Linear Concepts. Previewing the Activity. Lesson 3-1. Activity 3 Contents. Lesson 3-2. Lesson 3-3. Lesson 3-4. Lesson 3-5.

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1 ACTIVITY 3 Linear Concepts Previewing the Activit In this activit, ou will learn about literal equations and the forms of linear equations. You will compare point-slope, slope-intercept, and standard forms of linear equations. Activit 3 Contents Lesson 3-1 Lesson 3- Lesson 3-3 Lesson 3- Lesson 3-5 Lesson 3-6 Solving Literal Equations Slope-Intercept Form Point-Slope Form Standard Form Vertical and Horizontal Lines Piecewise-Defined Functions What kinds of measurements help meteorologists determine the strength of a storm? Embedded Assessment 3 Linear Equations Vocabular SPANISH COGNATES English Spanish linear linear Unpacking Embedded Assessment 3 Linear Equations: Storm Weather Read the Embedded Assessment on page 99 to get a preview of the skills and knowledge ou will need. Use the graphic organizer to categorize the terms ou alread know and the terms that are new to ou. Embedded Assessment 3 66 SpringBoard Mathematics Algebra 1, UNIT 1 Linear Equations and Functions

2 Linear Concepts ACTIVITY 3 Getting Read 1. Jeff can pack food boes according to the equation f = 10t, where f is the number of food boes and t is the number of hours. Venus can pack food boes according to the data in the table. Who can pack food boes faster? Boes Hours Solve the equation ( 3) 1= The temperature in Allendale is dropping according to the equation T(h) = h + 50, where h is the number of hours after noon. The temperature in South Webster is changing according to the values in the table. Which town is eperiencing a faster temperature drop? Hours after Noon 0 6 Temperature Which of the following equations has a graph that is a straight line? a. = - 6 b. = 8 + c. = -7 - d. = Michael is driving at a constant speed. After 30 minutes, he is 9 miles from home. After 5 minutes, he is 1 miles from home. Write a function D(t) that epresses the distance Michael is from home, in miles, t minutes after leaving home. ACTIVITY 3 Linear Concepts 67

3 LESSON 3-1 Solving Literal Equations LEARNING STRATEGIES Learning Targets Solve literal equations for a specified variable. Use a formula to determine unknown quantities. Identifing Subtasks Vocabular MATHEMATICS A literal equation has more than one variable and can be solved for a specific variable. Would the equation in the Eample be a literal equation if b was replaced b 8? A formula describes how two or more quantities are related. Formulas are important in man disciplines; geometr, phsics, economics, sports, and medicine are just a few eamples of fields in which formulas are widel used. A formula is an eample of a literal equation. Literal equations and formulas can be solved for a specific variable using the same procedures as equations containing one variable. Eample Solve the equation + b = 1 for. Step 1 Isolate the term that contains b subtracting b from both sides. + b = 1 + b b = 1 b = 1 b Step Isolate b dividing both sides b. Original equation Subtraction Propert of Equalit Combine like terms. Solution Eample = 1 b Division Propert of Equalit = 1 b = 1 b, or = 3 b Simplif. Solve the equation 5 z = 36 for. Identif the properties for each step. Step 1 Step Isolate the term that contains b subtracting z from both sides. 5 z = 36 5 z + z = 36 + z 5 = 36 +z 5 = 36 + z 5 5 Original equation Addition Propert of Equalit Combine like terms. Isolate b dividing both sides b 5. Solution This = 36 + z 5 68 SpringBoard Mathematics Algebra 1, UNIT 1 Linear Equations and Functions

4 Solving Literal Equations LESSON 3-1 Tr This Solve the equation for. a + 7 = 3 = a Check Your Understanding 1. Is the equation + = 5 6 a literal equation? Use the word variable in our eplanation.. Solve the equation = m + b for m. Eample The equation v = v 0 + at gives the velocit in meters per second of an object after t seconds, where v 0 is the object s initial velocit in meters per second and a is its acceleration in meters per second squared. Determine the acceleration for an object whose velocit after 15 seconds is 5 meters per second and whose initial velocit was 15 meters per second. Step 1 Step Solve the equation for a. v = v 0 + at v v 0 = v 0 v 0 + at v v 0 = at Original equation Subtraction Propert of Equalit: Subtract v 0 from both sides. Combine like terms. v v0 = at Division Propert of Equalit: t t Divide both sides b t. v v0 = a Simplif. t v v a = 0 Smmetric Propert of Equalit t Substitute 5 for v, 15 for v 0, and 15 for t. SYMBOL SENSE The variable v 0 has the subscript 0. A variable with a subscript of 0 usuall indicates an initial value. So, v 0 indicates the initial value of the velocit, or the velocit when the time t = 0. a = Solution = 3 m/s ACTIVITY 3 Linear Concepts 69

