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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 15
Chapter 4, Problem 15

In Exercises 15–18, use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d).] <IMAGE>
f(x)=−x4+x2f(x)=−x^4+x^2

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1
Identify the leading term of the polynomial function. For the function \(f(x) = -x^4 + x^2\), the leading term is \(-x^4\) because it has the highest power of \(x\).
Determine the degree of the polynomial, which is the exponent of the leading term. Here, the degree is 4, an even number.
Look at the leading coefficient, which is the coefficient of the leading term. In this case, it is \(-1\), a negative number.
Apply the Leading Coefficient Test: For an even degree polynomial, if the leading coefficient is negative, the ends of the graph both go down (toward negative infinity) as \(x\) approaches positive and negative infinity.
Use this end behavior to match the polynomial with the correct graph: the graph should show both ends falling downward, which corresponds to the behavior determined in the previous step.

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Key Concepts

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

Leading Coefficient Test

The Leading Coefficient Test helps determine the end behavior of a polynomial function by examining the degree and the leading coefficient. For even-degree polynomials, if the leading coefficient is positive, both ends of the graph rise; if negative, both ends fall. For odd-degree polynomials, the ends go in opposite directions depending on the sign of the leading coefficient.
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Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression. It influences the general shape and end behavior of the graph. For example, even-degree polynomials have similar end behaviors on both sides, while odd-degree polynomials have opposite end behaviors on each side.
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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 in matching the polynomial to its graph by predicting whether the graph rises or falls at the extremes.
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Related Practice
Textbook Question

Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=−x2−2x+8

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Write an equation that expresses each relationship. Then solve the equation for y. x varies jointly as y and z and inversely as the square of w.

Textbook Question

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

As x→1+, f(x)→x\(\to\)1^{+},\(\text{ }\)f(x)\(\to\) __

Textbook Question

Divide using long division. State the quotient, and the remainder, r(x). (x4+2x3−4x2−5x−6)/(x2+x−2)

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Textbook Question

Divide using long division. State the quotient, and the remainder, r(x).2x5−8x4+2x3+x22x3+1\(\frac{2x^5 - 8x^4 + 2x^3 + x^2}{2x^3 + 1}\)

Textbook Question

Write an equation that expresses each relationship. Then solve the equation for y. x varies jointly as y and z and inversely as the square root of w.