3%2d3 rate of change and slope - waynesville.k12.mo.us c3 l3...this rate of change means there was...

Find the rate of change represented in each table or graph.

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62/87,21��To find the rate of change, use the coordinates (3, 6) and (0, 2).�

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So, the rate of change is .

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62/87,21��To find the rate of change, use the coordinates (3, í6) and (5, 2).�

� So, the rate of change is 4.

CCSS SENSE-MAKING

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3-3 Rate of Change and Slope

��CCSS SENSE-MAKING Refer to the graph below.

a. Find the rate of change of prices from 2006 to 2008. Explain the meaning of the rate of change. b. Without calculating, find a two±year period that had a greater rate of change than 2006±2008. Explain. c. Between which years would you guess the new stadium was built? Explain your reasoning.

62/87,21��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 two±year period that had a greater rate of change than 2006±2008 was 1998±2000. 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. �

Determine whether each function is linear. Write yes or no. Explain.




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The rate of change, , is constant. So, the function is linear.

x y Slope Rate of Change

±7 5 - -±4 4

±1 3

2 2

5 1

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� The rate of change is not constant. So, the function is not linear.


8 8 - - 12 5 16 3 20 0 24 ±2

Find the slope of the line that passes through each pair of points.



Find the slope of the line that passes through each pair of points.��(5, 3), (6, 9)

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So, the slope is 6.

��(±4, 3), (±2, 1)

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So, the slope is í1.

��(6, ±2), (8, 3)

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So, the slope is .

��(1, 10), (±8, 3)



��(1, 10), (±8, 3)

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So, the slope is .

��(±3, 7), (±3, 4)

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Dividing by 0 is undefined. So, the slope is undefined.

��(5, 2), (±6, 2)

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So, the slope is 0.

Find the value of r so the line that passes through each pair of points has the given slope.



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

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� So, the line goes through (±4, ±17).

��(5, 2), (±7, r), m =

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� So, the line goes through (±7, ±8).

Find the rate of change represented in each table or graph.eSolutions Manual - Powered by Cognero



��

62/87,21��To find the rate of change, use the coordinates (10, 3) and (5, 2).


��

62/87,21��To find the rate of change, use the coordinates (1, 15) and (2, 9).

So, the rate of change is í6.





��

62/87,21��To find the rate of change, use the coordinates (±3, 7) and (3, ±1).�

�




��

62/87,21��To find the rate of change, use the coordinates (±4, í6) and (4, ±2).�

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SPORTS



��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. �

62/87,21��To find the rate of change, use the coordinates (2004, 5545) and (2008, 6830). �

� So, the rate of change is 321.25. This rate of change means there was an average increase of 321.25 women per year competing in triathlons.

RETAIL



��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 7±year±old car.

62/87,21��a. To find a linear function, find the y-LQWHUFHSW�DQG�VORSH��8VH�WKH�FRRUGLQDWHV��DQG��)LQG�WKH�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.





��

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The rate of change, í1, is constant. So, the function is linear.


4 ±1 - -

2 1 í1

0 3 í1 ±2 5 í1

±4 7 í1

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±7 11 - - ±5 14 ±3 17 ±1 20 0 23 3



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±0.2 0.7 - - 0 0.4 ±0.25

0.2 0.1 ±1

0.4 0.3 2

0.6 0.6 1.5



��


�

The rate of change, , is constant. So, the function is linear.


- -

1

2

Find the slope of the line that passes through each pair of points.��(4, 3), (±1, 6)

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So, the slope is .

��(8, ±2), (1, 1)



��(8, ±2), (1, 1)

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So, the slope is .

��(2, 2), (±2, ±2)

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So, the slope is 1.

��(6, ±10), (6, 14)

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��(5, ±4), (9, ±4)



��(5, ±4), (9, ±4)

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So, the slope is 0.

��(11, 7), (±6, 2)

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So, the slope is .

��(±3, 5), (3, 6)

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So, the slope is .

��(±3, 2), (7, 2)



��(±3, 2), (7, 2)

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So, the slope is 0.

��(8, 10), (±4, ±6)

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So, the slope is .

��(±8, 6), (±8, 4)

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��(±12, 15), (18, ±13)



��(±12, 15), (18, ±13)

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So, the slope is .

��(±8, ±15), (±2, 5)

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So, the slope is .

Find the value of r so the line that passes through each pair of points has the given slope.



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

62/87,21��

� So, the line goes through (±2, 66).

��(r, ±5), (3, 13), m = 8

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� So, the line goes through .



��(3, 5), (±3, r), m =

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�

So, the line goes through (±3, ).

��(±2, 8), (r, 4), m =

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� So, the line goes through (6, 4).

CCSS TOOLS Use a ruler to estimate the slope of each object.



CCSS TOOLS Use a ruler to estimate the slope of each object.��Refer to photo on Page 178.

62/87,21��Sample answer: about ±0.5

Use WR�FDOFXODWH�WKH�VORSH��%HFDXVH�LW�VORSHV�GRZQ�IURP�OHIW�WR�ULJKW��WKH�VORSH�LV�QHJDWLYH�

��Refer to photo on Page 178.

62/87,21��Sample answer: about ±1

Use WR�FDOFXODWH�WKH�VORSH��

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?

62/87,21��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 .





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So, the slope is .

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So, the slope is .

��MULTIPLE REPRESENTATIONS In this problem, you will investigate why the slope of a



��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 $GG�VHJPHQWV�WR�IRUP�ULJKW�WULDQJOHV�ABC and A¶B¶C¶�with right angles C and C¶. Describe and and and � F��9(5%$/�How are triangles ABC and A¶B¶C¶�related? What does that imply for the slope between any two distinct points on line ?

