Power Functions and Polynomial Functions

CHAPTER Power Functions and Polynomial Functions Estuaries form when rivers and streams meet the sea, resulting in a mix of salt and fresh water. On the coast of Georgia, large estuaries have formed where the Altamaha, Ogeechee, Satilla, Savannah, and St. Mary s rivers flow into the Atlantic Ocean. You will use polynomial functions to graph water salinity levels in estuaries..1 Power Ranges! Power Functions p. 97.2 Boxes Polynomial Functions p. 107. Humps and Bumps Graphs of Polynomial Functions p. 115.4 Mirror Mirror on the Wall... Symmetry p. 12.5 Cloned Zeros? Zeros and Multiplicity p. 11.6 Domains, Ends, and Extremes Extrema, Domain & Range, and End Behavior p. 141.7 Salinity Graphs of Polynomial Functions p. 147 Chapter l Power Functions and Polynomial Functions 95

96 Chapter l Power Functions and Polynomial Functions

.1 Power Ranges! Power Functions Objectives In this lesson you will: l Classify functions as 1 to 1 functions. l Determine the properties of the graphs of power functions. l Classify even and odd functions. l Transform basic power functions. Key Terms l 1 to 1 function l power function l even function l odd function Problem 1 Area and Volume 1. The area of a square is a function of its side length. a. Define a function A(s) to represent the area of a square with side length s. b. Graph the function A(s). c. What are the domain and range of A(s)? Lesson.1 l Power Functions 97

d. Graph the portion of the graph of A(s) that models the problem situation. e. What are the domain and range of A(s) in terms of the problem situation? Explain your reasoning. f. Describe the difference between the domain and range of A(s) and the domain and range of A(s) in terms of the problem situation. 2. The volume of a cube is a function of its side length. a. Define a function V(s) to represent the volume of a cube with side length s. 98 Chapter l Power Functions and Polynomial Functions

b. Graph the function V(s). c. What are the domain and range of V(s)? d. Graph the portion of the graph of V(s) that models the problem situation. e. What are the domain and range of V(s) in terms of the problem situation? Explain your reasoning. Lesson.1 l Power Functions 99

f. Describe the difference between the domain and range of V(s) and the domain and range of V(s) in terms of the problem situation.. What is the definition of a function? A 1 to 1 function is a function such that each element in the range has one and only one element in the domain. 4. Determine whether each function is a 1 to 1 function. Explain. a. A(s) b. A(s) in terms of the problem situation c. V(s) d. V(s) in terms of the problem situation 100 Chapter l Power Functions and Polynomial Functions

Problem 2 Power Functions A power function is a function of the form P(x) a x n, where n is a non-negative integer. For example, the functions f(x) x 2 and g(x) x are power functions. 1. Graph the power functions with a 1 and n 1,, 5. Label each graph. 2. What are the domain and range of each function in Question 1?. Describe what happens to the value of these functions as a. x increases toward. b. x decreases toward. Lesson.1 l Power Functions 101

4. Graph the power functions with a 1 and n 2, 4, 6. Label each graph. 5. What are the domain and range of each function in Question 4? 6. Describe what happens to the value of each function as: a. x increases toward. b. x decreases toward. An even function is a function f(x) such that for every value of x in the domain, f(x) f( x). An odd function is a function f(x) such that for every value of x in the domain, f(x) f( x). 7. Determine which power functions in Questions 1 and 4 are: a. even functions. b. odd functions. 102 Chapter l Power Functions and Polynomial Functions

Problem Translations and Dilations of Power Functions 1. Graph and label the functions f(x) x, g(x) x and h(x) x 4. 2. Based on the graphs in Question 1, how do you think the graph of P(x) x n k is different from the graph of the basic power function P(x) x n?. Determine whether each function in Question 1 is an even function, an odd function, or neither. Explain. Lesson.1 l Power Functions 10

4. Graph and label the functions f(x) x 4, g(x) (x 2) 4 and h(x) (x 4) 4. 5. Based on the graphs in Question 4, how do you think the graph of P(x) (x h) n is different from the graph of the basic power function P(x) x n? 6. Determine whether each function in Question 4 is an even function, an odd function, or neither. Explain. 104 Chapter l Power Functions and Polynomial Functions

