Book cover for College Algebra

College Algebra

Michael Sullivan

ISBN #9780321716811

9th Edition

5,201 Questions

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40,131 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section on Polynomial Functions and Models emphasizes the algebraic structure and graphical behavior of polynomials. Key points include understanding the general form of a polynomial, determining its degree, and identifying zeros along with their multiplicities to predict how the graph behaves at and between intercepts. Additionally, graph transformations, end behavior driven by the leading term, and the concept of turning points are discussed. The section also introduces the use of graphing utilities both for analyzing polynomial graphs and constructing cubic models from real-world data, underscoring the practical applications of these concepts.

Learning Objectives

1

Identify and classify polynomial functions by their algebraic form and degree.

2

Analyze polynomial graphs by determining intercepts, zeros (with multiplicity), turning points, and end behavior.

3

Use transformation techniques (shifting, stretching, compressing, reflecting) to graph polynomial functions.

4

Employ graphing utilities to approximate local extrema, zeros, and to model real-world data with cubic functions.

5

Explain how the multiplicity of a zero affects whether the graph crosses or touches the x-axis.

Key Concepts

CONCEPT

DEFINITION

Polynomial Function

A function of the form f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where aₙ, aₙ₋₁, …, a₀ are real numbers and n is a nonnegative integer. Its domain is all real numbers.

Degree

The highest exponent n in the polynomial; it largely determines the overall shape and behavior of the graph.

Monomial / Power Function

A monomial is a single-term polynomial; a power function is a monomial of the form f(x) = axⁿ, with n being a nonnegative integer.

Zero (or Root)

A real number r such that f(r) = 0. In the graph, these correspond to the x-intercepts.

Multiplicity

The number of times a factor (x - r) appears in the factorization of a polynomial. If the multiplicity is even, the graph touches the x-axis at r; if odd, it crosses.

End Behavior

The behavior of a polynomial function as x approaches positive or negative infinity. It is determined by the leading term (aₙxⁿ).

Turning Points

Points on the graph where the function changes direction (from increasing to decreasing or vice versa). A polynomial of degree n can have at most n - 1 turning points.

Example Problems

Example 1

Answers are given at the end of these exercises If you get a wrong answer read the pages listed in red. The intercepts of the equation $9 x^{2}+4 y=36$ are $(p p .159-160)$

Example 2

Answers are given at the end of these exercises If you get a wrong answer read the pages listed in red. Is the expression $4 x^{3}-3.6 x^{2}-\sqrt{2}$ a polynomial? If so, what is its degree? (pp. 39-47)

Example 3

Answers are given at the end of these exercises If you get a wrong answer read the pages listed in red. To graph $y=x^{2}-4,$ you would shift the graph of $y=x^{2}$______a distance of_____units.

Example 4

Answers are given at the end of these exercises If you get a wrong answer read the pages listed in red. Use a graphing utility to approximate (rounded to two decimal places) the local maximum value and local minimum value of $f(x)=x^{3}-2 x^{2}-4 x+5,$ for $-3<x<3 .

Example 5

Answers are given at the end of these exercises If you get a wrong answer read the pages listed in red. The $x$ -intercepts of the graph of a function $y=f(x)$ are the real solutions of the equation $f(x)=0$
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Step-by-Step Explanations

QUESTION

Identify the real zeros and their multiplicities for the function f(x) = x²(x - 2).

STEP-BY-STEP ANSWER:

Step 1: Note that the function is already factored: f(x) = x² · (x - 2).
Step 2: Set each factor equal to 0: x² = 0 gives x = 0 and (x - 2) = 0 gives x = 2.
Step 3: Determine the multiplicity: The factor x appears with an exponent of 2 (even multiplicity, so the graph touches the x-axis at x = 0), and (x - 2) appears once (odd multiplicity, so the graph crosses the x-axis at x = 2).
Final Answer: f(x) has zeros at x = 0 (multiplicity 2) and x = 2 (multiplicity 1).

Determining Zeros and Multiplicity

QUESTION

Describe the transformation steps needed to graph f(x) = 1 - x⁵.

STEP-BY-STEP ANSWER:

Step 1: Rewrite the function as f(x) = -x⁵ + 1.
Step 2: Recognize that x⁵ is a power function with an odd degree; its graph passes through the origin, with end behavior dictated by the sign of the leading coefficient.
Step 3: The negative sign indicates a reflection across the x-axis.
Step 4: The addition of 1 indicates a vertical shift upward by 1 unit.
Final Answer: The graph of f(x) = 1 - x⁵ is obtained by reflecting the graph of x⁵ across the x-axis and then shifting the resulting graph upward by 1 unit.

Graphing Using Transformations

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Common Mistakes

  • Assuming any algebraic expression with x as a variable is a polynomial, even if it contains negative or fractional exponents.
  • Misidentifying the degree of a polynomial after it is factored, especially when the factors are not multiplied out.
  • Overlooking the impact of multiplicity on the graph: not recognizing that an even multiplicity causes the graph to touch, not cross, the x-axis.
  • Ignoring the end behavior dictated by the leading term, which can lead to incorrect predictions about the graph for large values of x.
  • Failing to utilize transformations correctly, such as missing a reflection or vertical shift, which can distort the graph.