Textbook Question
In Exercises 19–24, (a) Use the Leading Coefficient Test to determine the graph's end behavior. (b) Determine whether the graph has y-axis symmetry, origin symmetry, or neither. (c) Graph the function.
Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 4, Problem 20
Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function.
Verified step by step guidance1
Identify the leading term of the polynomial function. The leading term is the term with the highest power of \(x\), which in this case is \$11x^{3}$.
Determine the degree of the polynomial, which is the exponent of the leading term. Here, the degree is 3, an odd number.
Look at the leading coefficient, which is the coefficient of the leading term. Here, the leading coefficient is 11, a positive number.
Apply the Leading Coefficient Test for end behavior: For an odd degree polynomial with a positive leading coefficient, as \(x \to \infty\), \(f(x) \to \infty\), and as \(x \to -\infty\), \(f(x) \to -\infty\).
Summarize the end behavior: The graph falls to the left (as \(x\) approaches negative infinity) and rises to the right (as \(x\) approaches positive infinity).
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3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Leading Coefficient Test
The Leading Coefficient Test uses the degree and leading coefficient of a polynomial to determine its end behavior. The sign and parity (odd or even) of the leading term dictate how the graph behaves as x approaches positive or negative infinity.
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End Behavior of Polynomial Functions
Degree of a Polynomial
The degree of a polynomial is the highest power of the variable in the expression. It indicates the general shape of the graph and influences the number of turning points and the end behavior of the polynomial function.
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05:16Standard Form of Polynomials
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 by the leading term and helps predict whether the graph rises or falls on either end.
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06:08End Behavior of Polynomial Functions
Related Practice
Textbook Question
Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function.
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Textbook Question
Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation.
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Textbook Question
Write an equation that expresses each relationship. Then solve the equation for y. x varies directly as z and inversely as the difference between y and w.
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Textbook Question
In Exercises 19–24, (a) Use the Leading Coefficient Test to determine the graph's end behavior. (b) Determine whether the graph has y-axis symmetry, origin symmetry, or neither. (c) Graph the function.
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Textbook Question
Write an equation that expresses each relationship. Then solve the equation for y. x varies directly as z and inversely as the sum of y and w.