7. The graph of the power function P(x) x i is shown. Sketch the graph of P(x) (x 2) i. y 4 2 1 4 2 1 1 1 2 4 x 2 4 8. Graph and label the functions f(x) x, g(x) 2 x and h(x) 1 2 x. 9. Based on the graphs in Question 8, how do you think the graph of P(x) ax n is different from the graph of the basic power function, P(x) x n? Lesson.1 l Power Functions 105

10. Determine whether each function in Question 1 is an even function, an odd function, or neither. Explain. Be prepared to share your methods and solutions. 106 Chapter l Power Functions and Polynomial Functions

.2 Boxes Polynomial Functions Objectives In this lesson you will: l Identify polynomial expressions and functions. l Write polynomial expressions and functions in ascending and descending order. l Add, subtract, and multiply polynomials. l Graph polynomial functions. Problem 1 Building a Box Key Terms l polynomial function l monomial l binomial l trinomial l degree of a term l degree of a polynomial l standard form of a polynomial function l quadratic, or second-degree polynomial l cubic, or third-degree polynomial l quartic, or fourth-degree polynomial l quintic, or fifth-degree polynomial To organize your tool shed, you plan on building some plywood boxes in the shape of rectangular prisms. The width of each box will be 6 inches less than its height. The length of each box will be 24 inches more than its height. 1. Define a variable for the height of each box. Then write expressions to represent the width and length. 2. Define a function to represent the volume of each box. Lesson.2 l Polynomial Functions 107

. Define a function to represent the surface area of each box. 4. Calculate the volume and surface area of each box with the height given. a. 10 inches b. 20 inches c. 40 inches A polynomial function is a function formed by adding and subtracting terms in the form ax b, where a is any real number and b is a non-negative integer. The root word poly means many and the root word nomial means terms. So, the term polynomial literally means many terms. The functions that you wrote for the volume and surface area of each box are polynomial functions. A monomial is a polynomial consisting of one term. A binomial is a polynomial consisting of two terms. A trinomial is a polynomial consisting of three terms. The degree of a term ax b is the value of the exponent b. The degree of a polynomial is the largest degree of the terms. The terms of a polynomial can be written in descending form, highest degree to lowest degree, or ascending form, lowest degree to highest degree. The standard form of a polynomial function is f(x) a n x n n 1 2 0 a n 1 x a 2 x a 1 x a 0 x Polynomial expressions, equations, and inequalities can also be written in standard form. 108 Chapter l Power Functions and Polynomial Functions

A quadratic polynomial is a second-degree polynomial. The function that you wrote for the surface area of each box is a quadratic polynomial. A cubic polynomial is a third-degree polynomial. The function that you wrote for the volume of each box is a cubic polynomial. A quartic polynomial is a fourth-degree polynomial. A quintic polynomial is a fifth-degree polynomial. 5. Graph the functions you wrote in Questions 2 and using a graphing calculator. Use a graphing window of 0 x 0 and 1000 y 000. Sketch each graph on the grid shown. 6. Describe each graph in Question 5. 7. Which portion of each graph models the problem situation? Explain. 8. How many zeros does each function have? Why? Lesson.2 l Polynomial Functions 109

9. Determine the zeros of each function. 10. How does the graph of a cubic function differ from the graph of each function? a. The graph of a quadratic function b. The graph of a linear function. Problem 2 Graphs of Polynomial Functions 1. For each function, complete the table provided and sketch the graph of the function by hand. Then determine the degree, the domain, the range, the zeros, and the y-intercept(s) using technology. a. f(x) x 4 x y Degree: Domain: Range: Zero(s): y-intercept(s): 110 Chapter l Power Functions and Polynomial Functions

b. f(x) x 2 x 4 x y Degree: Domain: Range: Zero(s): y-intercept(s): c. f(x) x 4x 2 x 4 x y Degree: Domain: Range: Zero(s): y-intercept(s): Lesson.2 l Polynomial Functions 111

d. f(x) x 4 1x 2 6 x y Degree: Domain: Range: Zero(s): y-intercept(s): Problem Operations with Polynomial Functions For each pair of polynomial functions in Questions 1 through 4: a. Rewrite each polynomial in descending order. b. State the degree of each polynomial. c. Calculate f(x) g(x). d. Calculate f(x) g(x). 1. f(x) 2 x 5x 2 4x 5 4x; g(x) x 2 x 4 x 6 a. b. c. d. 2. f(x) x 6 7x 2 x 7 4; g(x) x 2 x 4 4x 5 12 a. b. c. d. 112 Chapter l Power Functions and Polynomial Functions

