4.4 Scatter Plots and Lines of Fit 4.5 Analyzing Lines of Fit 4.6 Arithmetic Sequences 4.7 Piecewise Functions

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1 Writing Linear Functions. Writing Equations in Slope-Intercept Form. Writing Equations in Point-Slope Form.3 Writing Equations of Parallel and Perpendicular Lines. Scatter Plots and Lines of Fit.5 Analzing Lines of Fit. Arithmetic Sequences.7 Piecewise Functions Chapter Learning Target: Understand writing linear functions. Chapter Success Criteria: I can identif and write different forms of linear equations. I can interpret scatter plots and identif the correlation between data sets. I can analze lines of fit. I can write a function that represents an arithmetic sequence to solve real-life problems. Karaoke Machine (p. ) Old Faithful Geser (p. ) Helicopter Rescue (p. 9) SEE the Big Idea School Spirit (p. 8) Renewable Energ (p. 78)

2 Maintaining Mathematical Proficienc Using a Coordinate Plane Eample What ordered pair corresponds to point A? B C A G F D E Point A is 3 units to the left of the origin and units up. So, the -coordinate is 3 and the -coordinate is. The ordered pair ( 3, ) corresponds to point A. Use the graph to answer the question.. What ordered pair corresponds to point G?. What ordered pair corresponds to point D? 3. Which point is located in Quadrant I?. Which point is located in Quadrant IV? Rewriting Equations Eample Solve the equation 3 = 8 for. 3 = = 8 3 = 8 3 = 8 3 Write the equation. Subtract 3 from each side. Simplif. Divide each side b. Solve the equation for. = + 3 Simplif. 5. = = 7. = = 9. + = 7. = ABSTRACT REASONING Both coordinates of the point (, ) are multiplied b a negative number. How does this change the location of the point? Be sure to consider points originall located in all four quadrants. Dnamic Solutions available at BigIdeasMath.com 73

3 Mathematical Practices Mathematicall profi cient students tr simpler forms of the original problem. Problem-Solving Strategies Core Concept Solve a Simpler Problem When solving a real-life problem, if the numbers in the problem seem complicated, then tr solving a simpler form of the problem. After ou have solved the simpler problem, look for a general strateg. Then appl that strateg to the original problem. Monitoring Progress Using a Problem-Solving Strateg In the deli section of a grocer store, a half pound of sliced roast beef costs $3.9. You bu.8 pounds. How much do ou pa? Step Solve a simpler problem. Suppose the roast beef costs $3 per half pound, and ou bu pounds. $3 Total cost = lb Use unit analsis to write a verbal model. / lb = $ lb lb Rewrite $3 per = $ Simplif. In the simpler problem, ou pa $. Step Appl the strateg to the original problem.. You work 37 hours and earn $35.5. What is our hourl wage? pound as $ per pound. Total cost = $3.9.8 lb Use unit analsis to write a verbal model. / lb = $.38 lb.8 lb Rewrite $3.9 per = $.55 Simplif. In the original problem, ou pa $.55. pound as $.38 per pound.. You drive.5 miles and use 7.5 gallons of gasoline. What is our car s gas mileage (in miles per gallon)? Your answer is reasonable because ou bought about pounds. 3. You drive 3 miles in. hours. At the same rate, how long will it take ou to drive 5 miles? 7 Chapter Writing Linear Functions

4 . Writing Equations in Slope-Intercept Form Essential Question Given the graph of a linear function, how can ou write an equation of the line? Writing Equations in Slope-Intercept Form Work with a partner. Find the slope and -intercept of each line. Write an equation of each line in slope-intercept form. Use a graphing calculator to verif our equation. a. b. (, 3) (, ) 9 (, ) 9 9 (, ) 9 c. d. INTERPRETING MATHEMATICAL RESULTS To be proficient in math, ou need to routinel interpret our results in the contet of the situation. The reason for studing mathematics is to enable ou to model and solve real-life problems. 9 ( 3, 3) (3, ) 9 9 Mathematical Modeling (, ) (, ) Work with a partner. The graph shows the cost of a smartphone plan. a. What is the -intercept of the line? Interpret the -intercept in the contet of the problem. b. Approimate the slope of the line. Interpret the slope in the contet of the problem. c. Write an equation that represents the cost as a function of data usage. Cost per month (dollars) Smartphone Plan Data usage (megabtes) 9 Communicate Your Answer 3. Given the graph of a linear function, how can ou write an equation of the line?. Give an eample of a graph of a linear function that is different from those above. Then use the graph to write an equation of the line. Section. Writing Equations in Slope-Intercept Form 75