5 LESSON 3-1 Solving Literal Equations Tr This The equation t = 13p can be used to estimate the cooking time t in minutes for a stuffed turke that weighs p pounds. Solve the equation for p. Then find the weight of a turke that requires 85 minutes to cook. p = t ; about 13.6 pounds 13 MP 3. Reason abstractl. Formulas are written in the field of phsical science to epress relationships between quantities. Solve for the indicated variable in each formula. Name Formula Solve for Distance Pressure Kinetic energ Gravitational energ Bole s Law d = rt, where d is the distance an object travels, r is the average rate of speed, and t is the time traveled p = F, where p is the pressure on a A surface, F is the force applied, and A is the area of the surface k = 1 mv, where k is the kinetic energ of an object, m is its mass, and v is its velocit U = mgh, where U is the gravitational energ of an object, m is its mass, g is the acceleration due to gravit, and h is the object s height p 1 V 1 = p V, where p 1 and V 1 are the initial pressure and volume of a gas and p and V are the final pressure and volume of the gas when the temperature is kept constant r r = d t F F = pa m m = h h = k v U mg pv V V = p 1 1 Check Your Understanding. Solve the equation w+ i= s for c. c 5. Wh is it useful to solve a literal equation for a variable? Use a phsical science equation from Item 3 to justif our response. 70 SpringBoard Mathematics Algebra 1, UNIT 1 Linear Equations and Functions

6 Solving Literal Equations LESSON 3-1 Lesson 3-1 Practice Solve each equation for the indicated variable. 1. W = Fd, for d. P = W, for W t 3. P = W, for t t. ak r = on, for k MP6 5. Attend to precision. In baseball, the equation E = 9 R gives a I pitcher s earned run average E, where R is the number of earned runs the plaer allowed and I is the number of innings pitched. a. Solve the equation for I. State a propert or provide an eplanation for each step. b. Last season, a pitcher had an earned run average of.80 and allowed 70 earned runs. How man innings did the pitcher pitch last season? ACTIVITY 3 Linear Concepts 71

7 LESSON 3- Slope-Intercept Form LEARNING STRATEGIES Discussion Groups Learning Targets Write the equation of a line in slope-intercept form. Write a linear equation given a function. Analze ke features of a function given its graph. Think-Pair-Share When a diver descends in a lake or ocean, pressure is produced b the weight of the water on the diver. As a diver swims deeper into the water, the pressure on the diver s bod increases at a rate of about 1 atmosphere of pressure per 10 meters of depth. The table and graph represent the total pressure,, on a diver given the depth,, under water in meters. Vocabular MATHEMATICS A linear equation is an equation that can be written in the form A + B = C where A, B, and C are constants and A and B cannot both be zero Pressure (atm) Depth Under Water (meters) 1. Write an equation describing the relationship between the pressure eerted on a diver and the diver s depth under water. Vocabular What other forms of linear equations have ou encountered? How do these compare to slopeintercept form? COGNATES Linear and Linear Both the English linear and the Spanish linear come from the Latin linea, meaning a line. What English or Spanish words do ou know of that are similar to linear? = What is the slope of the line? What are the units of the slope? m = 0.1; the units are atmospheres per meter. 3. What is the -intercept? Eplain its meaning in this contet. (0, 1); the amount of pressure on the diver at the surface of the water (when depth = 0) Slope-Intercept Form of a Linear Equation = m + b where m is the slope of the line and (0, b) is the -intercept.. Eplain how the graph of = is related to its equation. The graph is the visual representation of all solutions (, ) to the equation = SpringBoard Mathematics Algebra 1, UNIT 1 Linear Equations and Functions

8 Slope-Intercept Form LESSON 3- Slope-Intercept Form of a Linear Function is f() = m + b where m is the slope of the line and (0, b) is the -intercept. MP7 5. Create a table of values for the function f() = Then plot the points, and graph the line. f() Eplain how to find the value of the slope from the table. What is the value of the slope of the line? Look at two points, and find the change in f() and the change in ; write the ratio of the change in f() over the change in ; the slope is Eplain how to find the value of the slope from the graph. What is the value of the slope? Find a point, and move from it to another point. Find the change in and the change in, and write it as a ratio; -. Vocabular MATHEMATICS The -values are the inputs, the values that are substituted into the equation. The -values are the outputs, the values that are the result of substituting the inputs into the linear equation. 8. Eplain how to find the -intercept from the table. What is the -intercept? Find the point where = 0; the -intercept is (0, 9). INTRODUCING THE STRATEGY: Discussion Groups Discussion Groups is a collaborative strateg that allows ou to share ideas, interpret concepts, and analze problem situations. 9. Eplain to our partner how to find the -intercept from the function. Identif the input and output in Item 9. Look at the constant term when the function is written in slopeintercept form. The constant is 9, so the -intercept is (0, 9). 10. Eplain how to find the -intercept from the graph. What is the -intercept? Find the point where the line intersects the -ais. That point is the -intercept; (0, 9). ACTIVITY 3 Linear Concepts 73

9 LESSON 3- Slope-Intercept Form Check Your Understanding 11. What are the slope and -intercept of the line described b the equation = 10? 5 1. Write the equation in slope-intercept form of the line that is represented b the data in the table Write the equation, in slope-intercept form, of the line with a slope of and a -intercept of (0, 5). 1. Write an equation of the line graphed in the section of this page. Monica gets on an elevator in a skscraper. The elevator starts to move at a rate of -0 ft/s. After 6 seconds on the elevator, Monica is 350 feet from the ground floor of the building. 15. The rate of the elevator is negative. What does this mean in the situation? What value in the slope-intercept form of an equation does this rate represent? The elevator is moving down; slope, or m 16. a. How man feet was Monica above the ground when she got on the elevator? Show how ou determined our answer (6) = 70 ft MP b. What value in the slope-intercept form does our answer to Part (a) represent? the -coordinate of the -intercept, or b 17. Model with mathematics. Write an equation in slope-intercept form for the motion of the elevator since it started to move. What do and represent? = ; represents the time in seconds since Monica got on the elevator, and represents the height of the elevator above the ground in feet. a. What does the -intercept represent? Monica s original height above the ground in feet b. Use the equation ou wrote to determine, at this rate, how long it will take after Monica enters the elevator for her to eit the elevator on the ground floor. Eplain how ou found our answer. At the ground floor, = 0. Solve the equation 0 = to find = 3.5 seconds. 7 SpringBoard Mathematics Algebra 1, UNIT 1 Linear Equations and Functions