. f(x) 4x 2 8x 11x; g(x) x 2 x 4 x 6 a. b. c. d. 4. f(x) 5x 7x 7x 4 ; g(x) 6x 7x 4 5 a. b. c. d. 5. Multiply each pair of polynomial expressions. Then write the product in the standard form of a polynomial. Take Note Remember to multiply expressions, you can use the multiplication table or apply the distributive property. a. 2 x x 4 ; x x 2 4 b. 5x 2 4x; 5x 4 2 x 2 x Be prepared to share your methods and solutions. Lesson.2 l Polynomial Functions 11

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. Humps and Bumps Graphs of Polynomial Functions Objectives In this lesson you will: l Graph higher order polynomial functions. l Compare and contrast graphs of polynomial functions. Problem 1 Cubic Functions The general form of a cubic function is f(x) a x a 2 x 2 a 1 x a 0. 1. Graph f(x) x and g(x) x. How does the sign of the coefficient of the leading term affect the graph of a cubic function? Lesson. l Graphs of Polynomial Functions 115

2. Graph f(x) x, g(x) x 2x 2 5, and h(x) x 2 x. How does the value of the constant term affect the graph of a cubic function?. Graph f(x) x 8, g(x) x 5x 2 x 5, and h(x) x 2 x 2. a. Determine the number of zeros of each cubic function. b. Determine the values for the zeros of each function. 116 Chapter l Power Functions and Polynomial Functions

c. Can a cubic function have no zeros? Explain. Problem 2 Quartic Functions The general form of a quartic function is f(x) a 4 x 4 a x a 2 x 2 a 1 x a 0. 1. Graph f(x) x 4 and g(x) x 4. How does the sign of the coefficient of the leading term affect the graph of a quartic function? 2. Graph f(x) x 4 1, g(x) x 4 2 x 2 6, and h(x) x 4 4x 2x 2 2x. How does the value of the constant term affect the graph of a quartic function? Lesson. l Graphs of Polynomial Functions 117

. Graph f(x) x 4, g(x) x 4 8x 2 16, h(x) x 4 16x 2 and k(x) x 4 10x 2 9. a. Determine the number of zeros of each quartic function. b. Determine the values for the zeros of each function. c. Can a quartic function have no zeros? Explain. 118 Chapter l Power Functions and Polynomial Functions

Problem Quintic Functions The standard form of a quintic function is f(x) a 5 x 5 a 4 x 4 a x a 2 x 2 a 1 x a 0. 1. Graph f(x) x 5 and g(x) x 5. How does the sign of the coefficient of the leading term affect the graph of a quintic function? 2. How do you think the value of the constant term affects the graph of a quintic function?. How many zeros can a quintic function have? 4. Can a quintic function have no zeros? Lesson. l Graphs of Polynomial Functions 119

Problem 4 Higher Order Polynomial Functions The function f(x) is an even degree polynomial function. 1. Describe how the sign of the coefficient of the leading term affects the graph of f(x). 2. Describe how the value of the constant term affects the graph of f(x).. Can f(x) have no zeros? 4. How many zeros can f(x) have? The function g(x) is an odd degree polynomial function. 5. Describe how the sign of the coefficient of the leading term affects the graph of g(x). 120 Chapter l Power Functions and Polynomial Functions

6. Describe how the value of the constant term affects the graph of g(x). 7. Can g(x) have no zeros? 8. How many zeros can g(x) have? Be prepared to share your methods and solutions. Lesson. l Graphs of Polynomial Functions 121

