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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 19
Chapter 4, Problem 19

Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. f(x)=5x3+7x2−x+9f(x)=5x^3+7x^2−x+9

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Identify the degree of the polynomial function \(f(x) = 5x^3 + 7x^2 - x + 9\). The degree is the highest power of \(x\), which is 3 in this case.
Determine the leading coefficient, which is the coefficient of the term with the highest degree. Here, the leading coefficient is 5.
Recall the Leading Coefficient Test rules for end behavior: For an odd degree polynomial, if the leading coefficient is positive, as \(x \to \infty\), \(f(x) \to \infty\) and as \(x \to -\infty\), \(f(x) \to -\infty\).
Apply the test to this polynomial: Since the degree is 3 (odd) and the leading coefficient is 5 (positive), the graph will rise to the right and fall to the left.
Summarize the end behavior: As \(x\) approaches positive infinity, \(f(x)\) approaches positive infinity; as \(x\) approaches negative infinity, \(f(x)\) approaches negative infinity.

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Key Concepts

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

Polynomial Functions

A polynomial function is an expression consisting of variables raised to whole-number exponents and coefficients, combined using addition, subtraction, and multiplication. Understanding the general form and degree of a polynomial helps in analyzing its graph and behavior.
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Introduction to Polynomial Functions

Leading Coefficient Test

The Leading Coefficient Test uses the degree and leading coefficient of a polynomial to determine the end behavior of its graph. Specifically, it predicts how the function behaves as x approaches positive or negative infinity based on whether the degree is even or odd and the sign of the leading coefficient.
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End Behavior of Polynomial Functions

End Behavior of Functions

End behavior describes how the values of a function behave as the input x becomes very large or very small. For polynomials, this is determined by the highest-degree term, which dominates the function's growth or decline at the extremes of the x-axis.
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End Behavior of Polynomial Functions
Related Practice
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. f(x)=x3−x2−9x+9f(x) = x^3 - x^2 - 9x + 9

Textbook Question

Use the graph of the rational function in the figure shown to complete each statement in Exercises 15–20.

As x→∞, f(x)→x\(\to\]\infty\),\(\text{ }\)f(x)\(\to\)_____

Textbook Question

Divide using synthetic division. (3x2+7x−20)÷(x+5)

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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

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=(x−1)2+2

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. x2−4x≥0x^2−4x≥0