202309142.05endbehaviorpart1part2

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St. Charles West High School**We aren't endorsed by this school

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BIO 2

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Mathematics

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Dec 18, 2024

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Lesson 8: End Behavior (Part 1)Let’s investigate the shape of polynomials.8.1: Notice and Wonder: A Different ViewWhat do you notice? What do you wonder?8.2: Polynomial End Behavior1. For your assigned polynomial, complete the column for the different values of.Discuss with your group what you notice.-1000-100-10-11101001000•Unit 2 Lesson 8117

2. Sketch what you think theend behaviorof your polynomial looks like, then checkyour work using graphing technology.Are you ready for more?Mai is studying the function. She makes a table ofvalues forwithand thinks that this function has large positiveoutput values in both directions on the-axis. Do you agree with Mai? Explain yourreasoning.118

8.3: Two Polynomial EquationsConsider the polynomial.1. Identify the degree of the polynomial.2. Which of the 6 terms,,,,,, or, is greatest when:a.b.c.d.3. Describe the end behavior of the polynomial.Lesson 8 SummaryWe know that if the expression for a polynomial functionwritten in factored form hasthe factor, thenis a zero of(that is,) and the pointis on thegraph of the function. But what about other values of? In particular, as we considervalues ofthat get larger and larger in either the negative or positive direction, whathappens to the values of?The answer to this question depends on the degree of the polynomial, because anynegative real number raised to an even power results in a positive number. For example, ifwe graph,andand zoom out, we see the following:Unit 2 Lesson 8119

For bothand, large positive values ofor large negative values ofeachresult in large positive values of. But for, large positive values ofresult in largepositive values of, while large negative values ofresult in large negative values of.Consider the polynomial. The leading term,, almostseems smaller than the other 3 terms. For certain values of, this is even true. But, forvalues offar away from zero, the leading term will always have the greatest value. Canyou see why?-50062,500,000,0003,750,000,000-5,000,0001,00066,245,001,000-100100,000,00030,000,000-200,0001,000129,801,000-1010,00030,000-2,0001,00039,00000001,00010001010,000-30,000-2,0001,000-21,000100100,000,000-30,000,000-200,0001,00069,801,00050062,500,000,000-3,750,000,000-5,000,0001,00058,745,001,000The value of the leading termdetermines theend behaviorof the function, that is, howthe outputs of the function change as we look at input values farther and farther from 0. Inthe case of, asgets larger and larger in the positive and negative directions, theoutput of the function gets larger and larger in the positive direction.Glossaryend behavior•120

Lesson 8 Practice Problems1. Match each polynomial with its end behavior. Some end behavior options may nothave a matching polynomial.A.B.C.D.1. Asgets larger and larger ineither the positive or negativedirection,gets larger and largerin the positive direction.2. Asgets larger and larger in thepositive direction,gets largerand larger in the positive direction.Asgets larger and larger in thenegative direction,gets largerand larger in the negative direction.3. Asgets larger and larger in thepositive direction,gets largerand larger in the negative direction.Asgets larger and larger in thenegative direction,gets largerand larger in the positive direction.4. Asgets larger and larger in eitherthe positive or negative direction,gets larger and larger in thenegative direction.2. Which polynomial function gets larger and larger in the negative direction asgetslarger and larger in the negative direction?A.B.C.D.Unit 2 Lesson 8 Practice Problems121

3. The graph of a polynomial functionisshown. Which statement about thepolynomial is true?A. The degree of the polynomial is even.B. The degree of the polynomial is odd.C. The constant term of the polynomial is even.D. The constant term of the polynomial is odd.4. Andre wants to make an open-top box by cutting out corners of a 22 inch by 28 inchpiece of poster board and then folding up the sides. The volumein cubic inchesof the open-top box is a function of the side lengthin inches of the square cutouts.a. Write an expression for.b. What is the volume of the box when?c. What is a reasonable domain forin this context?(From Unit 2, Lesson 1.)122