10 Slope-Intercept Form LESSON 3- c. Graph the equation. Elevator height (feet) Time (seconds) d. What feature on the graph gives ou the answer to Part b? Eplain. The -intercept gives the value of, the number of seconds that have elapsed, when = 0, which is when the elevator reaches the ground. Check Your Understanding 18. Write the equation 3 - = 16 in slope-intercept form. Eplain our steps. Then graph the equation, and identif the slope and -intercept. 19. Write an equation in slope-intercept form for the line that has a slope of and -intercept of (0, -5). 3 ACTIVITY 3 Linear Concepts 75

11 LESSON 3- Slope-Intercept Form Lesson 3- Practice 1. What are the slope, m, and -intercept, (0, b), of the line described b the equation = 1?. A flowering plant stands 6.5 inches tall when it is placed under a growing light. Its growth is 0.5 inches per da. Toda the plant is 11.5 inches tall. a. Write an equation in slope-intercept form for the height of the plant since it was placed under the growing light. b. In our equation, what do and represent? c. Use the equation to determine how man das ago the plant was placed under the light. d. Graph the equation. e. What does the -intercept on the graph represent? 3. Write an equation in slope-intercept form for the line that passes through the points (6, -3) and (0, ). MP Matt sells used books on the Internet. He has a weekl fee he has to pa for his website. He has graphed his possible weekl earnings, as shown. a. What is the weekl fee that Matt pas for his website? How do ou know? b. How much does Matt make for each book sold? How do ou know? c. Write the equation in slopeintercept form for the line in Matt s graph. d. How man books does Matt have to sell to make $30 for the week? Eplain. Earnings for the Week ($) 5. Make use of structure. Without graphing, describe the graph of each equation. Tell whether the line is ascending or descending from left to right and where the line crosses the -ais a. = 3 b. = 5 + c. = d. = Used Book Internet Business Number of Books Sold 76 SpringBoard Mathematics Algebra 1, UNIT 1 Linear Equations and Functions

12 Point-Slope Form LESSON 3-3 Learning Targets Write the equation of a line in point-slope form. Use point-slope form to solve problems that represent real-world situations. Another form of the equation of a line is the point-slope form. The point-slope form of the equation is found b solving the slope formula m = 1 for 1, b multipling both sides b 1. This form is 1 useful when ou know a point on the line and the slope. 1. Multipl both sides of the slope formula m = 1 b - 1 to find 1 the point-slope form of a linear equation. ( m 1) = 1 ( ) 1 1 m ( 1) = 1 = m ( ) 1 1 Point-Slope Form of a Linear Equation 1 = m( 1 ) where m is the slope of the line and ( 1, 1 ) is a point on the line. LEARNING STRATEGIES Close Reading Eample Write an equation of the line with a slope of 1 that passes through the point (, 5). Graph the line. Step 1 Step Substitute the given values into point-slope form. 1 = m( 1 ) 5 = 1 ( ) Graph 5 = 1 ( ). Plot the point (, 5), and use the slope to find another point ACTIVITY 3 Linear Concepts 77

13 LESSON 3-3 Point-Slope Form Tr This Write an equation of a line with a slope of that passes through the 3 point (, 7). 7 = + 3 ( ) Connect to Pre-AP Use Close Observation and Analsis to identif the ke information in the tet. The town of San Simon charges its residents for trash pickup and water usage on the same bill. Each month, the cit charges a flat fee for trash pickup and a fee of $0.5 per gallon for water used. In Januar, one resident used gallons of water and received a bill for $16.. If is the number of gallons of water used during a month, and represents the bill amount in dollars, write a point ( 1, 1 ). (, 16) 3. What does $0.5 per gallon represent? The rate of change of the bill. For ever additional gallon, there is an increase of $0.5. MP. Reason abstractl. Use point-slope form to write an equation that represents the bill cost in terms of the number of gallons of water used in a month. 16 = 0.5( ) 5. Write the equation in Item in slope-intercept form. What does the -intercept represent? = ; it represents the flat fee. 78 SpringBoard Mathematics Algebra 1, UNIT 1 Linear Equations and Functions

14 Point-Slope Form LESSON 3-3 Check Your Understanding 6. Determine the equation of the line given the point (86, 15) and the slope m = Violet has an Internet business selling paint sets. After an initial website fee each week, she makes a profit of $0.75 on each set she sells. If she sells 8 sets, she makes $.5. Write an equation representing her weekl possible earnings. What would be Violet s earnings for a week in which she sells 1 paint sets? MP3 8. Critique the reasoning of others. Jamilla and Ran were asked to write the equation of the line through the points (6, ) and (3, 5). Both Jamilla and Ran determined that the slope was Jamilla wrote the equation of the line as = 1 ( 6). Ran wrote the 3 equation of the line as 5 = 1 ( 3). 3 a. Rewrite each student s equation in slope-intercept form, and compare the results. Jamilla s line: = Ran s line: = The equations are the same. b. Whose equation was correct? Justif our response. Both equations were correct. The equations represent the same line. 9. Find the equation in point-slope form of the line shown in the graph. = 5( ) 10. Write the equation of the line in slopeintercept form. = ACTIVITY 3 Linear Concepts 79

