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Ch. 3 - Polynomial and Rational Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 3 - Polynomial and Rational FunctionsProblem 24
Chapter 4, Problem 24

Use an end behavior diagram, as shown below, to describe the end behavior of the graph of each polynomial function. ƒ(x)=4x7-x5+x3-1

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Identify the leading term of the polynomial function. For the function \(f(x) = 4x^7 - x^5 + x^3 - 1\), the leading term is \$4x^7\( because it has the highest power of \)x$.
Determine the degree and the leading coefficient of the polynomial. Here, the degree is 7 (which is odd), and the leading coefficient is 4 (which is positive).
Recall the end behavior rules for polynomials: For an odd degree with a positive leading coefficient, as \(x \to \infty\), \(f(x) \to \infty\), and as \(x \to -\infty\), \(f(x) \to -\infty\).
Use this information to sketch or describe the end behavior diagram: the graph falls to the left (goes down as \(x\) approaches negative infinity) and rises to the right (goes up as \(x\) approaches positive infinity).
Summarize the end behavior: \(\lim_{x \to -\infty} f(x) = -\infty\) and \(\lim_{x \to \infty} f(x) = \infty\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

End Behavior of Polynomial Functions

End behavior describes how the values of a polynomial function behave as x approaches positive or negative infinity. It is determined mainly by the leading term, which dominates the function for very large or very small x-values.
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Leading Term and Degree of a Polynomial

The leading term is the term with the highest power of x in a polynomial. The degree (highest exponent) and the coefficient of this term dictate the general shape and end behavior of the graph.
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Using End Behavior Diagrams

End behavior diagrams visually represent the direction of the graph's ends as x approaches infinity or negative infinity. They help summarize whether the graph rises or falls on each side based on the leading term's degree and sign.
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Related Practice
Textbook Question

Find a polynomial function ƒ(x) of least degree with real coefficients having zeros as given. √3, -√3, 2, 3

Textbook Question

Write each formula as an English phrase using the word varies or proportional. V = 1/3 πr2h, where V is the volume of a cone of radius r and height h

Textbook Question

Graph each quadratic function. Give the (a) vertex, (b) axis, (c) domain, and (d) range. ƒ(x) = (x - 2)2

Textbook Question

Use synthetic division to perform each division. x7+1 / x+1

Textbook Question

Use an end behavior diagram, as shown below, to describe the end behavior of the graph of each polynomial function. ƒ(x)=9x6-3x4+x2-2

Textbook Question

Solve each polynomial inequality. Give the solution set in interval notation.

(a) -x(x - 1)(x - 2) ≥ 0

(b) -x(x - 1)(x - 2) > 0

(c) -x(x - 1)(x - 2) ≤ 0

(d) -x(x - 1)(x - 2) < 0