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.4 Mirror Mirror on the Wall... Symmetry Objectives In this lesson you will: l Define axis symmetry. l Determine symmetry with respect to the y-axis. l Determine symmetry with respect to the x-axis. l Determine symmetry with respect to the origin. Key Terms l symmetric with respect to the y-axis l symmetric with respect to the x-axis l symmetric with respect to the origin Problem 1 Axis of Symmetry Take Note 1. Graph and label each quadratic function. Then, label the x-intercept(s), y-intercept, and vertex. Draw and label the axis of symmetry. For a quadratic function f(x) ax 2 bx c: 2a, f ( b The vertex is ( b 2a ) ). The equation of the axis of symmetry is x b 2a. a. f(x) x 2 9 Lesson.4 l Symmetry 12

b. g(x) 2 x 2 x 6 c. h(x) x 2 4x 4 2. Determine each value and plot the corresponding point on the graph in Question 1. a. f(x) x 2 9 i. f(1) ii. f( 1) iii. f(5) iv. f( 5) b. g(x) 2 x 2 x 6 i. g(1.75) ii. g( 2.25) iii. g(.25) iv. g(.75) 124 Chapter l Power Functions and Polynomial Functions

c. h(x) x 2 4x 4 i. h( ) ii. h( 1) iii. h(1) iv. h( 5). What do you notice about the value of each function for points that are equal distance from the axis of symmetry? 4. Prove if f(x) is a quadratic function of the form f(x) a x 2 bx c, then f ( b 2a d ) f ( b 2a d ) when d is a real number. Lesson.4 l Symmetry 125

Problem 2 Symmetry with Respect to the y-axis The graph of a function f is symmetric with respect to the y-axis if and only if f(x) f( x). The graph of the polynomial function f(x) x 2 9 has an axis of symmetry that coincides with the y-axis. So, the graph of f(x) is symmetric with respect to the y-axis. 1. Determine whether the graph of each polynomial function is symmetric with respect to the y-axis. Justify your answer algebraically. a. g(x) 2 x 2 x 6 b. f(x) x 4 8x 2 16 c. h(x) 2 x x 2 2 x 6 d. g(x) x 6 x 6 126 Chapter l Power Functions and Polynomial Functions

2. Determine whether the graph of each polynomial function shown is symmetric with respect to the y-axis. y 8 6 g(x) f(x) 4 2 h(x) 8 6 4 2 2 2 4 6 8 x 4 6 8. Can the graph of an odd degree polynomial function be symmetric with respect to the y-axis? Explain. Problem Symmetry with Respect to the x-axis The graph of a relation r is symmetric with respect to the x-axis if and only if r(x) r(x). 1. Can the graph of a function be symmetric with respect to the x-axis? Explain. 2. Determine whether the graph of each relation is symmetric with respect to the x-axis. Justify your answer algebraically. a. r(x) x Lesson.4 l Symmetry 127

b. r(x) x. Determine whether the graph of each relation shown is symmetric with respect to the x-axis. y 8 r 2 (x) 6 4 r 1 (x) 2 8 6 r (x) 4 2 2 4 2 4 6 8 x 6 8 Problem 4 Symmetry with Respect to the Origin The graph of a function f is symmetric with respect to the origin if and only if f(x) f( x). 1. Determine whether the graph of each polynomial function is symmetric with respect to the origin. Justify your answer algebraically. a. g(x) x 2 x 2 x 5 128 Chapter l Power Functions and Polynomial Functions

b. g(x) x 2 x 4 c. g(x) x 5 x 2. Determine whether the graph of each polynomial function shown is symmetric with respect to the origin. y 8 g(x) 6 4 2 h(x) f(x) 8 6 4 2 2 2 4 6 8 x 4 6 8. Can the graph of an even degree polynomial function be symmetric with respect to the origin? Explain. Be prepared to share your methods and solutions. Lesson.4 l Symmetry 129

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.5 Cloned Zeros? Zeros and Multiplicity Objectives In this lesson you will: l Determine the number of real zeros of polynomial functions. l Calculate the zero values of polynomial functions. l Determine the multiplicity of the zeros of polynomial functions. Key Term l multiplicity Problem 1 Linear and Quadratic Functions A linear function is a first-degree polynomial function. A quadratic function is a second-degree polynomial function. 1. Calculate the zero value of each linear function algebraically. Round your answer to the nearest hundredth if necessary. a. f(x) x 2 b. f(x) 2.x 7.6 2. Calculate the zero values of each quadratic function by factoring. a. f(x) x 2 x 2 b. f(x) x 2 9 Lesson.5 l Zeros and Multiplicity 11

. Calculate the zero values of each quadratic function using the quadratic formula. a. f(x) x 2 x 2 b. f(x) x 2 5x 7 4. Calculate the zero values of each quadratic function using a graphing calculator. a. f(x) x 2 2x 6 b. f(x) x 2 5x 12 12 Chapter l Power Functions and Polynomial Functions