15 LESSON 3-3 Point-Slope Form Check Your Understanding 11. Eplain the process ou would use to write an equation of a line in point-slope form when given two points on the line. 1. Write the point-slope equation of the line through the points ( 3, 5) and (9, 1). 80 SpringBoard Mathematics Algebra 1, UNIT 1 Linear Equations and Functions

16 Point-Slope Form LESSON 3-3 Lesson 3-3 Practice 1. Write an equation of the line with a slope of 0.5 that passes through the point ( 1, 8).. Find the slope and a point on the line for the lines with the following equations. a. 9 = - 3 ( ) b. 3 = ( + ) 3 3. Write the equation of the line through the points ( 3, 3) and (7, 5) in slope-intercept form. What is the -intercept?. Ja pas a flat fee each month for basic cable service. He also pas $3.50 for each movie he orders during the month. Last month, he ordered 5 movies, and his total bill came to $5. a. Write an equation in point-slope form that represents the total bill,, in terms of the number of movies,. b. Write the equation in slope-intercept form. c. What is the monthl fee for basic cable service? How do ou know? d. Net month, Ja plans to order 7 movies. What will be his total bill for the month? e. This month, Ja s total bill is $ How man movies did he order this month? 5. Attend to precision. The equation 160 = 0( 1) represents MP6 the height in feet,, of a hot-air balloon minutes after the pilot started her stopwatch. a. Is the hot-air balloon rising or descending? Justif our answer. b. At what rate is the hot-air balloon rising or descending? Be sure to use appropriate units. c. What was the height of the balloon when the pilot started her stopwatch? ACTIVITY 3 Linear Concepts 81

17 LESSON 3- Standard Form LEARNING STRATEGIES KWHL Chart Learning Targets Write the equation of a line in standard form. Use standard form to solve problems that represent real-world situations. Think-Pair-Share William plas a game on his device where his avatar earns gold coins for completing quests. The coins he earns can be redeemed for etra lives or for costumes for the avatar. The graph shows the relationship between the number of avatar costumes and etra lives he can bu with 50 coins. 10 Avatar Costumes and Etra Lives Number of etra lives Number of avatar costumes 1. Epress the -intercept as an ordered pair. What does this represent for William's situation? (15, 0); The -intercept represents the greatest number of avatar costumes William can bu with 50 coins.. Epress the -intercept as an ordered pair. What does this represent for William's situation? (9, 0); The -intercept represents the greatest number of etra lives that William can bu with 50 coins. 3. Eplain what the slope of the line means in the contet of the problem. Sample answer: The slope of the line represents the change in possible lives. For ever 5 avatar costumes he bus, William will be able to bu 3 less lives than the maimum 9.. Write an equation for the graph of the line shown. = How man avatar costumes can William purchase if he bus 6 etra lives? 5 6. What does the coordinate (10, 3) mean in terms of avatar costumes and etra lives William can purchase? If William bus 10 avatar costumes, he will onl have enough coins left over for 3 etra lives. 8 SpringBoard Mathematics Algebra 1, UNIT 1 Linear Equations and Functions

18 Standard Form 7. How man coins are needed for one costume? 30 coins 8. How man coins are needed for one etra life? LESSON 3-50 coins Standard Form of a Linear Equation A + B = C where A 0, A and B are not both zero, and A, B, and C are integers whose greatest common factor is Reason abstractl. You can use the coefficients of this form of an MP equation to find the -intercept, -intercept, and slope. a. Determine the -intercept and the -intercept. -intercept: C ( A,0 ), -intercept ( 0, C ) b. Write A + B = C in slope-intercept form to find the slope. B = A + B C B ; the slope is - A B. c. Find the slope-intercept form of the line for William's situation, then write it in standard form. = 3 + 9; standard form: = 5 5 The definition of standard form states that both A and B are not 0. However, one of A or B ma be equal to Write the standard form if A = 0. B = C a. Suppose A = 0, B = 1, and C = 3. Write the equation of the line in standard form. 0 = 3, or = 3 b. Graph the line on the grid in the section. Describe the graph. What is the slope? It is a horizontal line; 0 ACTIVITY 3 Linear Concepts 83

19 LESSON 3- Standard Form 11. Write the standard form if B = 0. A = C a. Suppose A = 1, B = 0, and C = 6. Write the equation of the line in standard form. + 0 = 6, or = 6 b. Graph the equation on the grid in the section. Describe the graph. What is the slope? It is a vertical line; undefined Check Your Understanding 1. Write the equation = 6 in standard form Susheila is making a large batch of granola to sell at a school fundraiser. She needs to bu walnuts and almonds to make the granola. Walnuts cost $3 per pound, and almonds cost $ per pound. She has $30 to spend on these ingredients. a. Write an equation that represents the different amounts of walnuts,, and almonds,, that Susheila can bu. 3 + = 30 b. Graph the - and -intercepts on the coordinate plane. Use these to help ou graph the line. Connect to Pre-AP Engage in Higher-Order Questioning to determine what the -intercept represents in Item 10. Pounds of Almonds Pounds of Walnuts c. If Susheila bus pounds of walnuts, how man pounds of almonds can she bu? 9 pounds 8 SpringBoard Mathematics Algebra 1, UNIT 1 Linear Equations and Functions

