Structure in Graphs of Polynomial Functions


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Structure in Graphs of Polynomial Functions

Student Outcomes

  • Students graph polynomial functions and describe end behavior based upon the degree of the polynomial.

New York State Common Core Math Algebra II, Module 1, Lesson 15

Worksheets for Algebra II, Module 1, Lesson 15

Classwork

Opening Exercises
Sketch the graph of 𝑓(𝑥) = 𝑥2. What will the graph of 𝑔(𝑥) = 𝑥4 look like? Sketch it on the same coordinate plane. What will the graph of ℎ(𝑥) = 𝑥6 look like?

Example 1

Sketch the graph of 𝑓(𝑥) = 𝑥3. What will the graph of 𝑔(𝑥) = 𝑥5 look like? Sketch this on the same coordinate plane. What will the graph of ℎ(𝑥) = 𝑥7 look like? Sketch this on the same coordinate plane.

Exercise 1

a. Consider the following function, 𝑓(𝑥) = 2𝑥4 + 𝑥3 − 𝑥2 + 5𝑥 + 3, with a mixture of odd and even degree terms. Predict whether its end behavior will be like the functions in the Opening Exercise or more like the functions from Example 1. Graph the function 𝑓 using a graphing utility to check your prediction.
b. Consider the following function, 𝑓(𝑥) = 2𝑥5 − 𝑥4 − 2𝑥3 + 4𝑥2 + 𝑥 + 3, with a mixture of odd and even degree terms. Predict whether its end behavior will be like the functions in the Opening Exercise or more like the functions from Example 1. Graph the function 𝑓 using a graphing utility to check your prediction.
c. Thinking back to our discussion of 𝑥-intercepts of graphs of polynomial functions from the previous lesson, sketch a graph of an even-degree polynomial function that has no 𝑥-intercepts. d. Similarly, can you sketch a graph of an odd-degree polynomial function with no 𝑥-intercepts?

Exercise 2

The Center for Transportation Analysis (CTA) studies all aspects of transportation in the United States, from energy and environmental concerns to safety and security challenges. A 1997 study compiled the following data of the fuel economy in miles per gallon (mpg) of a car or light truck at various speeds measured in miles per hour (mph).
The data are compiled in the table below.
a. Plot the data using a graphing utility. Which variable is the independent variable?
b. This data can be modeled by a polynomial function. Determine if the function that models the data would have an even or odd degree.
c. Is the leading coefficient of the polynomial that can be used to model this data positive or negative?
d. List two possible reasons the data might have the shape that it does.

Relevant Vocabulary

EVEN FUNCTION: Let 𝑓 be a function whose domain and range is a subset of the real numbers. The function 𝑓 is called even if the equation 𝑓(𝑥) = 𝑓(−𝑥) is true for every number 𝑥 in the domain. Even-degree polynomial functions are sometimes even functions, like 𝑓(𝑥) = 𝑥10, and sometimes not, like 𝑔(𝑥) = 𝑥2 − 𝑥.

ODD FUNCTION: Let 𝑓 be a function whose domain and range is a subset of the real numbers. The function 𝑓 is called odd if the equation 𝑓(−𝑥) = −𝑓(𝑥) is true for every number 𝑥 in the domain.

Odd-degree polynomial functions are sometimes odd functions, like 𝑓(𝑥) = 𝑥 11, and sometimes not, like ℎ(𝑥) = 𝑥3 − 𝑥2.




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