5. For each polynomial function, rewrite the polynomial in standard form. Then state itsdegree and constant term.a.b.(From Unit 2, Lesson 6.)6. Kiran wroteas an example of a function whose graph has-intercepts at. What was his mistake?(From Unit 2, Lesson 7.)7. A polynomial function,, has-intercepts atand. What is onepossible factor of?(From Unit 2, Lesson 7.)Unit 2 Lesson 8 Practice Problems123

Lesson 9: End Behavior (Part 2)Let’s describe the end behavior of polynomials.9.1: It’s a Cover UpMatch each of the graphs to the polynomial equation it represents. For the graph withouta matching equation, write down what must be true about the polynomial equation.ABCD1.2.3.9.2: The Case of Unexpected End Behavior1. Write an equation for a polynomial with the following properties: it has even degree,it has at least 2 terms, and, as the inputs get larger and larger in either the negativeor positive directions, the outputs get larger and larger in the negative direction.Pause here so your teacher can review your work.2. Write an equation for a polynomial with the following properties: it has odd degree, ithas at least 2 terms, as the inputs get larger and larger in the negative direction theoutputs get larger and larger in the positive direction, and as the inputs get largerand larger in the positive direction, the outputs get larger and larger in the negativedirection.•124

Are you ready for more?In the given graph all of the horizontal intercepts are shown. Find a function with thisgeneral shape and the same horizontal intercepts.9.3: Which is Greater?andare each functions ofdefined byand.1. Describe the end behavior ofand.2. For, which function do you think has greater values? Be prepared to share yourreasoning with the class.Unit 2 Lesson 9125

Lesson 9 SummaryWhat happens when we multiply a number by a negative number? If the original numberwas positive, the product is negative. But if the original number was negative, the productis positive. The sign of the new number is the opposite of the original number.Now let’s consider the polynomial functionsand. For any non-zeroreal number, the output ofis positive while the output ofis negative. The signs of allthe output values forare the opposite of those of. The difference between these twofunctions is also easy to see when we look at their graphs.This is the effect of a negative leading coefficient: the end behavior of the polynomial is theopposite of what it would be if the leading coefficient were positive. For polynomials ofodd degree, we can see that a negative leading coefficient has the same effect on the endbehavior.126

Here are the graphs of, which has a leading term of, and, which has a leading term of. They have the same zeros,but opposite end behavior, because they have opposite signs on their leading coefficients.Unit 2 Lesson 9127

Lesson 9 Practice Problems1. Match the polynomial with its end behavior.A.B.C.D.1. Asgets larger and larger in eitherthe positive or negative direction,gets larger and larger in thepositive direction.2. Asgets larger and larger in thepositive direction,gets largerand larger in the positive direction.Asgets larger and larger in thenegative direction,gets largerand larger in the negative direction.3. Asgets larger and larger in thepositive direction,gets largerand larger in the negative direction.Asgets larger and larger in thenegative direction,gets largerand larger in the positive direction.4. Asgets larger and larger in eitherthe positive or negative direction,gets larger and larger in thenegative direction.2. State the degree and end behavior of. Explain or showyour reasoning.128

3. The graph of a polynomial functionis shown. Selectallthe true statements aboutthe polynomial.A. The degree of the polynomial is even.B. The degree of the polynomial is odd.C. The leading coefficient is positive.D. The leading coefficient is negative.E. The constant term of the polynomial is positive.F. The constant term of the polynomial is negative.4. Write the sum ofandas a polynomial in standard form.(From Unit 2, Lesson 4.)5. State the degree and end behavior of. Explain or showyour reasoning.(From Unit 2, Lesson 8.)Unit 2 Lesson 9 Practice Problems129

6. Selectallthe polynomial functions whose graphs have-intercepts at.A.B.C.D.E.F.(From Unit 2, Lesson 7.)130