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 22
Chapter 4, Problem 22

Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. f(x)=11x4−6x2+x+3

Verified step by step guidance
1
Identify the degree of the polynomial function. The degree is the highest power of \(x\) in the polynomial. For \(f(x) = 11x^{4} - 6x^{2} + x + 3\), the degree is 4.
Determine the leading coefficient, which is the coefficient of the term with the highest degree. Here, the leading coefficient is 11.
Recall the Leading Coefficient Test rules for end behavior: For even degree polynomials, if the leading coefficient is positive, both ends of the graph go up; if negative, both ends go down. For odd degree polynomials, if the leading coefficient is positive, the left end goes down and the right end goes up; if negative, the left end goes up and the right end goes down.
Since the degree is 4 (even) and the leading coefficient is 11 (positive), conclude that as \(x \to \infty\), \(f(x) \to \infty\), and as \(x \to -\infty\), \(f(x) \to \infty\).
Summarize the end behavior: The graph rises to the right and rises to the left.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
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 the end behavior of its graph. It states that the sign and parity (even or odd) of the leading term dictate how the function 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 the same end behavior 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 is determined by the leading term and helps predict whether the graph rises or falls on each end, which is essential for sketching or analyzing the function.
Recommended video:
06:08
End Behavior of Polynomial Functions
Related Practice
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)=2(x+2)2−1

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 + x ≥ 0

3
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

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. 2x2+3x>02x^2+3x>0

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. g(x)=(x+3)/x(x+4)

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. 3x2 − 5x ≤ 0

3
views