Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

4. Polynomial Functions

Understanding Polynomial Functions

College Algebra

4. Polynomial Functions

Understanding Polynomial Functions

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1:16 minutes

Problem 21

Textbook Question

Use an end behavior diagram, , , , or , to describe the end behavior of the graph of each polynomial function. See Example 2. ƒ(x)=5x^5+2x^3-3x+4

Verified step by step guidance

Identify the leading term of the polynomial function, which is the term with the highest power of x. In this case, the leading term is \(5x^5\).

Determine the degree of the polynomial. The degree is the highest power of x in the polynomial, which is 5 in this case.

Since the degree of the polynomial is odd (5), the end behavior of the graph will be different on each side.

Look at the leading coefficient, which is 5. Since it is positive, the graph will rise to the right.

Combine the information: For an odd-degree polynomial with a positive leading coefficient, the end behavior is that the graph falls to the left and rises to the right.

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Key Concepts

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

End Behavior of Polynomials

The end behavior of a polynomial function describes how the function behaves as the input values (x) approach positive or negative infinity. This behavior is primarily determined by the leading term of the polynomial, which is the term with the highest degree. For example, in the polynomial f(x) = 5x^5 + 2x^3 - 3x + 4, the leading term is 5x^5, indicating that as x approaches infinity, f(x) will also approach infinity, and as x approaches negative infinity, f(x) will approach negative infinity.

Leading Coefficient Test

The leading coefficient test helps predict the end behavior of a polynomial function based on the sign and degree of the leading term. If the leading coefficient is positive and the degree is odd, the graph will rise to the right and fall to the left. Conversely, if the leading coefficient is negative and the degree is odd, the graph will fall to the right and rise to the left. This test is crucial for sketching the overall shape of the polynomial graph.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. It plays a significant role in determining the number of turning points and the overall shape of the graph. For instance, a polynomial of degree 5, like f(x) = 5x^5 + 2x^3 - 3x + 4, can have up to 4 turning points and will exhibit specific end behaviors based on its degree and leading coefficient.