5 . Lesson Core Vocabular linear model, p. 78 Previous slope-intercept form function rate What You Will Learn Write equations in slope-intercept form. Use linear equations to solve real-life problems. Writing Equations in Slope-Intercept Form Using Slopes and -Intercepts to Write Equations Write an equation of each line with the given slope and -intercept. a. slope = 3; -intercept = b. slope = ; -intercept = a. = m + b Write the slope-intercept form. = 3 + Substitute 3 for m and for b. An equation is = 3 +. b. = m + b Write the slope-intercept form. = + ( ) Substitute for m and for b. = Simplif. An equation is =. Using Graphs to Write Equations Write an equation of each line in slope-intercept form. a. (, 3) b. (, ) STUDY TIP You can use an two points on a line to find the slope. STUDY TIP After writing an equation, check that the given points are solutions of the equation. (, 3) a. Find the slope and -intercept. Let (, ) = (, 3) and (, ) = (, 3). m = = 3 ( 3) =, or 3 (, ) Because the line crosses the -ais at (, 3), the -intercept is 3. So, the equation is = 3 3. b. Find the slope and -intercept. Let (, ) = (, ) and (, ) = (, ). m = = = 3, or 3 Because the line crosses the -ais at (, ), the -intercept is. So, the equation is = Chapter Writing Linear Functions

6 Using Points to Write Equations Write an equation of each line that passes through the given points. a. ( 3, 5), (, ) b. (, 5), (8, 5) REMEMBER If f is a function and is in its domain, then f() represents the output of f corresponding to the input. a. Find the slope and -intercept. m = 5 ( 3) = Because the line crosses the -ais at (, ), the -intercept is. So, an equation is =. Writing a Linear Function b. Find the slope and -intercept. 5 ( 5) m = = 8 Because the line crosses the -ais at (, 5), the -intercept is 5. So, an equation is = 5. Write a linear function f with the values f() = and f() = 3. Step Write f() = as (, ) and f() = 3 as (, 3). Step Find the slope of the line that passes through (, ) and (, 3). 3 m = =, or Step 3 Write an equation of the line. Because the line crosses the -ais at (, ), the -intercept is. = m + b Write the slope-intercept form. = + Substitute for m and for b. A function is f() = +. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Write an equation of the line with the given slope and -intercept.. slope = 7; -intercept =. slope = ; -intercept = 3 Write an equation of the line in slope-intercept form. 3.. (, ) (, 3) (, ) (5, 3) 5. Write an equation of the line that passes through (, ) and (, ).. Write a linear function g with the values g() = 9 and g(8) = 7. Section. Writing Equations in Slope-Intercept Form 77

