3-3 Rate of Change and Slope. Find the rate of change represented in each table or graph.

Transcription

1 Find the rate of change represented in each table or graph. To find the rate of change, use the coordinates (3, 6) and (0, 2). So, the rate of change is. To find the rate of change, use the coordinates (3, 6) and (5, 2). So, the rate of change is 4. esolutions Manual - Powered by Cognero Page 1

2 CCSS SENSE-MAKING Refer to the graph below. a. Find the rate of change of prices from 2006 to Explain the meaning of the rate of change. b. Without calculating, find a twoyear period that had a greater rate of change than Explain. c. Between which years would you guess the new stadium was built? Explain your reasoning. a. To find the rate of change, use the coordinates (2008, 28.73) and (2006, 26.66). So, the rate of change is 1.035, which means there was an average increase in ticket price of $1.035 per year. b. Sample answer: The twoyear period that had a greater rate of change than was There was a steeper segment, which means a greater rate of change. c. Sample answer: I would guess that the new stadium was built between 1998 and 2000, because the ticket prices show a sharp increase. esolutions Manual - Powered by Cognero Page 2

3 Determine whether each function is linear. Write yes or no. Explain. x y Slope Rate of Change The rate of change,, is constant. So, the function is linear. x y Slope Rate of Change The rate of change is not constant. So, the function is not linear. esolutions Manual - Powered by Cognero Page 3

4 Find the slope of the line that passes through each pair of points. (5, 3), (6, 9) So, the slope is 6. (4, 3), (2, 1) So, the slope is 1. (6, 2), (8, 3) So, the slope is. (1, 10), (8, 3) esolutions Manual - Powered by Cognero Page 4

5 (1, 10), (8, 3) So, the slope is. (3, 7), (3, 4) Dividing by 0 is undefined. So, the slope is undefined. (5, 2), (6, 2) So, the slope is 0. esolutions Manual - Powered by Cognero Page 5

6 Find the value of r so the line that passes through each pair of points has the given slope. (4, r), (8, 3), m = 5 So, the line goes through (4, 17). (5, 2), (7, r), m = So, the line goes through (7, 8). esolutions Manual - Powered by Cognero Page 6

7 Find the rate of change represented in each table or graph. To find the rate of change, use the coordinates (10, 3) and (5, 2). So, the rate of change is. To find the rate of change, use the coordinates (1, 15) and (2, 9). So, the rate of change is 6. esolutions Manual - Powered by Cognero Page 7

8 Find the rate of change represented in each table or graph. To find the rate of change, use the coordinates (3, 7) and (3, 1). So, the rate of change is. esolutions Manual - Powered by Cognero Page 8

9 To find the rate of change, use the coordinates (4, 6) and (4, 2). So, the rate of change is. esolutions Manual - Powered by Cognero Page 9

10 SPORTS What was the annual rate of change from 2004 to 2008 for women participating in college lacrosse? Explain the meaning of the rate of change. To find the rate of change, use the coordinates (2004, 5545) and (2008, 6830). So, the rate of change is This rate of change means there was an average increase of women per year competing in triathlons. esolutions Manual - Powered by Cognero Page 10

11 RETAIL The average retail price in the spring of 2009 for a used car is shown in the table below. a. Write a linear function to model the price of the car with respect to age. b. Interpret the meaning of the slope of the line. c. Assuming a constant rate of change, predict the average retail price for a 7yearold car. a. To find a linear function, find the y- slope, or rate of change. Find the y-intercept. So, a linear function to model the price of the car with respect to age is p = 1221t + 19,820. b. The slope of $1221 represents how much the car value depreciates by each year. c. So, the average retail price for a 7-year-old car is $11,273. esolutions Manual - Powered by Cognero Page 11

12 Determine whether each function is linear. Write yes or no. Explain. x y Slope Rate of Change The rate of change, 1, is constant. So, the function is linear. x y Slope Rate of Change The rate of change is not constant. So, the function is not linear. esolutions Manual - Powered by Cognero Page 12

13 x y Slope Rate of Change The rate of change is not constant. So, the function is not linear. esolutions Manual - Powered by Cognero Page 13

14 Rate of x y Slope Change The rate of change,, is constant. So, the function is linear. Find the slope of the line that passes through each pair of points. (4, 3), (1, 6) So, the slope is. (8, 2), (1, 1) esolutions Manual - Powered by Cognero Page 14