20 Standard Form LESSON 3-1. Refer to the graph ou made in Item 1b. What is the -intercept? What does it represent? (10, 0); if Susheila bus no almonds and spends all the mone on walnuts, she can bu 10 pounds of walnuts. 15. Write an equation in standard form for the line shown = Check Your Understanding 16. Write an equation in standard form for the line that is represented b the data in the table Write an equation in standard form for the line with a slope of 7 that passes through the point (1, ). ACTIVITY 3 Linear Concepts 85

21 LESSON 3- Standard Form Lesson 3- Practice 1. Determine the -intercept, -intercept, and slope of the line described b = 1.. Write each equation in standard form. a. 8 = b. = Write an equation in standard form for each line. a b Pedro walks at a rate of miles per hour and runs at a rate of 8 miles per hour. Each week, his eercise program requires him to cover a total distance of 0 miles with some combination of walking and running. MP1 a. Write an equation that represents the different amounts of time Pedro can walk,, and run,, each week. b. Graph the equation. c. What is the -intercept? What does this tell ou? 5. Make sense of problems. Keisha bought a discount pass at a movie theater. It entitles her to a special discounted admission price for ever movie she sees. Keisha wrote an equation that gives the total cost of seeing movies. In standard form, the equation is 7 = 31. a. What was the cost of the pass? b. What is the discounted admission price for each movie? 86 SpringBoard Mathematics Algebra 1, UNIT 1 Linear Equations and Functions

22 Vertical and Horizontal Lines LESSON 3-5 Learning Targets Identif the slopes of horizontal and vertical lines. Write equations of horizontal and vertical lines. Graph and analze functions and relations. Juan is getting lunch at Mi d, a make our own salad store where the cost of a salad with unlimited toppings is $ Let c represent the cost of a salad with t toppings. Write ordered pairs in the form (t, c) to represent when Juan gets no toppings, 1 topping, toppings, 3 toppings, and toppings. LEARNING STRATEGIES Using a Pattern Think-Pair-Share (1, 7); (, 7); (3, 7); (, 7). Plot the ordered pairs from Item 1 on the coordinate plane. MP Make use of structure. What patterns do ou notice in the points ou plotted in Item? Sample answer: The points form a horizontal line.. How does the cost of a salad change depending on the number of toppings? What does this mean about the slope of the line? Sample answer: The cost of the salad doesn't change. This means that the slope is Write an equation relating the cost c of a salad to the number of toppings t. c = 7 + (0)t or c = 7 ACTIVITY 3 Linear Concepts 87

23 LESSON 3-5 Vertical and Horizontal Lines 6. A horizontal line passes through the point (5, 3). a. What are three more points on the line? Vocabular MATHEMATICS A horizontal line has the same -values for ever point and a slope of 0. Notice that horizontal contains the word horizon. Where might ou see the horizon in the real world? Answers will var. Points should each have -3 as the -value. b. Write the equation of the line. = 7 c. What is the slope of the line? 0 7. Rewrite the ordered pairs from Item 1 with the coordinates reversed. (7, 1); (7, ); (7, 3); (7, ) 8. Plot the ordered pairs from Item 7 on the coordinate plane Describe the values ou plotted in Item 8. How do the relate to the -ais? Sample answer: The points form a line that is parallel to the -ais. 88 SpringBoard Mathematics Algebra 1, UNIT 1 Linear Equations and Functions

24 Vertical and Horizontal Lines LESSON 3-5 INTRODUCING THE STRATEGY: Think-Pair-Share Think-Pair-Share is a strateg that allows ou to first think through a problem on our own and then pair with a partner to discuss ideas and concludes with sharing our results with the class. 10. Determine the slope of the line of the data from Item 7. Sample answer: There is no slope. 11. Write the equation of the data from Item 7. = 7 Vocabular MATHEMATICS A vertical line has the same -values and a slope that is undefined. The equation of a vertical line is written as = k, where k is a constant. 1. Does the equation of a vertical line represent a function? Wh or wh not? No; Sample answer: It does not represent a function because there is more than one -value when =. Functions onl have one -value for each value of. MATH TIP Recall that in a function, each input has a unique output. Check Your Understanding 13. Write the equation of the line shown in the graph Write an equation of a vertical line that passes through the point ( 7, 3). ACTIVITY 3 Linear Concepts 89

25 LESSON 3-5 Vertical and Horizontal Lines Lesson 3-5 Practice MP 1. Reason abstractl. What is the equation of the line that passes through the points ( 3, ) and (8, )? Does the equation describe a function? Eplain.. Write the equation of a line with undefined slope that passes through the point ( 15, 13). 3. Do the points shown represent a linear function? Justif our answer Write the equation of the line parallel to the line = 30 that passes through the point ( 1, 17). 5. What are the equations of the horizontal and vertical lines that intersect at the point ( 5, 6)? 90 SpringBoard Mathematics Algebra 1, UNIT 1 Linear Equations and Functions