5. How many real zeros can a linear function have? Explain. 6. How many real zeros can a quadratic function have? Explain. 7. Graph and label each quadratic function: f(x) x 2 4x 4; g(x) 4x 2 20x 25; h(x) x 2 18x 27 Lesson.5 l Zeros and Multiplicity 1

8. Calculate the zero value of each function. a. f(x) x 2 4x 4 b. g(x) 4x 2 20x 25 c. h(x) x 2 18x 27 9. Rewrite each function in factored form. a. f(x) x 2 4x 4 b. g(x) 4x 2 20x 25 c. h(x) x 2 18x 27 10. A quadratic function f(x) ax 2 bx c can be written in factored form as f(x) a(x r) 2. What do you know about the graph of this function and its zero value(s)? Explain. If a quadratic function can be factored as f(x) a(x r ) 2, then the zero r is a double zero and has a multiplicity of two. The multiplicity is determined by the exponent of the parentheses. So, f(x) a(x r) has a multiplicity of one, f(x) a(x r ) 2 has a multiplicity of two, f(x) a(x r ) has a multiplicity of three, and so on. 14 Chapter l Power Functions and Polynomial Functions

Problem 2 Zeros of Higher Degree Polynomial Functions 1. Graph and label the functions h(x) x 4x 2 x 4 and f(x) (x 4)(x 1)(x 1). 2. What must be true about h(x) and f(x)?. Prove that f(x) h(x) algebraically. 4. What are the zeros of h(x) and f(x)? 5. What is the multiplicity of each zero of h(x) and f(x)? Lesson.5 l Zeros and Multiplicity 15

6. Calculate the zero value of each function by factoring. State the multiplicity of each zero. a. f(x) x 4 8x 2 16 b. g(x) 4x 4 20x x 2 5x c. h(x) x 5 7x 4 10x 2x 11x 5 (Hint: (5, 0) is a zero) 16 Chapter l Power Functions and Polynomial Functions

7. Graph and label the function f(x) x 4 8x 2 16. Then, label each zero of the function. a. Describe the behavior of the graph at the zeros. b. Describe the behavior of the graph of a polynomial function at a zero with a multiplicity of two. c. How many times does this graph change direction? How many turning points does it have? 8. Graph and label the function g(x) 4x 4 20x x 2 5x. Then, label each zero of the function. a. Describe the behavior of the graph at the zeros. b. Describe the behavior of the graph of a polynomial function at a zero with a multiplicity of one. c. How many times does this graph change direction? How many turning points does it have? Lesson.5 l Zeros and Multiplicity 17

9. Graph and label the function h(x) x 5 7x 4 10 x 2 x 2 11x 5. Then, label each zero of the function. a. Describe the behavior of the graph at the zeros. b. Describe the behavior of the graph of a polynomial function at a zero with a multiplicity of three. c. How many times does this graph change direction? How many turning points does it have? 10. Sketch and label the graph of f( x) ( x r 1 )(x r 2 ) 2 (x r ) with 0 r 1 r 2 r. y 4 2 1 1 1 2 4 1 2 4 5 6 7 x 18 Chapter l Power Functions and Polynomial Functions

Problem Summarize Complete each table. Degree of Polynomial Function Odd nth degree Possible Number of Zeros Possible Number of Turning Points Even nth degree Multiplicity of a Zero Sketch of Behavior at Zero Corresponding Factors 1 2 Be prepared to share your methods and solutions. Lesson.5 l Zeros and Multiplicity 19

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.6 Domains, Ends, and Extremes Extrema, Domain & Range, and End Behavior Objectives In this lesson you will: l Identify and determine extreme points of polynomial functions. l Identify and determine end behavior of polynomial functions. l Determine the domain and range of polynomial functions. Key Terms l absolute minimum l absolute maximum l relative maximum l relative minimum l extreme points, extremum, or extrema l end behavior Problem 1 Extreme Points The absolute minimum of a polynomial function is the lowest value of the function. The absolute maximum of a polynomial function is the highest value of the function. 1. Graph and label the third degree polynomial functions f(x) x, f(x) x 4x 2 x and f(x) x 6x 2 8x 24. Lesson.6 l Extrema, Domain & Range, and End Behavior 141