15 (8, 2), (1, 1) So, the slope is. (2, 2), (2, 2) So, the slope is 1. (6, 10), (6, 14) Dividing by 0 is undefined. So, the slope is undefined. (5, 4), (9, 4) esolutions Manual - Powered by Cognero Page 15

16 (5, 4), (9, 4) So, the slope is 0. (11, 7), (6, 2) So, the slope is. (3, 5), (3, 6) So, the slope is. (3, 2), (7, 2) esolutions Manual - Powered by Cognero Page 16

17 (3, 2), (7, 2) So, the slope is 0. (8, 10), (4, 6) So, the slope is. (8, 6), (8, 4) Dividing by 0 is undefined. So, the slope is undefined. (12, 15), (18, 13) esolutions Manual - Powered by Cognero Page 17

18 (12, 15), (18, 13) So, the slope is. (8, 15), (2, 5) So, the slope is. esolutions Manual - Powered by Cognero Page 18

19 Find the value of r so the line that passes through each pair of points has the given slope. (12, 10), (2, r), m = 4 So, the line goes through (2, 66). (r, 5), (3, 13), m = 8 So, the line goes through. esolutions Manual - Powered by Cognero Page 19

20 (3, 5), (3, r), m = So, the line goes through (3, ). (2, 8), (r, 4), m = So, the line goes through (6, 4). esolutions Manual - Powered by Cognero Page 20

21 CCSS TOOLS Use a ruler to estimate the slope of each object. Refer to photo on Page 178. Sample answer: about 0.5 Use Refer to photo on Page 178. Sample answer: about 1 Use Because it slopes up from right to left, the slope is negative. DRIVING When driving up a certain hill, you rise 15 feet for every 1000 feet you drive forward. What is the slope of the road? The rise is 15 feet and the run is 1000 feet. Because you are driving up a hill, the slope is positive. So, the slope of the road is. esolutions Manual - Powered by Cognero Page 21

22 Find the slope of the line that passes through each pair of points. So, the slope is. Dividing by 0 is undefined. So, the slope is undefined. esolutions Manual - Powered by Cognero Page 22

23 So, the slope is. In this problem, you will investigate why the slope of a esolutions Manual - Powered by Cognero Page 23

24 MULTIPLE REPRESENTATIONS In this problem, you will investigate why the slope of a line through any two points on that line is constant. a. VISUAL Sketch a line that contains points A, B, A, and B on a coordinate plane. b. GEOMETRIC ABC and ABCwith right angles C and C. Describe and and and How are triangles ABC and ABCrelated? What does that imply for the slope between any two distinct points on line? a.. Place four points on the line and label them from left to right A, B, A, and B. Draw a line straight down from B and straight across from A to form a right angle and label the vertex of the angle C. Repeat the same process from B and Ato form a right angle at a point C. and and cabcabcare similar because line is a transversal of parallel segments, so ABC ABC since corresponding angles of parallel lines are congruent. C C because all right angles are congruent. Therefore, by AA Similarity, ABC ABC. The ratios formed by corresponding sides of similar triangles are equal. The slope of a line is defined as. So, the slope of line through points A and B, is equal to, the slope of line through points A and B. BASKETBALL The table shown below shows the average points per game (PPG) Michael Redd has scored in each of his first 9 seasons with the NBAs Milwaukee Bucks. Season PPG esolutions Manual - Powered by Cognero Page 24

25 a. Make a graph of the data. Connect each pair of adjacent points with a line. b. Use the graph to determine in which period Michael Redds PPG increased the fastest. Explain your reasoning. c. Discuss the difference in the rate of change from season 1 through season 4, from season 4 through season 7, from season 7 through season 9. a. Graph the season on the x-axis and the points per game on the y-axis. Then, plot the ordered pairs from the table and connect the points with straight lines. b. To find the period that Michael Redds PPG increased the fastest, look for the steepest slope or find the slope and locate the one with the highest rate of change.. x y Slope Rate of Change esolutions Manual - Powered by Cognero Page 25

26 From Season 1 to Season 2, the line is the steepest. So, that is the period Michael Redds PPG increased the fastest. c. Rate of x y Slope Change The rate of change was much more dramatic or steeper in the first four years, it leveled off the next three seasons, and was negative and steeper the last two seasons. esolutions Manual - Powered by Cognero Page 26

27 REASONING Why does the Slope Formula not work for vertical lines? Explain. Let (x 1, y 1 ) = (2, 2) and (x 2, y 2 ) = (4, Division by 0 is undefined. Thus, the slope is undefined. This occurs when the difference in the xvalues for vertical esolutions Manual - Powered by Cognero Page 27

