Write in simplest form. New Vocabulary rate of change

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1 -. Plan bjectives To find rates of change from tables and graphs To find slope Eamples Finding Rate of Change Using a Table Finding Rate of Change Using a Graph Finding Slope Using a Graph Finding Slope Using Points Horizontal and Vertical Lines - What You ll Learn To find rates of change from tables and graphs To find slope... And Wh To find the rate of change of an airplane s altitude, as in Eample Rate of Change and Slope Check Skills You ll Need G for Help Lessons - and - Evaluate each function rule for Write in simplest form () 7. 7 () New Vocabular rate of change slope Math Background Rate of change is also known as grade or pitch, or rise over run. Change is often smbolized in mathematics b delta for which the smbol is this Greek letter:. Some calculators use in their TABLE menu. More Math Background: p. C Lesson Planning and Resources See p. E for a list of the resources that support this lesson. Bell Ringer Practice Check Skills You ll Need For intervention, direct students to: Relations and Functions Lesson -: Eample Etra Skills and Word Problem Practice, Ch. Subtracting Rational Numbers Lesson -: Eample Etra Skills and Word Problem Practice, Ch. Part Finding Finding Rates Rates of Change of Change Vocabular Tip A rate is a comparison of two quantities measured in different units. The diagram at the right shows the side view of a ski lift.. What is the vertical change from A to B? From B to C? From C to D? ; ; 8 Chapter Linear Equations and Their Graphs. What is the horizontal change from A to B? From B to C? From C to D? ; ; Activit: Eploring Rate of Change. Find the ratio of the vertical change to the horizontal 8 change for each section of Horizontal Position (feet) the ski lift. ; ;. Which section is the steepest? How does the ratio for that section compare to the ratios of the other sections? B to C; it has the highest ratio. In the graph above, AB and BC have different rates of change. Rate of change allows ou to see the relationship between two quantities that are changing. If one quantit depends on the other, then the following is true. rate of change Vertical Position (feet) 8 A change in the dependent variable change in the independent variable B C D 8 Special Needs L Have students describe the slope of a model ramp in terms of an measurements the can make. Discuss how these measurements can be used to describe the rate at which the vertical dimension changes in relationship to the horizontal dimension. learning stle: tactile Below Level L Students ma be confused about which point to label (, ) when using the slope formula. Recalculate the slopes in Eample to show that labeling does not affect the slope. learning stle: verbal

2 Cost of Renting a Computer Number of Das Rental Charge $ $7 $9 $ $ Finding Rate of Change Using a Table Business For the data at the left, is the rate of change for each pair of consecutive das the same? What does the rate of change represent? change in cost rate of change change in number of das Cost depends on the number of das The rate of change for each consecutive pair of das is. The rate of change is the same for all the data. It costs $ for each da a computer is rented after the first da. a. Find the rate of change using Das and. Critical Thinking Does finding the rate of change for just one pair of das mean that the rate of change is the same for all the data? Eplain. No; the rate of change for each consecutive pair of das does not have to be the same. The graphs of all the ordered pairs (number of das, cost) in Eample lie on a line as shown at the right. So, the data are linear. You can use a graph to find a rate of change. Recall that the independent variable is plotted on the horizontal ais and the dependent variable is plotted on the vertical ais. vertical change change in the dependent variable rate of change horizontal change change in the independent variable Cost (dollars) 8 Das. Teach Guided Activit Activit Since the segments are neither vertical nor horizontal, some students ma need help understanding the phrases vertical change and horizontal change. Additional Eamples For the data in the table, is the rate of change for each pair of consecutive mileage amounts the same? Yes Fee for Miles Driven Miles Fee $ $ $ $ Finding Rate of Change Using a Graph Airplane Altitude The graph shows the altitude Airplane Landing of an airplane as it comes in for a landing. Find the rate of change. Eplain what this rate of change means. (, ) rate of vertical change d change in altitude change (8, ) horizontal change d change in time 8 Time (seconds) 8 Use two points. Divide the vertical change b the horizontal change. 8 Simplif. The rate of change is 8. The airplane descends 8 feet each second. Find the rate of change of the data in the graph. Altitude (feet) A Moving Automobile Time (hours) mi/h Distance Traveled (miles) Lesson - Rate of Change and Slope 9 Advanced Learners L Draw a horizontal line on a grid. Lead students in a discussion of how the slope of that line changes as it is rotated counterclockwise. learning stle: verbal English Language Learners ELL Man students confuse the terms vertical and horizontal. Ask students to imagine being in a place where all the can see is sk and either land or water. Relate the boundar between the sk and the land or the water (Horizon) to the term horizontal. learning stle: verbal 9

