Graphing Linear Nonproportional Relationships Using Slope and y-intercept

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1 L E S S O N. Florida Standards The student is epected to: Functions.F.. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (, ) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Functions.F.1. Interpret the equation = m + b as defining a linear function, whose graph is a straight line; give eamples of functions that are not linear. MP.6.1 Precision Graphing Linear Nonproportional Relationships Using Slope and -intercept Engage ESSENTIAL QUESTION How can ou graph a line using the slope and -intercept? Sample answer: First, plot the point that contains the -intercept. Then use the slope to find another point on the line and draw a line through the points. Motivate the Lesson Ask: In the equation = + 1, what is the slope? What is the -intercept? Eplore Discuss with students what it means for a line to have a slope of. If the line passes through the origin (b = 0), what would be another point on the line? Eplain Animated Math Eploring Linear Graphs Students eplore graphs of linear relationships b changing the values of m and b using interactive sliders. ADDITIONAL EXAMPLE 1 Graph each equation. A = + B = m.hrw.com Interactive Whiteboard Interactive eample available online 107 Lesson. A B m.hrw.com EXAMPLE 1 Questioning Strategies In Step of part A, wh do ou count up and right? Sample answer: The numerator and denominator of the slope are both positive, so ou count up and to the right. In Step of part B, wh do ou count down 5 and right or up 5 and left? Sample answer: The slope of the line is negative, so ou count down and to the right or up and to the left. Focus on Math Connections Help students make the connection between the slopes of the graphed lines and the coefficient of in the equations in parts A and B of this eample. If slope is rise run and the slope is, then the rise is and the run is. Engage with the Whiteboard Have students select a point on the line and locate a second point using the slope of the line. Avoid Common Errors Make sure students correctl distinguish the slope and the -intercept. The could write out the slope and -intercept for each equation, or underline the slope and circle the -intercept. EXAMPLE Questioning Strategies How man calories does Ken burn each hour b walking briskl? 00 calories; calories left to burn decreases b 00 with each hour of walking. What does the point (, 1500) represent on the graphed line? After walking briskl for hours, Ken still needs to burn 1500 calories.

DO NOT EDIT--Changes must be made through File info CorrectionKe=B? Reflect.F.. A line with a negative slope will fall from left to right. How can ou graph a line using the slope and -intercept?

Note that the line passes through all three points: (, ), (0, ), and (, ). No; the line whose slope has the larger absolute value will be steeper.

The equation = -00 + 00 represents the number of calories Ken has left to burn after hours of walking which burns 00 calories per hour.