28 OPEN ENDED Use what you know about rate of change to describe the function represented by the table. To find the rate of change, use the coordinates (4, 9.0) and (6, 13.5). So, the rate of change is inches of growth per week. esolutions Manual - Powered by Cognero Page 28

29 CHALLENGE Find the value of d so the line that passes through (a, b) and (c, d) has a slope of. Set the slope formula equal to d using the coordinates (a, b) and (c, d). So, the value of d so the line that passes through (a, b) and (c, d) has a slope of. WRITING IN MATH Explain how the rate of change and slope are related and how to find the slope of a line. Studentsanswers may vary. Sample answer: The slope and rate of change are ratios. Rate of change is a ratio that describes how much one quantity changes with respect to a change in another quantity. The slope of a line is the ratio of the change in the ycoordinates to the change in the xcoordinates. esolutions Manual - Powered by Cognero Page 29

30 CCSS ARGUMENTS Kyle and Luna are finding the value of a so the line that passes through (10, a) and (2, 8) has a slope of. Is either of them correct? Explain. Luna is correct because she found the change in y divided by the change in x. Kyle is incorrect because he found the change in x divided by the change in y.. The cost of prints from an online photo processor is given by C(p ) = p. $29.99 is the cost of the membership, and p is the number of 4inch by 6inch prints. What does the slope represent? A cost per print B cost of the membership C cost of the membership and 1 print D number of prints The slope is It represents the cost per print, or choice A. Choice B is the cost of membership or $ Choice C is the cost of the membership and one print or $ $0.13 = $ Choice D is the number of prints or p. So, the correct choice is A. esolutions Danita Manual bought - Powered a computer by Cognero for $1200 and its value depreciated linearly. After 2 years, the value was $250. What Page was30

31 Danita bought a computer for $1200 and its value depreciated linearly. After 2 years, the value was $250. What was the amount of yearly depreciation? F $950 G $475 H $250 J $225 Since the deprecation is linear, the computer value dropped the same amount every year. Over 2 years it dropped from $1200 to $250, or $950. To find the amount for one year, divide $950 by 2, which is $475. So, the correct choice is G. SHORT RESPONSE The graph represents how much the Wright Brothers National Monument charges visitors. How much does the park charge each visitor? The graph shows a constant rate of change, so the cost per visitor is constant. Look at the cost for one visitor. Find 1 visitor on the x-axis. Move your finger up to the corresponding point, then left to the y-axis. There, you see a 4. So, the cost for 1 visitor is $4. esolutions Manual - Powered by Cognero Page 31

32 PROBABILITY At a gymnastics camp, 1 gymnast is chosen at random from each team. The Flipstars Gymnastics Team consists of 5 elevenyearolds, 7 twelveyearolds, 10 thirteenyearolds, and 8 fourteenyearolds. What is the probability that the age of the gymnast chosen is an odd number? A B C D The probability of choosing a girl with an age that is an odd number is. So the correct choice is C. Solve each equation by graphing. 3x + 18 = 0 The related function is y = 3x The graph intersects the x-axis at 6. So, the solution is = 0 esolutions Manual - Powered by Cognero Page 32

33 8x 32 = 0 The related function is y = 8x 32. The graph intersects the x-axis at 4. So, the solution is 4. 0 = 12x 48 The related function is y = 12x 48. The graph intersects the x-axis at 4. So, the solution is 4. Find the x- and y-intercepts of the graph of each linear function. The x-coordinate of the point at which the graph of an equation crosses the x-axis is the x-intercept. The y- coordinate of the point at which the graph of an equation crosses the y-axis is the y-intercept. The graph crosses the x-axis at (2, 0) and the y-axis at (0, 2). So the x-intercept is 2 and the y-intercept is 2. esolutions Manual - Powered by Cognero Page 33

34 The x-intercept is the point where the line crosses the x-axis. The y-value at that point is 0. So, the x-intercept is 1. The y-intercept is the point where the line crosses the y-axis. The x-value at that point is 0. So, the y-intercept is 2. HOMECOMING Dance tickets are $9 for one person and $15 for two people. If a group of seven students wishes to go to the dance, write and solve an equation that would represent the least expensive price p of their tickets. Let p be the price, x be the number of couplestickets and y be the number of single tickets. If seven students are going, then they can form three couples with one student left over. So, the least expensive price p of their tickets is $54. Find each quotient. esolutions Manual - Powered by Cognero Page 34

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