Name: ____________________________________ Date: __________________

COMMON CORE ALGEBRA II, UNIT #10 – POLYNOMIAL AND RATIONAL FUNCTIONS – LESSON #1.5 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2018

INVESTIGATING END BEHAVIOR IN POLYNOMIALS COMMON CORE ALGEBRA II

Polynomials are functions that have the form 1 2n n ny ax bx cx where the powers are all non-negative integers. Examples of polynomials include:

22 3 7y x x 5 213

2y x x x 34 5y x

The graphical behavior of polynomials, especially their turning points and zeroes, is rich and interesting and will be explored future lessons. In this lesson we look more deeply at their end behavior:

Exercise #1: Given the function 213 4

2y x x .

(a) Sketch the graph below for the indicated window. The end behavior of all quadratic functions is either to point upward (go towards infinity) or point downwards (go towards negative infinity). Next, let's investigate cubic functions. Exercise #3: Two cubic polynomials as shown below along with their equations

END BEHAVIOR OF FUNCTIONS

The end behavior of a function is how it acts graphically as x gets very large (either positive or negative). Symbolically, it is how f x behaves as x or x .

(b) Describe the end behavior of the function as: x (going to the right): x (going to the left):

(a) How does the behavior of cubics differ from quadratics?

(b) Why is the end behavior of f x

the opposite of that of g x ?

COMMON CORE ALGEBRA II, UNIT #10 – POLYNOMIAL AND RATIONAL FUNCTIONS – LESSON #1.5 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2018

We will look now at quartic functions (also known as degree four polynomials).

Exercise #4: The quartic function 4 3 22 7 95 154 240f x x x x x is shown graphed below.

(a) Describe the end behavior of the quartic function: as x :

as x :

(b) Does the end behavior of the quartic function resemble that of a quadratic function or cubic function more? Explain.

(c) Using the axes and window indicated above, sketch the function 4 3 24 23 54 125g x x x x x . Why

do the ends of this quartic function point in the opposite direction of those of f x ?

As should be evident from the polynomial examples we have seen, the end behavior of a polynomial is determined by two factors: (a) the degree of the polynomial (its highest power) and the (b) leading coefficient of the polynomial. Exercise #5: Fill in the following statements to make them true: (a) When the degree of the polynomial is even, the ends of the polynomial point in the ___________ direction. (b) When the degree of the polynomial is odd, the ends of the polynomial point in the ____________ direction. (c) When the leading coefficient changes ___________, the ends of the polynomial change ____________. Exercise #6: For each polynomial graph below, state whether its degree is even or odd and whether the leading coefficient is positive or negative. (a) (b)

Degree: Leading Coefficient:

Degree: Leading Coefficient:

Name: ____________________________________ Date: __________________

COMMON CORE ALGEBRA II, UNIT #10 – POLYNOMIAL AND RATIONAL FUNCTIONS – LESSON #1.5 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2018

INVESTIGATING END BEHAVIOR IN POLYNOMIALS COMMON CORE ALGEBRA II HOMEWORK

FLUENCY

1. For each of the following polynomials, sketch a graph on the axes provided using the indicated graphing window. Then, state the degree of the polynomial, the value of its leading coefficient, and its end behavior.

(a) 3 20.5 12f x x x x

(b) 4 226 25f x x x

(c) 5 4 3 22 5 46 79 170 200f x x x x x x

Degree of polynomial: ____________

Leading coefficient: ______________ End Behavior:

as x : as x :

Degree of polynomial: ____________

Leading coefficient: ______________ End Behavior:

as x : as x :

Degree of polynomial: ____________

Leading coefficient: ______________ End Behavior:

as x : as x :

COMMON CORE ALGEBRA II, UNIT #10 – POLYNOMIAL AND RATIONAL FUNCTIONS – LESSON #1.5 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2018

2. Given the polynomial shown, which of the following could be its highest-powered term?

(1) 43

4x

(2) 52x

(3) 63x

(4) 71

2x

3. For the polynomial function f x , it is known that as x , f x , and as x , f x .

Which of the following must be true about this polynomial? (1) It has an even degree and an odd leading coefficient.

(2) It has an odd degree and a negative leading coefficient.

(3) It has positive degree and a positive leading coefficient.

(4) It has an even degree and a negative leading coefficient.

APPLICATIONS 4. The profile of a hill has the shape shown in the graph below. If the profile was modeled using a polynomial,

then it would likely have (1) a highest power that is odd and a leading

coefficient that is negative.

(2) a highest power that is even and a leading coefficient that is negative.

(3) a highest power that is odd and a leading coefficient that is positive.

(4) a highest power that is even and a leading coefficient that is positive.

REASONING 5. A polynomial has a factored form of 2 6 1 4 10f x x x x x . Describe its end behavior

(both directions) and why it occurs.