/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 19 Apply the Leading Coefficient Te... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 19

Apply the Leading Coefficient Test, describe the right-hand and left-hand behavior of the graph of the polynomial function. $$ f(x)=\frac{1}{5} x^{3}+4 x $$

Short Answer

Expert verified
The graph of the function \( f(x) = \frac{1}{5}x^{3}+4x \) falls to the left and rises to the right.

Step by step solution

01

Identify the leading term

The leading term in a polynomial function is the term with the highest exponent. In our function \( f(x) = \frac{1}{5}x^{3}+4x \), the leading term is \( \frac{1}{5}x^{3} \).
02

Apply the Leading Coefficient Test

The Leading Coefficient Test states that if the leading coefficient of the polynomial function is positive and the degree of the function is odd, as in our case, then the function rises to the right and falls to the left.
03

Describe the behavior of the graph

Based on the Leading Coefficient Test, we can conclude that the graph of the function \( f(x) = \frac{1}{5}x^{3}+4x \) falls to the left and rises to the right.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Function
A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Each term in a polynomial consists of a coefficient and a variable raised to a power, known as the exponent. For example, in the function \( f(x) = \frac{1}{5}x^{3} + 4x \), we see two terms: \( \frac{1}{5}x^{3} \) and \( 4x \).
Each term is a building block of the polynomial, and they can be combined in different ways to form an infinite number of polynomials.
  • The term with the highest exponent is often crucial in determining the behavior of the polynomial function.
  • The degree of the polynomial is defined by the highest exponent present.
Understanding these elements helps in analyzing the graph and solving the function.
Right-Hand Behavior
Right-hand behavior refers to how the graph of a function behaves as the input variable, often \( x \), increases towards positive infinity. In simpler terms, it's what happens on the far-right end of the graph.
For the polynomial function \( f(x) = \frac{1}{5}x^{3} + 4x \), we apply the Leading Coefficient Test to determine this behavior. This test reveals what happens based on the degree of the polynomial (odd or even) and the sign of the leading coefficient (positive or negative).
  • If both degree and leading coefficient are odd and positive respectively, the function's graph "rises to the right."
  • That means, as \( x \) gets larger, the value of \( f(x) \) also goes up.
Recognizing right-hand behavior is essential for predicting and graphing polynomial functions.
Left-Hand Behavior
Left-hand behavior explores what happens as \( x \) approaches negative infinity, or the far-left end of the graph. Just like the right-hand behavior, we use the leading term of the polynomial function to predict this.
For \( f(x) = \frac{1}{5}x^{3} + 4x \), the degree is odd, and the leading coefficient \( \frac{1}{5} \) is positive. According to the Leading Coefficient Test, an odd degree with a positive leading coefficient causes the graph to "fall to the left."
  • This means as \( x \) becomes more negative, \( f(x) \) reduces and heads downward.
Understanding this aspect helps to draw and interpret the entire behavior of polynomial graphs more accurately.
Leading Term
The leading term in a polynomial is crucial as it significantly impacts the function's overall behavior. This term carries the highest power of the variable. In \( f(x) = \frac{1}{5}x^{3} + 4x \), \( \frac{1}{5}x^{3} \) is the leading term because it has the highest exponent, 3.
Here's why the leading term matters:
  • The exponent of the leading term, also known as the degree, determines the polynomial's general shape and end behaviors.
  • The coefficient of the leading term decides whether the graph opens upwards or downwards.
The leading term isn't just a part of the polynomial; it's the anchor that holds down many essential character traits of the graph.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Think About It Sketch the graph of each polynomial function. Then count the number of real zeros of the function and the numbers of relative minima and relative maxima. Compare these numbers with the degree of the polynomial. What do you observe? (a) \(f(x)=-x^{3}+9 x\) (b) \(f(x)=x^{4}-10 x^{2}+9\) (c) \(f(x)=x^{5}-16 x\)

Population The populations \(P\) (in thousands) of Horry County, South Carolina, from 1980 through 2010 can be modeled by $$P=20.6+85.5 e^{0.0360 t}$$ where \(t\) represents the year, with \(t=0\) corresponding to \(1980 .\) (Source: U.S. Census Bureau) (a) Use the model to complete the table. (b) According to the model, when will the population of Horry County reach \(350,000 ?\) (c) Do you think the model is valid for long-term predictions of the population? Explain.

Page Design A page that is \(x\) inches wide and \(y\) inches high contains 30 square inches of print. The top and bottom margins are each 1 inch deep, and the margins on each side are 2 inches wide (see figure). (a) Write a function for the total area \(A\) of the page in terms of \(x .\) (b) Determine the domain of the function based on the physical constraints of the problem. (c) Use a graphing utility to graph the area function and approximate the page size for which the least amount of paper is used.

Writing an Equation Write the equation for a quadratic function \(f\) (with integer coefficients) that has the given zeros. Assume that \(b\) is a positive integer. (a) \(\pm \sqrt{b} i\) (b) \(a \pm b i\)

Graphical Analysis In Exercises \(63-66,\) use a graphing utility to graph the rational function. State the domain of the function and find any asymptotes. Then zoom out sufficiently far so that the graph appears as a line. Identify the line. $$f(x)=\frac{x^{2}+5 x+8}{x+3}$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.