7 Solving Real-Life Problems A linear model is a linear function that models a real-life situation. When a quantit changes at a constant rate with respect to a quantit, ou can use the equation = m + b to model the relationship. The value of m is the constant rate of change, and the value of b is the initial, or starting, value of. Modeling with Mathematics Ecluding hdropower, U.S. power plants used renewable energ sources to generate 5 million megawatt hours of electricit in 7. B, the amount of electricit generated had increased to 9 million megawatt hours. Write a linear model that represents the number of megawatt hours generated b non-hdropower renewable energ sources as a function of the number of ears since 7. Use the model to predict the number of megawatt hours that will be generated in 7.. Understand the Problem You know the amounts of electricit generated in two distinct ears. You are asked to write a linear model that represents the amount of electricit generated each ear since 7 and then predict a future amount.. Make a Plan Break the problem into parts and solve each part. Then combine the results to help ou solve the original problem. Part Define the variables. Find the initial value and the rate of change. Part Write a linear model and predict the amount in Solve the Problem Part Let represent the time (in ears) since 7 and let represent the number of megawatt hours (in millions). Because time is defined in ears since 7, 7 corresponds to = and corresponds to = 5. Let (, ) = (, 5) and (, ) = (5, 9). The initial value is the -intercept b, which is 5. The rate of change is the slope m. Part 7 corresponds to =. m = = = 5 =.8 Megawatt hours = Initial + Rate of Years (millions) value change since 7 = = Write the equation. = 5 +.8() Substitute for. = 333 Simplif. The linear model is = The model predicts non-hdropower renewable energ sources will generate 333 million megawatt hours in 7.. Look Back To check that our model is correct, verif that (, 5) and (5, 9) are solutions of the equation. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 7. The corresponding data for electricit generated b hdropower are 8 million megawatt hours in 7 and 77 million megawatt hours in. Write a linear model that represents the number of megawatt hours generated b hdropower as a function of the number of ears since Chapter Writing Linear Functions

8 . Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. COMPLETE THE SENTENCE A linear function that models a real-life situation is called a.. WRITING Eplain how ou can use slope-intercept form to write an equation of a line given its slope and -intercept. Monitoring Progress and Modeling with Mathematics In Eercises 3 8, write an equation of the line with the given slope and -intercept. (See Eample.) 3. slope:. slope: -intercept: 9 -intercept: 5 5. slope: 3. slope: 7 -intercept: -intercept: 7. slope: 3 8. slope: 3 -intercept: 8 -intercept: In Eercises 9, write an equation of the line in slope-intercept form. (See Eample.) ( 3, ) (, ) (3, 3) In Eercises 3 8, write an equation of the line that passes through the given points. (See Eample 3.) 3. (3, ), (, ). (, 7), (, 5) (, ) (, 3) (, ) (, ) (, ) In Eercises 9, write a linear function f with the given values. (See Eample.) 9. f() =, f() =. f() = 7, f(3) =. f() = 3, f() =. f(5) =, f() = 5 3. f( ) =, f() =. f() = 3, f( ) = 3 In Eercises 5 and, write a linear function f with the given values. 5. f() 3. f() 7. ERROR ANALYSIS Describe and correct the error in writing an equation of the line with a slope of and a -intercept of 7. = ERROR ANALYSIS Describe and correct the error in writing an equation of the line shown. slope = 5 = 3 5 = 3 5 (, ) (5, ) 5. (, ), (, ). (, ), (, ) = (, 5), (.5, ) 8. (, 3), ( 5,.5) Section. Writing Equations in Slope-Intercept Form 79