62/87,21��a.�'UDZ�D�VODQWHG�OLQH�DQG�ODEHO�LW�� . 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 A¶�to form a right angle at a point C¶.

� � E�� and �DUH�KRUL]RQWDO��DQG� and �DUH�YHUWLFDO� � c��6DPSOH�DQVZHU��7ULDQJOHV�ABC�DQG��A¶B¶C¶�are similar because line is a transversal of parallel segments

�DQG� , so ӘABC ӘA¶B¶C¶ since corresponding angles of parallel lines are congruent. ӘC ӘC¶�because all right angles are congruent. Therefore, by AA Similarity, ǻABC a ǻA¶B¶C¶. 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 NBA¶s Milwaukee Bucks. �

Season PPG 1 2.2 2 11.4 3 15.1 4 21.7 5 23.0 6 25.4 7 26.7 8 22.7 eSolutions Manual - Powered by Cognero


� 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 Redd¶s 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.

8 22.7 9 21.2

62/87,21��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 Redd¶s PPG increased the fastest, look for the steepest slope or find the slope and locate the one with the highest rate of change. . �


1 2.2 - - 2 11.4 9.2

3 15.1 3.7

4 21.7 6.6

5 23.0 1.3

6 25.4 2.4

7 26.7 1.3



� From Season 1 to Season 2, the line is the steepest. So, that is the period Michael Redd¶s PPG increased the fastest. � c.�8VH�WKH�JUDSK�DQG�WKH�WDEOH�WR�DQDO\]H��

� 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.

8 22.7 ±4

9 21.2 ±1.5


1 2.2 - - 4 21.7 6.5

7 26.7 1.7

9 21.2 ±2.8

REASONING



��REASONING Why does the Slope Formula not work for vertical lines? Explain.

62/87,21��&RQVLGHU�WKH�IROORZLQJ�JUDSK��

� Let (x1, y 1) = (±2, ±2) and (x2, y 2) = (4, ±��)LQG�WKH�VORSH��

� Division by 0 is undefined. Thus, the slope is undefined. This occurs when the difference in the x±values for vertical OLQHV�LV�DOZD\V��ZKLFK�LV�LQ�WKH�GHQRPLQDWRU�RI�WKH�VORSH�IRUPXOD��

OPEN ENDED



��OPEN ENDED Use what you know about rate of change to describe the function represented by the table.�

62/87,21��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.



��CHALLENGE Find the value of d so the line that passes through (a, b) and (c, d) has a slope of .

62/87,21��

Set the slope formula equal to �DQG�VROYH�IRU�d using the coordinates (a, b) and (c, d).

�

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So, the value of d so the line that passes through (a, b) and (c, d) has a slope of �LV� .

��WRITING IN MATH Explain how the rate of change and slope are related and how to find the slope of a line.

62/87,21��Students¶�answers 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 y±coordinates to the change in the x±coordinates.

CCSS ARGUMENTS a a



��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.

�

62/87,21��Luna is correct because she found the change in y divided by the change in x. Kyle is incorrect because he found thechange in x divided by the change in y . � 6ORSH�LV�GHILQHG�DV� .

��The cost of prints from an online photo processor is given by C(p ) = 29.99 + 0.13p . $29.99 is the cost of the membership, and p is the number of 4±inch by 6±inch 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

62/87,21��The slope is 0.13. It represents the cost per print, or choice A. Choice B is the cost of membership or $29.99. ChoiceC is the cost of the membership and one print or $29.99 + $0.13 = $30.12. Choice D is the number of prints or p . So, the correct choice is A.

��Danita bought a computer for $1200 and its value depreciated linearly. After 2 years, the value was $250. What waseSolutions Manual - Powered by Cognero


��Danita bought a computer for $1200 and its value depreciated linearly. After 2 years, the value was $250. What wasthe amount of yearly depreciation? F $950 G $475 H $250 J $225

62/87,21��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 choiceis G.

��SHORT RESPONSE The graph represents how much the Wright Brothers National Monument charges visitors. How much does the park charge each visitor?

62/87,21��The graph shows a constant rate of change, so the cost per visitor is constant. Look at the cost for one visitor. Find 1visitor 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.

PROBABILITY



��PROBABILITY At a gymnastics camp, 1 gymnast is chosen at random from each team. The Flipstars GymnasticsTeam consists of 5 eleven±year±olds, 7 twelve±year±olds, 10 thirteen±year±olds, and 8 fourteen±year±olds. What isthe probability that the age of the gymnast chosen is an odd number?

A

B

C

D

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�

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

62/87,21��The related function is y = 3x + 18.

The graph intersects the x-axis at í6. So, the solution is í6.

��8x ± 32 = 0



��8x ± 32 = 0

62/87,21��The related function is y = 8x í 32.

The graph intersects the x-axis at 4. So, the solution is 4.

��0 = 12x ± 48

62/87,21��The related function is y = 12x í 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.

��

62/87,21��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 (í2, 0) and the y-axis at (0, 2). So the x-intercept is í2 and the y-intercept is 2.



��

62/87,21��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 í1. 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 wishesto go to the dance, write and solve an equation that would represent the least expensive price p of their tickets.

62/87,21��Let p be the price, x be the number of couples¶�tickets 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.��

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3%2d3 rate of change and slope - waynesville.k12.mo.us c3 l3...this rate of change means there was...

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