3 Additional Eamples Below is a graph of the distance traveled b a motorccle from its starting point. Find the rate of change. Eplain what this rate of change means. m/s; the motorccle travels meters each second Motorccle Ride 8 7 Distance (m) Math Tip The idea of a rise being negative is difficult for some students to understand. Point out that a fall is a negative rise. 8 8 Time (sec) Visual Learners Remind students to read coordinates (, ) as sub, sub. Some students ma have difficult writing the coordinates in the correct place in the formula. Have them circle the -coordinates, and bo the -coordinates. Then draw arrows going the same direction between the -coordinates and between the -coordinates to help keep the order correct. Use the following as an eample: (, ) (, 7 ) Part Finding Slope Real-World Connection The grade of a road is the ratio of rise to run epressed as a percent. For eample, a road with % grade is at a 8 angle with level ground. nline Visit: PHSchool.com Web Code: ate-77 Some roads are steeper than others. A steeper road has a greater rate of change. The slope of a line is its rate of change. vertical change slope rise horizontal change run Finding Slope Using a Graph Find the slope of each line. a. right units slope rise slope rise run run () () The slope of the line is. The slope of the line is Find the slope of each line. a. (, ) (, ) (, ) (, ) You can use an two points on a line to find its slope. You use subscripts to distinguish between two points. In the diagram, (, ) are the coordinates of P, and (, ) are the coordinates of Q. To find the slope of ou can use the following formula. * PQ ), up units (, ) (, ) (, ) down units right units (, ). Q(, ) P(, ) Ke Concepts Formula Slope slope rise where -, run Chapter Linear Equations and Their Graphs

4 A B C D E A B C D E A B C D E A B C D E A B C D E B C D E Test-Taking Tip To set up the subtraction in the slope formula, think of moving from the coordinates of B to the coordinates of A. A(, ) 7 ( ) B(, 7) Keep in mind that when calculating slope, the -coordinate ou use first in the denominator must belong to the same ordered pair as the -coordinate ou use first in the numerator. Finding Slope Using Points Gridded Response Find the slope of the line through A(-, ) and B(, 7). slope 8 Simplif. * ) The slope of AB is or 8. Fill in the grid with /. Find the slope of the line through each pair of points. d b a. C(, ) and D(, 7) P(-, ) and Q(, -) c. M(a, b) and N(c, d) c a You can also analze the graphs of horizontal and vertical lines. The net eample shows wh the slope of a horizontal line is, and the slope of a vertical line is undefined. Horizontal and Vertical Lines Find the slope of each line. a. slope (, ) Substitute (, ) for (, ) (, ) and (, ) for (, ). Simplif. The slope of the horizontal line is. slope (, ) () Simplif. (, ) Find the slope of each line. a. 7 () Substitute (, 7) for (, ) and (, ) for (, ). Substitute (, ) for (, ) and (, ) for (, ). Division b zero is undefined. So, the slope of the vertical line is undefined. / / / Lesson - Rate of Change and Slope undefined Resources Dail Notetaking Guide - L Dail Notetaking Guide - Adapted Instruction L Additional Eamples Find the slope of each line. a. ; a. Find the slope of the line through E(, -) and F(-, -). Find the slope of each line. a. ; undefined a. (, ) down units (, ) right units right unit up units (, ) (, ) (, ) Closure (, ) (, ) (, ) Ask: What are two was to find the slope of a line? Count the units of rise and run between two points on a graph of the line, Find the difference of the -coordinates, and, the difference of the - coordinates and write the quotient of the difference of the -coordinates divided b the difference of the -coordinates.