2 DO NOT EDIT--Changes must be made through File info CorrectionKe=B? Reflect.F.. A line with a negative slope will fall from left to right. How can ou graph a line using the slope and -intercept? Online Assessment and Intervention EXAMPL 1 EXAMPLE STEP STEP STEP STEP = O - O Analzing a Graph Animated Math - + O + Math Talk (0, ) + (, ) Draw a line through the points. Note that the line passes through all three points: (, ), (0, ), and (, ). No; the line whose slope has the larger absolute value will be steeper. In Eample 1, the line in part B is steeper than the line in part A even though its slope is negative. Lesson. _MFLESE05671_UM0L.indd /0/1 11:5 AM EXAMPLE.F.. Ken has a weekl goal of burning 00 calories b taking brisk walks. The equation = represents the number of calories Ken has left to burn after hours of walking which burns 00 calories per hour. Is a line with a positive slope alwas steeper than a line with a negative slope? Eplain. 1 m.hrw.com (, ) Math On the Spot (, 1) (0, -1) - The -intercept is b =. Plot the point that contains the -intercept: (0, ). Man real-world situations can be represented b linear relationships. You can use graphs of linear relationships to visualize situations and solve problems. m.hrw.com Draw a line through the points. The slope is m = -_5. Use the slope to find a second point. From (0, ), count down 5 and right, or up 5 and left. The new point is (, ) or (, ).. m.hrw.com B Graph = - _5 +. = _1 + 1.F.1. The -intercept is b = -1. Plot the point that contains the -intercept: (0, -1). The slope is m = _. Use the slope to find a second point. From (0, -1), count up and right. The new point is (, 1).. Math On the Spot Graph = _ - 1. Graph each equation. m.hrw.com Recall that = m + b is the slope-intercept form of the equation of a line. In this form, it is eas to see the slope m and the -intercept b. So ou can use this form to quickl graph a line b plotting the point (0, b) and using the slope to find a second point. A Draw Conclusions How can ou use the slope of a line to predict the wa the line will be slanted? Eplain. A line with positive slope will rise from left to right. ESSENTIAL QUESTION Using Slope-intercept Form to Graph a Line Houghton Mifflin Harcourt Publishing Compan 1. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph. Also.F.1. TIME HOURS 0:0 1:0 CALORIES Houghton Mifflin Harcourt Publishing Compan. Graphing Linear Nonproportional Relationships Using Slope and -intercept A Graph the equation = Write the slope as a fraction Using the slope as helps in drawing a more accurate graph. Plot the point for the -intercept: (0, 00) = -600 = m = 1 STEP STEP Use the slope to locate a second point. From (0, 00), count down 900 and right. The new point is (, 1500). STEP 10 Draw a line through the two points. Calories remaining LESSON DO NOT EDIT--Changes must be made through File info CorrectionKe=B Time (h) Unit _MFLESE05671_UM0L.indd 10 10/0/1 11:5 AM PROFESSIONAL DEVELOPMENT Integrate Mathematical Practices MP.6.1 This lesson provides an opportunit to address this standard. It calls for students to communicate precisel, including communicating through the use of smbols and graphs. In Eample, students begin with a real-world situation represented b a linear equation, the find the -intercept and slope, and then represent the equation with a graph. Math Background This lesson shows how to graph equations of lines that are not horizontal or vertical. For horizontal lines, there is a -intercept, but no -intercept (unless the horizontal line is the -ais), and the slope is zero. The equation for a horizontal line is of the form = b, where b is a fied real number. For vertical lines, there is an -intercept, but no -intercept (unless the vertical line is the -ais), and the slope is undefined. The equation for a vertical line is of the form = a, where a is a fied real number. Graphing Linear Nonproportional Relationships Using Slope and -intercept 10

3 ADDITIONAL EXAMPLE A shipping compan charges a fied amount plus a certain amount per pound to ship a package. The total cost, in dollars, to ship a package is given b the equation = + 5, where is the weight of the package in pounds. A Graph the equation. B What is the weight of a package that can be shipped for $17? lb Total cost ($) O 1 5 Weight (lb) Interactive Whiteboard Interactive eample available online m.hrw.com Connect Multiple Representations Guide students in making connections between the situation, the equation, the slope, the -intercept, and the graph. The relationship in Eample can be represented in words, with an equation (and statements eplaining what each variable represents), and b the graph that is shown in part A. Ask students to epress the relationship in their own words without an equation. For eample: Ken has a goal to burn 00 calories each week b walking fast; for each hour that he walks, he burns 00 calories. Focus on Modeling Have students identif the -intercept of the line representing this new situation. Ask them to eplain what that -intercept means. Have students identif the slope of the line representing this new situation and to eplain what that new slope means. Have students epress the relationship in the problem in words without an equation. Elaborate Talk About It Summarize the Lesson Ask: How can ou graph a line using the slope and -intercept? You plot the -intercept on the -ais. Then ou use the slope to find another point and draw a line through the two points. GUIDED PRACTICE Engage with the Whiteboard In Eercises 1 and, have students change the sign of the slope and graph the new equation. Net have them change the sign of the -intercept and graph this equation as well. Avoid Common Errors Eercises 1 Remind students that the -intercept of a line is the number on the -ais where the line intersects the -ais. Eercise Students ma think that an point with a whole number -coordinate makes sense. Have students find the -coordinate when = 1. Eplain that after weeks ou have 10 cards and after weeks ou have 1 cards. Since the cards are onl bought once a week and ou bu at a time, there is no point at which ou will have 1 cards. 109 Lesson.