2. Consider the graphs of the third-degree polynomial functions in Question 1 and other graphs of third-degree polynomial functions that you have seen. a. Do any of the graphs of third-degree polynomial functions have an absolute minimum or maximum? b. Is it possible for a third-degree polynomial function to have an absolute minimum or maximum? Explain. c. Do any of the graphs in Question 1 have points that are similar to a vertex? Explain.. Consider the graphs of the fourth-degree polynomial functions that you have seen. a. Do any of the graphs of fourth-degree polynomial functions have an absolute minimum or maximum? b. Do the graphs of fourth-degree polynomial functions have points that are similar to a vertex? Explain. 142 Chapter l Power Functions and Polynomial Functions

c. Do all fourth-degree polynomial functions have an absolute maximum or minimum? Explain. 4. Points that are similar to vertices are called relative maximums or relative minimums. Explain why. The set of absolute maximums, absolute minimums, relative maximums, and relative minimums is the set of extreme points, extremum, or extrema. 5. How many extrema can each polynomial function have? a. A quadratic function b. A cubic function c. A quartic function d. A polynomial function of degree n 6. What happens to the values of a polynomial function on either side of a minimum? Maximum? Calculating the extrema of many polynomial functions algebraically is beyond the scope of this course. However, a graphing calculator can be used to approximate the extrema. Lesson.6 l Extrema, Domain & Range, and End Behavior 14

7. Determine the zeros, the extrema, and the intervals of increase and decrease of each polynomial function using a graphing calculator. a. f(x) x(x 2)(x 2) b. f(x) (x 2) 2 (x ) 2 c. f(x) x 4 x 9x 2 x 10 8. Complete the table. Degree of Polynomial Function Odd nth degree Even nth degree Possible Number of Absolute Extrema Possible Number of Relative Extrema 144 Chapter l Power Functions and Polynomial Functions

Problem 2 End Behavior The end behavior is a description of the appearance of a graph to the extreme left and the extreme right. 1. Consider the graphs of the odd-degree polynomial functions that you have seen. a. Describe what happens at either end of the graph. Why? b. How does changing the sign of the leading coefficient change the end behavior? 2. Consider the graphs of the even-degree polynomial functions that you have seen. a. Describe what happens at either end of the graph. Why? b. How does changing the sign of the leading coefficient change the end behavior?. Will this end behavior of a polynomial function change if any of the other coefficients are much larger or smaller than the leading coefficient? Explain. 4. Describe the graph of each polynomial function without graphing. Include the number of zeros the number of extrema, and the end behavior. a. f(x) x 2 (x 2) 2 (x ) 2 Lesson.6 l Extrema, Domain & Range, and End Behavior 145

b. f(x) 2x(x 2)(x ) 2 (x 10) Problem Domains and Ranges Consider the graphs of all the polynomial functions that you have seen. 1. What is the domain of a polynomial function? Explain. 2. What is the range of an odd-degree polynomial function? Explain.. What is the range of an even-degree polynomial function? Explain. Be prepared to share your methods and solutions. 146 Chapter l Power Functions and Polynomial Functions

.7 Salinity Graphs of Polynomial Functions Objectives In this lesson you will: l Model data and situations with polynomial functions. l Use the characteristic of the graphs to match them to polynomial functions. Key Term l salinity Problem 1 Estuaries An estuary is a partially enclosed coastal body of water that has one or more rivers or streams flowing into it. Because of the nature of an estuary, the water's salinity varies frequently. Salinity is the measure of saltiness, or dissolved salt content in water. Salinity in an estuary changes due to location, tidal functions, seasonal weather changes, and volume of freshwater runoff. Typically, salinity levels are highest near the mouth of a river where ocean and freshwater converge and lowest upstream where fresh water flows. Ecologists routinely measure the estuary salinity because of its impact on plants, animals, and people. Too much salinity can reduce vegetation in surrounding areas. The table shown represents four different monitoring sites in North Carolina and their average salinity variance over 24 days. Lesson.7 l Graphs of Polynomial Functions 147