5 . Practice Assignment Guide A B -, 7-9,, A B 7-, -9, -, - C Challenge -7 Test Prep 7-7 Mied Review 7-87 Homework To check students understanding of ke skills and concepts, go over Eercises,,,, 8. Teaching Tip Eercises Discuss with students which is the independent variable and which is the dependent variable. Error Prevention! Eercise Students ma think the rate of change is. Caution them to look carefull at the scales of the graph. Practice and Problem Solving Eamples, (page 9) The following summarizes what ou have learned about slope. Ke Concepts Summar Slopes of Lines EXERCISES A G Practice b Eample for Help The rate of change is constant in each table and graph. Find the rate of change. Eplain what the rate of change means for each situation... Time Temperature ; the (hours) (ºF) temperature increases F each 7 hour. 7 A line with positive slope slants upward from left to right. A line with a slope of is horizontal. Cost People (dollars) A line with negative slope slants downward from left to right. A line with an undefined slope is vertical. For more eercises, see Etra Skill and Word Problem Practice..9; the cost per person is $.9. GPS Enrichment Guided Problem Solving Reteaching Adapted Practice Practice Name Class Date Practice - Find the slope of each line.... Rate of Change and Slope... Find the slope of the line that passes through each pair of points. 7. (, ), (, ) 8. (7, ), (, ) 9. (, ), (, ). (-, ), (, -). (, ), (, 7). (-, -), (, ). (-, -), (, -). (, -), (, 9). (, ), (8, -) L L L L L Real-World Connection A jump from, feet gives a skdiver about seconds of free fall at more than mi/h.. A Tank of Gas gal/mi. Electricit for TV Use Hours of Use Miles Traveled ; there are lb of carbon emissions for ever h of television use.. Descent of a Skdiver. Price of regano Weight (ounces) Time (seconds) ; the cost of oregano is $ for ; the skdiver descends ft/s. ounces. Fuel in Tank (gallons) Height After Parachute pens (feet) Carbon Emissions (pounds) Cost (dollars) Emissions: Generating Pearson Education, Inc. All rights reserved. Find the rate of change. Eplain what the rate of change means for each situation.. Points Scored 7. Distance Sound 8. for -point Baskets Travels in Air Time (seconds) Number of Baskets Points Scored Distance (miles) Find the slope of the line that passes through each pair of points. 9. (, ), (, 7). (-, ), (, -). (-, ), (, -). (, ), (, -). (, -), (, -). (, ), (, ). (, 7), (, ). (7, 9), (, 9) 7. (-, -), (-, ) Speed (ft/s) Speed 7 Time (seconds) Chapter Linear Equations and Their Graphs

6 Eample (page ) Eample (page ) Find the slope of each line Find the slope of the line that passes through each pair of points.. (, ), (, ). (, ), (, ). (, 8), (8, ). (-, ), (, -). (-, ), (, -). (-, ), (, -) Eercise Suggest to students that the substitute (, ) for (, ). Careers Eercise Encourage students to find other was construction workers use slope, such as pitch of a roof or slant of a drivewa. Ask students wh construction workers need to know the slope of these items in order to build them. Eample (page ). (-8, ), (, ) 9 7. (, ), (, ) 8. (-, -), (-9, ) 9. (, ), (, ). (-7, ), (7, 8). (, -), (, -) State whether the slope is zero or undefined... undefined pages. K 7. Eercises B Appl Your Skills. (, ), (-, ). (, ), (, -). Q, R, (-, ) undefined undefined Find the rate of change for each situation. M 9 7. A bab is 8 in. long at birth and 7 in. long at ten months. in./month 8. The cost of group museum tickets is $8 for four people and $78 for ten people. $/person 9. You drive mi in one hour and mi in four hours. mi/hr Find the slope of the line passing through each pair of points.. (-7, ), (7, 8). Q, R, R. (,.), (-,.)., 8R, (, -). Q, R,(-, ). (.,.), (, -.) undefined Through the given point, draw the line with the given slope. 9. See margin.. K(, ) 7. M(, ) 8. Q(-, ) 9. R(, -) slope - slope slope slope Q R Height (in.) Plant Growth A B C Time (weeks). a. Biolog Which line in the graph at the left is the steepest? C During the -week period, which plant had the greatest rate of change? The least rate of change? How do ou know? C greatest; A least; the slope. a. Find the slope of the line through A(, -) and B(, -) using A for (, ) and B for (, ). Find the slope of the line in part (a) using B for (, ) and A for (, ). c. Critical Thinking Eplain wh it does not matter which point ou use for (, ) and which point ou use for (, ) when ou calculate a slope. See margin. c. Answers ma var. Sample: Lesson - Rate of Change and Slope