4 B After how man hours of walking will Ken have 600 calories left to burn? After how man hours will he reach his weekl goal? STEP Locate 600 calories on the -ais. Read across and down to the -ais. Ken will have 600 calories left to burn after 6 hours. Ken will reach his weekl goal when the number of calories left to burn is 0. Because ever point on the -ais has a -value of 0, find the point where the line crosses the -ais. Calories remaining Ken will reach his goal after hours of brisk walking. 6 Time (h) Guided Practice Graph each equation using the slope and the -intercept. (Eample 1) 1. = 1_ -. = - + 1_ slope = -intercept = - slope = - -intercept = - O - O Houghton Mifflin Harcourt Publishing Compan What If? Ken decides to modif his eercise plans from Eample b slowing the speed at which he walks. The equation for the modified plan is = Graph the equation. 5. How does the graph of the new equation compare with the graph in Eample? The new graph has the same -intercept but a slope of 00 instead of -00. Calories remaining Time (h) 6. Will Ken have to eercise more or less to meet his goal? Eplain. The calories left to burn will decrease more slowl with each hour of eercise, so it will take longer for Ken to meet his goal. 7. Suppose that Ken decides that instead of walking, he will jog, and that jogging burns 600 calories per hour. How do ou think that this would change the graph? Sample answer: The -intercept would not change, but the slope would become -600, which is much steeper. The line would intersect the -ais when = hours. Math Talk What do the slope and the -intercept of the line represent in this situation? The slope is 00, which means the number of calories left to burn decreases b 00 calories for ever hour of walking. The -intercept is 00, which means Ken s weekl goal is to burn 00 calories b walking. Online Assessment and Intervention m.hrw.com? -. A friend gives ou two baseball cards for our birthda. Afterward, ou begin collecting them. You bu the same number of cards once each week. The equation = + describes the number of cards,, ou have after weeks. (Eample ) a. Find and interpret the slope and the -intercept of the line that represents this situation. Graph = +. Include ais labels. Slope = ; -intercept = ; ou start with cards and add cards each week. b. Discuss which points on the line do not make sense in this situation. Then plot three more points on the line that do make sense. The points with coordinates that are not whole numbers; You will not bu part of a baseball card and ou are buing onl once a week. ESSENTIAL QUESTION CHECK-IN. Wh might someone choose to use the -intercept and the slope to graph a line? Sample answer: You can easil identif the slope m and -intercept b from the slope-intercept form = m + b and quickl use them to locate two points that determine the line. - Baseball cards Weeks Houghton Mifflin Harcourt Publishing Compan Lesson Unit DIFFERENTIATE INSTRUCTION Kinesthetic Eperience Some students ma still be having difficult remembering that lines with positive slopes go up as one moves from left to right and lines with negative slopes go down as one moves from left to right. Draw a line on the board that has a positive or negative slope. Have students etend their right arms to model the slope of the line. Have students sa whether the slope of their arms is positive or negative and then eplain wh. Repeat with other lines. Then sa the words positive or negative and have students use their arms to model a line with such a slope. Cooperative Learning Have students work in pairs. Have each pair choose one student to be the Equation Creator and the other student to be the Equation Grapher. The Equation Creator should create a linear equation in slope-intercept form with integer coefficients that are between -5 and 5. The Equation Grapher should check that the equation is in that form. The Equation Grapher should then graph the equation and the Equation Creator should check that the graph is correct. Encourage students to use the -intercept and the slope to graph the equations. After the Equation Grapher has graphed one or two equations, have students switch roles. Additional Resources Differentiated Instruction includes Reading Strategies Success for English Learners ELL Reteach Challenge PRE-AP Graphing Linear Nonproportional Relationships Using Slope and -intercept 110

5 . LESSON QUIZ Online Assessment and Intervention Online homework assignment available m.hrw.com.f.1.,.f.. 1. Graph the equation = Maria is ordering comic books online. The equation = + represents the total cost in dollars,, including shipping, for ordering number of comic books. a. Graph the equation. b. If the total cost including shipping is $60, how man comic books is Maria ordering?. Mr. Goldstein is driving to Houston. The equation = represents the numbers of miles that he still has to travel after driving for hours. Find and interpret the slope and -intercept of the line that represents this situation. Lesson Quiz available online Evaluate GUIDED AND INDEPENDENT PRACTICE.F.1.,.F.. Concepts & Skills Eample 1 Using Slope-Intercept Form to Graph a Line Eample Analzing a Graph Practice Eercises 1, 5, 1, 16 Eercises, 5, 1 Eercise Depth of Knowledge (D.O.K.) 5 Strategic Thinking MP..1 Modeling 6 11 Skills/Concepts MP.5.1 Using Tools 1 Skills/Concepts MP..1 Reasoning 1 1 Strategic Thinking MP..1 Logic Strategic Thinking MP..1 Reasoning Additional Resources Differentiated Instruction includes: Leveled Practice worksheets m.hrw.com Answers a. Total cost ($) O Number of comic books b. 7 comic books. Slope = -5; -intercept = 70; he is driving 5 miles per hour and at the beginning, had 70 miles to drive. 111 Lesson.