Time (days) East Cribbing Loosing Creek Research Creek Zeke s Basin Average 1 27.9 1 29.7 25. 28.48 2 27.9 0.8 29.1 25.6 28.5 28.2 0.9 29. 25.8 28.55 4 0.5 1.4 0.2 27.2 29.8 5 29.6 0. 29. 27.6 29.2 6 28. 0.2 29.4 26 28.78 7 27.9 0.1 29.1 24.7 27.95 8 27.9 0.2 29.1 24.9 28.0 9 28.6 1 29.2 26.2 28.75 10 0.1 1.4 29.9 28. 29.9 11 29.9 1. 0.2 27.9 29.8 12 0 1.1 29.9 28.2 29.8 1 29.5 1 29.6 27.5 29.4 14 29.5 1.2 29.8 27.4 29.48 15 29.5 1.2 29.4 27.4 29.8 16 29.4 1.1 29.8 27.4 29. 17 29.2 1.1 29.7 27.4 29.5 18 29.1 0. 29.7 27.8 29.2 19 29 0.9 29. 27.6 29.2 20 29 0.9 29.6 27.6 29.28 21 28.9 0.9 29.6 27.5 29.2 22 28.8 0.8 29.6 27.4 29.15 2 28.8 0.7 29.7 27.2 29.1 24 28.7 0.8 29.2 27. 29 1. Is it possible to identify any trends from the table? 2. What are the advantages and disadvantages of including the column for average salinity? 148 Chapter l Power Functions and Polynomial Functions

The following graph displays the data in the table. 4 y Salinity in Bodies of Water North Carolina Salinity (parts per thousand) 2 1 0 29 28 27 26 25 East Cribbing Loosing Creek Research Creek Zeke s Basin Average 24 x 0 5 10 15 20 25 Time (days). At which sites(s) does the salinity appear to remain fairly constant? 4. At which site(s) does the salinity appear to fluctuate? 5. Describe the salinity of each estuary after 12 days. Lesson.7 l Graphs of Polynomial Functions 149

Problem 2 Individual Estuaries The following graph displays the salinity of East Cribbing. 1 y Salinity of East Cribbing 0.5 Salinity (parts per thousand) 0 29.5 29 28.5 28 27.5 x 0 5 10 15 20 25 Time (days) 1. Connect the points with a smooth curve. Describe the shape of the curve to at least the fifteenth day. 2. Look at the points between the second and the eleventh day. What type of polynomial function would best model these points? Explain.. Look at the points after 15 days. What type of polynomial function would best model these points? Explain. 150 Chapter l Power Functions and Polynomial Functions

The following graph displays the average salinity. 0.5 y Average Salinity 0 Salinity (parts per thousand) 29.5 29 28.5 28 27.5 0 5 10 15 20 25 Time (days) 4. Connect the points with a smooth curve. Describe the shape of the curve to at least the fifteenth day. x 5. Look at all the points. What type of polynomial function would best model these points? Explain. 6. Look at the points after 15 days. What type of polynomial function would best model these points? Explain. Lesson.7 l Graphs of Polynomial Functions 151

The following graph displays the salinity of Loosing Creek. 1.6 y Salinity of Loosing Creek 1.4 Salinity (parts per thousand) 1.2 1 0.8 0.6 0.4 0.2 0 0 5 x 10 15 20 25 Time (days) 7. Connect the points with a smooth curve. Describe the shape of the curve to at least the fifteenth day. 8. Look at the y-scale of the graph. How does the y-scale impact the appearance of the salinity values? 9. Look at the points between the first and sixteenth day. What type of polynomial function would best model these points? Explain. 10. Look at the points after 15 days. What type of polynomial function would best model these points? Explain. 152 Chapter l Power Functions and Polynomial Functions

The following graph displays the salinity of Research Creek. 0.4 y Salinity of Research Creek 0.2 Salinity (parts per thousand) 0 29.8 29.6 29.4 29.2 29 0 5 10 15 20 25 Time (days) x 11. Connect the points with a smooth curve. Describe the shape of the curve. 12. Look at all the points. What type of polynomial function would best model these points? Explain. Lesson.7 l Graphs of Polynomial Functions 15

The following graph displays the salinity of Zeke s Basin. 28.5 y Salinity of Zeke s Basin 28 Salinity (parts per thousand) 27.5 27 26.5 26 25.5 25 24.5 x 0 5 10 15 20 25 Time (days) 1. Connect the points with a smooth curve. Describe the shape of the curve. 14. Look at all the points. What type of polynomial function would best model these points? Explain. Be prepared to share your methods and solutions. 154 Chapter l Power Functions and Polynomial Functions