26 Piecewise-Defined Functions LESSON 3-6 Learning Targets Determine a reasonable domain and range of a linear function. Evaluate a function at specific inputs within the function s domain. Graph piecewise-defined functions, including absolute-value functions. LEARNING STRATEGIES Creating Representations Olivia works at a veterinarian s office, and her job is to feed the dogs that are housed there. A complete feeding chart is posted on the office wall. Barko Dog Food Feeding Chart Weight of Dog (pounds) Dail Amount of Barko (ounces) over 100 pounds 60 ounces plus 1 ounce for each additional 10 pounds of weight Adult dogs that weigh less than 0 pounds are classified as small dogs. Dogs that weigh 0 to 100 pounds are classified as mid-size dogs. Finall, dogs that weigh more than 100 pounds are classified as large dogs. 1. Write a function that epresses A(w), the ounces of dog food, in terms of w, the dog s weight in pounds, for mid-size dogs. Aw ( ) = ( w - 0) or Aw ( ) = 0.5w a. Remember that this function is true onl for mid-size dogs. Describe the appropriate input values for the function. Input values from 0 pounds up to 100 pounds are appropriate. b. Use our description of the appropriate input values to epress the domain of the function using set notation. domain = { w: 0 w 100 } ACTIVITY 3 Linear Concepts 91

27 LESSON 3-6 Piecewise-Defined Functions Olivia knows that she also has several large dogs to feed. When she looks at the chart, she reads the instruction 60 ounces plus 1 ounce for each additional 10 pounds of weight. SYMBOL SENSE The smbols <, >,, and are inequalit smbols. < means "less than." > means "greater than." means "less than or equal to." means "greater than or equal to." 3. How much additional dog food should large dogs be fed for each pound of weight greater than 100 pounds? MP3 Large dogs should receive 0.1 ounce of dog food for each additional pound of weight over 100 pounds.. Critique the reasoning of others. A 10-pound Great Dane has arrived for a short sta at the kennel. Olivia sas that the dog should be fed 6 ounces of food per da. Her friend Chase sas that the dog requires 100 ounces of food per da. Who is correct? Justif our response. Olivia; because the Great Dane weighs 0 pounds more than the 100-pound minimum for large dogs, it should receive additional ounces of food, or 60 + = 6 ounces altogether. 5. Write a function that epresses the amount of food, A(w), in ounces that a large dog should be fed, as a function of w, the weight of the dog in pounds. Determine the domain and range of the function. A(w) = (w - 100) or A(w) = 0.1w + 50 domain: {w w >100} ; range: {A(w) A(w) > 60} Olivia realizes that there are three different algebraic feeding rules to follow because dogs are different sizes. She organizes her feeding rules in a list so that she can quickl refer to them whenever she has to decide how much food to feed a dog. 6. Complete Olivia s list b writing the appropriate function. Indicate the domain b writing the appropriate inequalit smbols. A(w) = w, when 0 < w < 0 A(w) = 0.5w + 10, when 0 w 100 A(w) = 0.1w + 50, when w > The vet has a German shepherd named Ma, and Olivia knows that the vet feeds Ma 63 ounces of food each da. Olivia also knows that the vet feeds her cocker spaniel Min 3 ounces of food each da. If the two dogs are being correctl fed, what is each dog s weight? Eplain our reasoning. Since Ma receives 63 ounces of dog food, Ma must be a large dog; solving the equation 63 = 0.1w + 50 for w, w = 130 pounds. Since Min receives 3 ounces of food, Min must be a mid-size dog; solving the equation 0.5w + 10 = 3 for w, w = pounds. 9 SpringBoard Mathematics Algebra 1, UNIT 1 Linear Equations and Functions

28 Piecewise-Defined Functions LESSON 3-6 Olivia decides to make a table that lists the weight of the dogs she will be feeding, in the order that she will feed them. A portion of Olivia s table is shown. 8. Complete the table using the rules ou wrote in Item 6. Weight of Dog (pounds) Amount of Dog Food (ounces) Vocabular MATHEMATICS A piecewise-defined function is a function that is defined differentl for different disjoint intervals in its domain. Tr using this term in our response to Item 9c. The functions in Item 6 can be written as piecewise-defined functions. Ever possible weight w has eactl one feeding amount A assigned to it, but the rule for determining that feeding amount changes for dogs of different sizes. The different feeding rules, along with their domains, are considered to be the pieces of a single piecewise-defined function. 9. Olivia decides to write a piecewise-defined function to represent the functions she wrote in Item 6. a. Eplain how ou know that each equation in Item 6 represents a function. For each input value, there is eactl one output value. Each equation represents a linear function. b. Complete the piecewise-defined function for the feeding rules. w, when 0 < w < 0 Aw ( ) = 0.5w + 10, when 0 w w + 50, when w > 100 c. Eplain wh the equation from Part (b) is also a function. Use the domain of each rule to help justif our answer. Each rule represents a function with the same domain as the corresponding part of the piecewise-defined function. Since none of the domains overlap, each input value still has eactl one output value. Therefore, the equation represents a function. ACTIVITY 3 Linear Concepts 93

