Skip to main content
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 21
Chapter 4, Problem 21

Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. f(x)=5x4+7x2−x+9f(x)=5x^4+7x^2−x+9

Verified step by step guidance
1
Identify the leading term of the polynomial function. The leading term is the term with the highest power of \(x\). In this case, it is \$5x^4$.
Determine the degree of the polynomial, which is the exponent of the leading term. Here, the degree is 4, which is an even number.
Look at the leading coefficient, which is the coefficient of the leading term. Here, the leading coefficient is 5, a positive number.
Apply the Leading Coefficient Test: For an even degree polynomial with a positive leading coefficient, the end behavior is such that as \(x \to \infty\), \(f(x) \to \infty\), and as \(x \to -\infty\), \(f(x) \to \infty\).
Summarize the end behavior based on the test: both ends of the graph will rise upwards toward positive infinity.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

Key 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. It states that the sign of the leading coefficient and whether the degree is even or odd dictate how the graph behaves as x approaches positive or negative infinity.
Recommended video:
06:08
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 influences the 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.
Recommended video:
Guided course
05:16
Standard 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 helps predict whether the graph rises or falls on the far left and right sides, based on the leading term's degree and coefficient.
Recommended video:
06:08
End Behavior of Polynomial Functions
Related Practice
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+2x<0x^2+2x<0

3
views
Textbook Question

In Exercises 17–24, a) List all possible rational roots. b) List all possible rational roots. c) Use the quotient from part (b) to find the remaining roots and solve the equation. 6x3+25x2−24x+5=0

Textbook Question

Divide using synthetic division. (4x3−3x2+3x−1)÷(x−1)

4
views
Textbook Question

Find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. f(x)=x/(x+4)

1
views
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. y−1=(x−3)2

3
views
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.