Analyzing the Options- **Option 1:** $ x \rightarrow \infty, f(x) \rightarrow \infty; x \rightarrow -\infty, f(x) \rightarrow -\infty $- **Option 2:** $ x \rightarrow \infty, f(x) \rightarrow -\infty; x \rightarrow -\infty, f(x) \rightarrow \infty $- **Option 3:** $ x \rightarrow \infty, f(x) \rightarrow \infty; x \rightarrow -\infty, f(x) \rightarrow \infty $- **Option 4:** $ x \rightarrow \infty, f(x) \rightarrow -\infty; x \rightarrow -\infty, f(x) \rightarrow -\infty $#### Correct IdentificationGiven the graph:- As $ x \rightarrow \infty $, $ f(x) \rightarrow \infty $- As $ x \rightarrow -\infty $, $ f(x) \rightarrow \infty $Thus, the correct end behavior is represented by:**Option 3:** $ x \rightarrow \infty, f(x) \rightarrow \infty; \quad x \rightarrow -\infty, f(x) \rightarrow \infty $

We are asked to find the end behaviour of the graph.

Answered: Correctly identify the end behavior of…

Correctly identify the end behavior of the graph of , S(1) → 00; I -00, / (z) →0 . S (1) -00; I-00, f (r) →-0 /(1) X -x, ()→0

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 the End Behavior of Polynomial Functions

#### Introduction
When studying the behavior of polynomial functions, particularly as they extend towards positive and negative infinity, it is essential to correctly identify their end behavior. This concept will aid you in anticipating the general direction and shape of the polynomial graph at extreme values of $ x $.

#### End Behavior Analysis
Consider the graph of the function $ f(x) $ depicted below:

![Graph](image-link) 

The associated graph is a polynomial function characterized by its ‘W’-shaped curve. Next, let's discuss the behavior of $ f(x) $ as $ x $ approaches positive and negative infinity.

#### Visual Representation
The graph showcases:
- $ x $-axis ranging from $ -4 $ to $ +4 $
- $ y $-axis ranging from $ -4 $ to $ +4 $

The function appears to continue increasing indefinitely as $ x $ moves away from the origin in both positive and negative directions.

#### Analyzing the Options
- **Option 1:** $ x \rightarrow \infty, f(x) \rightarrow \infty; x \rightarrow -\infty, f(x) \rightarrow -\infty $
- **Option 2:** $ x \rightarrow \infty, f(x) \rightarrow -\infty; x \rightarrow -\infty, f(x) \rightarrow \infty $
- **Option 3:** $ x \rightarrow \infty, f(x) \rightarrow \infty; x \rightarrow -\infty, f(x) \rightarrow \infty $
- **Option 4:** $ x \rightarrow \infty, f(x) \rightarrow -\infty; x \rightarrow -\infty, f(x) \rightarrow -\infty $

#### Correct Identification
Given the graph:

- As $ x \rightarrow \infty $, $ f(x) \rightarrow \infty $
- As $ x \rightarrow -\infty $, $ f(x) \rightarrow \infty $

Thus, the correct end behavior is represented by:
**Option 3:** $ x \rightarrow \infty, f(x) \rightarrow \infty; \quad x \rightarrow -\infty, f(x) \rightarrow \infty $

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