29 LESSON 3-6 Piecewise-Defined Functions Check Your Understanding A wholesale grocer store has the following sale on mied nuts: Vocabular ACADEMIC To identif means to show what something is. When ou identif the domain and range in Items 10 and 11, remember to use set notation. SALE! Mied nuts Less than lbs: $/lb 10 lbs: $6 + $0.50/lb 10. Write a function to represent the cost C() of buing less than pounds of mied nuts. Identif the domain and range. 11. Write a function to represent the cost C() of buing to 10 pounds of mied nuts. Identif the domain and range. 1. Make a table to show the cost of, 5, and 8 pounds of mied nuts. 13. Use our answers to Items 10 and 11 to write a piecewise-defined function to represent the cost C() of pounds of mied nuts. 1. Olivia must feed a dog that weighs 57 pounds. a. Which piece of the feeding function should Olivia use? Eplain our answer. 0.5w + 10; the given weight, 57 pounds, is between 0 and 100 pounds. b. Use our answer to Part (a) to evaluate A(57). A(57) = 38.5 CONNECT TO TECHNOLOGY To graph a piecewisedefined function in our calculator, enter the function into Y =, in dot mode, using parentheses to indicate the domain intervals. For eample f = - ( ) - 1, when 1 +, when 1 would be entered as Y 1 = ( 1)( < 1) + ( + ) ( 1). Go to the Test menu for inequalit smbols. MP7 15. Make use of structure. When graphing a piecewise-defined function, it is necessar to graph each piece of the function onl for its appropriate interval of the domain. Graph the feeding function on the aes. When finished, our graph should consist of three line segments. w Weight of dog (pounds) 9 SpringBoard Mathematics Algebra 1, UNIT 1 Linear Equations and Functions Amount of dog food (ounces) A

30 Piecewise-Defined Functions LESSON Wh is the graph of a piecewise-defined function more useful than three separate graphs? The graph of the piecewise-defined function includes each classification of size, so it is not necessar to first determine which graph to use based on the classification of size. It also combines all of the same information on one graph, so it requires less space. 17. How can ou conclude that the graph represents a function? No part of a line segment lies above or below part of another line segment. This means each w-value is assigned eactl one A-value. Check Your Understanding Use the piecewise-defined function for Items 18 and , when 3 < 1 f( ) =, when 1 < 1, when < < Find f(1) and f(). Eplain which piece of the function ou used to find each value. 19. Sketch a graph of the function. Absolute-value functions are a special tpe of piece-wise defined function. The presence of the absolute values in a function suggests that the function behaves differentl for different values of the variable. 0. Graph the function f() = Use our graph of the absolute-value function f() = 3 to write the function without absolute value. 3, when 0 f ( ) = {-3, when 0 ACTIVITY 3 Linear Concepts 95

31 LESSON 3-6 Piecewise-Defined Functions Check Your Understanding. How does writing an absolute-value function as a piecewise-defined function help ou understand the graph of the function? Sample answer: Writing the function f() = 3 as a piece-wise defined function helped me see the constraints of the absolute value function the same wa that creating a graph would. 3. Write the function f() = + 8 as a piece-wise defined function without using absolute value. Lesson 3-6 Practice MP 1. Model with mathematics. If Pam works more than 0 hours per week, her hourl wage for ever hour over 0 is 1.5 times her normal hourl wage of $8.5. Write a piecewise-defined function that gives Pam s weekl pa P(h) in terms of the number of hours h that she works. Then graph the function. Graph the piecewise-defined function. >. =, when 0 f( ),when < 3. = 3,when 1 f( ) - + 3, when 1 For Items and 5, consider the piecewise-defined function 1 - = 10, when 3 f( ) -- 1, when > 3.. Find f( ). 5. Find f(6). 96 SpringBoard Mathematics Algebra 1, UNIT 1 Linear Equations and Functions

32 Linear Concepts ACTIVITY 3 ACTIVITY 3 PRACTICE Write our answers on notebook paper. Show our work. Lesson 3-1 Solve each equation for the indicated variable. 1. w = gm, for m. Q= 1 P+ 15, for P 3. I = V, for R R. = m + b, for m 5. The equation f = d + e + t can be used to find an athlete s final score f in an Olmpic trampoline event, where d is the difficult score, e is the eecution score, and t is the time of flight score. An athlete s final score is His difficult score is 1.6, and his eecution score is.9. Solve the equation for t to find the athlete s time of flight score. Lesson 3- Use the following information for Items 6 10: A tank at a factor has 6 gallons of water remaining after a leak occurs. Water is leaking out of the tank at a rate of 1 gallon per hour. 6. Write an equation that epresses the amount of water in the tank, A, as a function of the number of hours, h, since the leak occurred. Use function notation. 7. Graph the function. 8. How man gallons of water were in the tank before the leak occurred? Eplain how ou know b comparing features of our equation and our graph. 9. Use the equation that ou wrote for Item 6 to determine the amount of water that will be left in the tank after 13 hours. 10. Suppose that the rate of leakage increased at the point when the water had been leaking for 1 hours. The table shows the amount of water in the tank starting at 1 hours. Time Since Leak Occurred (h) Amount of Water in the Tank (gal) Write a new function, N(h), that models the amount of water in the tank for the last minutes of leakage. Lesson 3-3 For Items 11 and 1, find the point-slope equation of the line with the given slope and point. 11. m = 5; (-3, 6) 1. m = 5 ; (0, 9) 13. Write the point-slope equation of the line through (-8, ) and (-6, -). Use the following information for Items 1 and 15: Irina deposited some mone she earned working a summer job into a savings account. Since then, she has added $75 to the account at the beginning of ever month. At the end of her fifth month of saving, she had deposited a total of $1095 in the account. 1. Write the point-slope equation of the line that models the amount of mone in Irina s account, A, as a function of the number of months, m, that have passed since her initial investment. 15. Use the equation that ou wrote in Item 1 to find the amount that Irina will have deposited in her account b the end of the 15th month. ACTIVITY 3 Linear Concepts 97