7 . Assess & Reteach Lesson Quiz G nline Homework Help Visit: PHSchool.com Web Code: ate-. Construction An etension ladder has a label that sas, Do not place base of GPS ladder less than ft from the vertical surface. What is the greatest slope possible if the ladder can safel etend to reach a height of ft? f 8 ft? ;. Writing If two points on a line have positive coordinates, is the slope necessaril positive? Eplain. See margin.. Find the rate of change for the data in the table. $ Tickets Cost per ticket $7. Find the rate of change for the data in the graph. calories per hour Calories Burned. Find the slope of the line. (, ). Find the slope of the line through (, -) and (-, ). 7. State whether the slope is zero or undefined. a. undefined; a. $9 7 $ 8 $ 8 (, ) Calories Burned doing Aerobics Time (hours) G For help with direct variation and the constant of variation see p a. Answers ma var. Sample: (, ), Q, R Answers ma var. Sample: (, ), Q, R Real-World for Help Connection n a carousel, the outer horses cover more distance than the inner horses, giving them a faster speed. Chapter Linear Equations and Their Graphs Geometr Find the slope of the sides of each figure... J PQ : Q R ; QR: ; JK: ; KL: ; P RS: ; SP: K ML: ; MJ: M S L. a. Graph the direct variation -. See margin. What is the constant of variation? c. What is the slope? d. What is the relationship between the constant of variation and the slope? equal 7. a. pen-ended Name two points on a line with a slope of. See left. Name two points on a line with a slope of. Each pair of points lies on a line with the given slope. Find or. 8. (, ), (, 8); slope - 9. (, ), (, 8); slope. (, ), (, 7); slope. (, ), (, 8); slope. (-, ), (, ); slope. (, ), (, ); undefined slope Reasoning In Eercises, tell whether each statement is true or false. If false, give a countereample.. Countereamples ma var.. A rate of change must be either positive or zero. False; it can be neg. or undefined; for instance, the rate of change for. All horizontal lines have the same slope. true (, ) and (, ) is negative.. A line with slope alwas passes through the origin. false; ± 7. Two lines ma have the same slope. true 8. The slope of a line that passes through Quadrant III must be negative. false; 9. A line with slope never passes through point (, ). false;. Two points with the same -coordinate are alwas on the same vertical line. true. Business The graph shows how much it costs to rent carousel equipment. 8 a. Estimate the slope of the line. What does that number mean? ; $/h Customers pa $ for a ride. What is the average number of customers needed to cover the rental costs? about customers per hour. Error Analsis A friend sas the slope of a line passing through (, 7) and (, 9) is equal to the ratio 7 9. What is our friend s error? run rise Friend found rise instead of run. Rental Cost (dollars) Rent for a Carousel 7 8 Hours pages Eercises. No; for eample, the line passing through two points such as (, ) and (, ) has a slope of. a.

8 C Challenge.Yes; AB and BC have the same slope. 7.Yes; GH and HI have the same slope. 8. No; DE and EF do not have the same slope. 9. No; PQ and QR do not have the same slope. Standardized Test Test Prep Prep Multiple Choice Short Response Find the slope of the line passing through each pair of points.. (a, -b), (-a, -b). (-m, n), (m, -n) m n d b. (a, b), (c,d) c a For 9. See left. Do the points in each set lie on the same line? Eplain our answer.. A(, ), B(, ), C(-, ) 7. G(, ), H(-, ), I(7, 7) 8. D(-, ), E(, -), F(, ) 9. P(, ), Q(-, ), R(, ) 7. G(, -), H(-, -), I(, ) 7. S(-, ), T(, ), X(-, ) Yes; GH and HI have the No; ST and TX do not have same slope. the same slope. 7. A line has slope. Through which two points could this line pass? C A. (, 9), (8, ) B. (, 8), (, ) C. (8, ), (, ) D. (, ), (, 7) 7. A horizontal line passes through (, ). Which other point does the line contain? G F. (, ) G. (, ) H. (, ) J. (, ) 7. Josie found the slope of the line shown at the right. She used the points (, -) and (, ) and () wrote 8 to find the slope. Which of the following describes her error? A A. She mied up and in the denominator. B. She subtracted instead of added. C. She used an -value in the numerator of the fraction. D. She did not combine the negative signs correctl. 7. The steepness, or grade, of a road is epressed as a percent. If a road rises feet for ever horizontal feet, what is the slope of the road? What percent grade is this? Show our work. [] 8 ; the slope is 8. 8.%; the grade is.%. [] finds grade onl Alternative Assessment Give each student a laminated coordinate plane and a piece of uncooked spaghetti. Instruct students to place the spaghetti on the graph to represent a line, and then to find the slope of the line. Repeat. Test Prep Resources For additional practice with a variet of test item formats: Standardized Test Prep, p. 9 Test-Taking Strategies, p. Test-Taking Strategies with Transparencies Math Tip Eercise 7 You ma want to point out that on a horizontal line, all the points are the same distance from the -ais. So all the -coordinates are the same. Mied Mied Review Review G for Help Lesson - Lesson - Write a function rule for each situation. 7. the total cost of renting a movie for n das if it costs $./da c.n 77. the total profit if supplies and wages cost $, and each item q sells for $.9 p.9q Solve (t + ) ( - ) 9 8. n - 7(n - 9) 7 8. (7 - q) - 8. (p + ) Lesson - Find each probabilit based on one roll of a number cube. 8. P() 8. P(even number) 8. P( or ) 87. P(integer) lesson quiz, PHSchool.com, Web Code: ata- Lesson - Rate of Change and Slope