6 Houghton Mifflin Harcourt Publishing Compan Image Credits: Steve Williams/ Houghton Mifflin Harcourt Name Class Date. Independent Practice.F.1.,.F.. Spring length (in.) m.hrw.com 5. Science A spring stretches in relation to the weight hanging from it according to the equation = where is the weight in pounds and is the length of the spring in inches. a. Graph the equation. Include ais labels. b. Interpret the slope and the 1 -intercept of the line. The slope, 0.75, means 1 that the spring stretches Weight (lb) b 0.75 inch with each additional pound of weight. The -intercept, 0.5, is the unstretched length of the spring in inches. Online Assessment and Intervention c. How long will the spring be if a -pound weight is hung on it? Will the length double if ou double the weight? Eplain 1.75 inches; no; the length with a -pound weight is.5 in., not.5 in. Look for a Pattern Identif the coordinates of four points on the line with each given slope and -intercept. Sample answers are given. 6. slope = 5, -intercept = slope = -1, -intercept = (0, -1), (1, ), (, 9), (, 1) (0, ), (1, 7), (, 6), (, 5). slope = 0., -intercept = slope = 1.5, -intercept = - (0, 0.), (1, 0.5), (, 0.7), (, 0.9) (0, -), (1, -1.5), (, 0), (, 1.5) 10. slope = - 1_, -intercept = 11. slope = _, -intercept = -5 (0, ), (, ), (, ), (6, 1) (0, -5), (, -), (6, -1), (9, 1) 1. A music school charges a registration fee in addition to a fee per lesson. Music lessons last 0.5 hour. The equation = represents the total cost of lessons. Find and interpret the slope and -intercept of the line that represents this situation. Then find four points on the line. Slope = 0, so the cost per lesson is $0; -intercept = 0, so the registration fee is $0; sample answers: (0, 0), (1, 70), (, 110), (, 150). 1. A public pool charges a membership fee and a fee for each visit. The equation = + 50 represents the cost for visits. a. After locating the -intercept on the coordinate plane shown, can ou move up three gridlines and right one gridline to find a second point? Eplain. Yes.; Since the horizontal and vertical gridlines each represent 5 units, moving up gridlines and right 1 gridline represents a slope of 75, or. 5 b. Graph the equation = Include ais labels. Then interpret the slope and -intercept. m = so $ is the charge per visit; b = 50 so the membership fee is $50. c. How man visits to the pool can a member get for $00? 50 visits FOCUS ON HIGHER ORDER THINKING - - O Cost ($) 1. Eplain the Error A student sas that the slope of the line for the equation = 0-15 is 0 and the -intercept is 15. Find and correct the error. The coefficient of, -15, is the slope, not the constant term. The constant term is the -intercept, Critical Thinking Suppose ou know the slope of a linear relationship and a point that its graph passes through. Can ou graph the line even if the point provided does not represent the -intercept? Eplain. Yes; ou can plot the point and use the slope to find a second point. Then draw a line through the two points. 16. Make a Conjecture Graph the lines =, = -, and = +. What do ou notice about the lines? Make a conjecture based on our observation. The lines appear to be parallel. Parallel lines have the same slope but different -intercepts Pool visits Work Area Houghton Mifflin Harcourt Publishing Compan Lesson Unit EXTEND THE MATH PRE-AP Activit available online m.hrw.com Activit Have students use graphing calculators to graph on the same set of aes three linear equations whose graphs have the same slope. For eample, have them graph =, = +, and = -. Then have them graph, on a new pair of aes, another set of linear equations whose graphs have the same slope, but different from the slope of the first set of lines. For eample, have them graph = -, = - +, and = Ask students to make a conjecture about lines with the same slopes (the are parallel). Have them tr out other sets of linear equations to test their conjectures. Graphing Linear Nonproportional Relationships Using Slope and -intercept 11

Essential Question: How can you represent a linear function in a way that reveals its slope and y-intercept?

Essential Question: How can you represent a linear function in a way that reveals its slope and y-intercept? COMMON CORE 5 Locker LESSON Slope-Intercept Form Common Core Math Standards The student is epected to: COMMON CORE F-IF.C.7a Graph linear... functions and show intercepts... Also A-CED.A., A-REI.D. Mathematical