33 ACTIVITY 3 Linear Concepts Lesson Write an equation in standard form that represents the data on the table: /3 6 Items 17 0 are based on the following information: As part of his fitness routine, Andre hikes or jogs a total of 75 miles. He hikes at a rate of 3 miles per hour and jogs at a rate of 5 miles per hour. 17. Write an equation that represents the different amounts of time that Andre can hike,, and jog,, each month to meet his fitness goals. 18. Graph the equation. 19. If Andre plans on hiking for 10 hours during one month, how man hours should he spend jogging in order to meet his goal? 0. What is the -intercept? What does it represent? Eplain.. Write the equation of the line parallel to the line = -5 that passes through the point (-8, -9). 5. A horizontal line and a vertical line intersect at the point (19, ). What are the equations of the lines? Lesson Speed Cell Wireless offers a plan of $0 for the first 00 minutes and an additional $0.50 for ever minute over 00. Let t represent the total talk time in minutes. Write a piecewise-defined function to represent the cost C(t). 7. A man walks for 5 minutes at a rate of 3 mi/h, then jogs for 75 minutes at a rate of 5 mi/h, then rests for 30 minutes, and finall walks for 90 minutes at a rate of 3 mi/h. Write a piecewisedefined function epressing the distance D(t) he traveled as a function of time t. MP 8. Model with mathematics. A state income ta law reads as follows: Lesson Write the equation of a line with zero slope that passes through the point (8, -3).. The table shows points on a line. Does the line represent a linear function? Use the equation of the line to justif our answer a. What is the equation of the line that passes through the points (-0, ) and (1, )? b. Does the equation describe a function? Eplain wh there is no variable in the equation. Annual Income (dollars) Ta < $000 $0 ta $000 - $6000 % of income over $000 > $6000 $80 plus 5% of income over $6000 Write a piecewise-defined function to represent the income ta law. For Items 9 and 30, consider the following piecewisedefined function: 1 + 3, < f( ) = when , when 1 9. Graph the function. 30. Find a. f(-5) b. f(3) 98 SpringBoard Mathematics Algebra 1, UNIT 1 Linear Equations and Functions

34 Linear Equations: Storm Weather EMBEDDED ASSESSMENT 3 When measuring the wind speed of a tropical cclone, meteorologists don t want to get hit b fling debris. The use a measure the can get in the comfort of their TV studio. Luckil for the meteorologists, the barometric pressure is related to the wind speed linearl. Suppose scientists compared the wind speeds and barometric pressure readings of two tpical tropical cclones. For one tropical cclone, the barometric pressure was 1000 mb (measured in millibars), and it had maimum wind speeds of 100 km/h (measured in kilometers per hour). The second tropical cclone had a barometric pressure of 960 mb and maimum wind speeds of 180 km/h. 1. Develop a rule for an barometric pressure that ou could use to predict the wind speed.. Use our rule to predict the wind speed of a tropical cclone with a barometric pressure of 980 mb. Using what ou know about linearit, eplain wh our prediction is reasonable. 3. Interpret the intercept for wind speed in this contet as it pertains to the weather.. Eplain the meaning of slope in this contet. UNIT 1 Linear Equations and Functions 99

35 EMBEDDED ASSESSMENT 3 Linear Equations: Storm Weather Scoring Guide Problem Solving Computation & Eecution Communication Modeling Proficient Developing Emerging Incomplete The selected strategies would, if eecuted correctl, result in a correct and efficient answer. Important patterns, relationships, and/or properties in the problem are identified eplicitl. Solutions are accurate with at most one computational error. Solutions are accurate with no conceptual errors. Work clearl supports all solutions. Mathematical language is used to support reasoning in a clear, concise, and accurate wa. Use of nonmathematical language is clear and concise. Models are full developed and relevant to the given contet. The selected strategies would, if eecuted correctl, result in a correct answer. Important patterns, relationships, and/or properties in the problem are identified, but not eplicitl. Solutions contain at most two computational errors or omissions. Solutions contain at most one conceptual error. Work clearl supports all solutions with at most one area of inconsistenc or lack of clarit. Mathematical language is used to support reasoning in a mostl clear, concise, and accurate wa. Use of nonmathematical language is mostl clear and concise. Models are mostl developed and relevant to the given contet. The selected strategies would not result in a correct answer. Important patterns, relationships, and/or properties in the problem are somewhat identified. Solutions contain at most three computational errors or omissions. Solutions contain at most two conceptual errors. Work supports all or some solutions with some areas of inconsistenc or lack of clarit and ma be disorganized. Mathematical language lacks clarit, conciseness, and/or accurac. Use of nonmathematical language lacks clarit, conciseness, and/or accurac. Models are partiall developed and somewhat relevant to the given contet. No strategies are selected. Important patterns, relationships, and/or properties in the problem are not identified. Solutions contain more than three computational errors or omissions. Solutions contain three or more conceptual errors. Work does not support most solutions and ma be disorganized. Mathematical language is not used. Nonmathematical language is not used. Models are undeveloped and irrelevant to the given contet. 100 SpringBoard Mathematics Algebra 1