9 9. MODELING WITH MATHEMATICS In 9, the world record for the men s mile was 3.9 minutes. In 98, the record time was 3.8 minutes. (See Eample 5.) a. Write a linear model that represents the world record (in minutes) for the men s mile as a function of the number of ears since 9. b. Use the model to estimate the record time in and predict the record time in. 3. MODELING WITH MATHEMATICS A recording studio charges musicians an initial fee of $5 to record an album. Studio time costs an additional $75 per hour. a. Write a linear model that represents the total cost of recording an album as a function of studio time (in hours). b. Is it less epensive to purchase hours of recording time at the studio or a $75 music software program that ou can use to record on our own computer? Eplain. 3. WRITING A line passes through the points (, ) and (, 5). Is it possible to write an equation of the line in slope-intercept form? Justif our answer. 3. THOUGHT PROVOKING Describe a real-life situation involving a linear function whose graph passes through the points (, ) (, 8) MAKING AN ARGUMENT Your friend claims that given f() and an other value of a linear function f, ou can write an equation in slope-intercept form that represents the function. Your cousin disagrees, claiming that the two points could lie on a vertical line. Who is correct? Eplain. 35. ANALYZING A GRAPH Line is a reflection in the -ais of line k. Write an equation that represents line k. 3. HOW DO YOU SEE IT? The graph shows the approimate U.S. bo office revenues (in billions of dollars) from to, where = represents the ear. U.S. Bo Office Revenue 8 8 Year ( ) Revenue (billions of dollars) a. Estimate the slope and -intercept of the graph. b. Interpret our answers in part (a) in the contet of the problem. c. How can ou use our answers in part (a) to predict the U.S. bo office revenue in 8? (, ) (3, ) 33. REASONING Recall that the standard form of a linear equation is A + B = C. Rewrite this equation in slope-intercept form. Use our answer to find the slope and -intercept of the graph of the equation + 5 = 9. Maintaining Mathematical Proficienc 37. ABSTRACT REASONING Show that the equation of the line that passes through the points (, b) and (, b + m) is = m + b. Eplain how ou can be sure that the point (, b m) also lies on the line. Reviewing what ou learned in previous grades and lessons Solve the equation. (Section.3) 38. 3( 5) = = ( 3). (3d + 3) = 7 + d. 5( 3n) = (n ) Use intercepts to graph the linear equation. (Section 3.). + = = 5. = 5. 7 = 8 Chapter Writing Linear Functions

10 . Writing Equations in Point-Slope Form Essential Question How can ou write an equation of a line when ou are given the slope and a point on the line? Writing Equations of Lines Work with a partner. Sketch the line that has the given slope and passes through the given point. Find the -intercept of the line. Write an equation of the line. a. m = b. m = USING A GRAPHING CALCULATOR To be proficient in math, ou need to understand the feasibilit, appropriateness, and limitations of the technological tools at our disposal. For instance, in real-life situations such as the one given in Eploration 3, it ma not be feasible to use a square viewing window on a graphing calculator. Writing a Formula Work with a partner. The point (, ) is a given point on a nonvertical line. The point (, ) is an other point on the line. Write an equation that represents the slope m of the line. Then rewrite this equation b multipling each side b the difference of the -coordinates to obtain the point-slope form of a linear equation. Writing an Equation Work with a partner. For four months, ou have saved $5 per month. You now have $75 in our savings account. a. Use our result from Eploration to write an equation that represents the balance A after t months. b. Use a graphing calculator to verif our equation. Communicate Your Answer Account balance (dollars) A 5 (, ) Savings Account. How can ou write an equation of a line when ou are given the slope and a point on the line? 5. Give an eample of how to write an equation of a line when ou are given the slope and a point on the line. Your eample should be different from those above. 5 5 (, 75) (, ) t Time (months) Section. Writing Equations in Point-Slope Form 8

11 . Lesson Core Vocabular point-slope form, p. 8 Previous slope-intercept form function linear model rate What You Will Learn Write an equation of a line given its slope and a point on the line. Write an equation of a line given two points on the line. Use linear equations to solve real-life problems. Writing Equations of Lines in Point-Slope Form Given a point on a line and the slope of the line, ou can write an equation of the line. Consider the line that passes through (, 3) and has a slope of. Let (, ) be another point on the line where. You can write an equation relating and using the slope formula with (, ) = (, 3) and (, ) = (, ). m = = 3 Write the slope formula. Substitute values. ( ) = 3 Multipl each side b ( ). The equation in point-slope form is 3 = ( ). Core Concept Point-Slope Form Words A linear equation written in the form = m( ) is in point-slope form. The line passes through the point (, ), and the slope of the line is m. (, ) slope (, ) Algebra = m( ) passes through (, ) Using a Slope and a Point to Write an Equation Check 3 = ( + 8) 3 3 =? ( 8 + 8) = Write an equation in point-slope form of the line that passes through the point ( 8, 3) and has a slope of. = m( ) Write the point-slope form. 3 = [ ( 8)] Substitute for m, 8 for, and 3 for. 3 = ( + 8) Simplif. The equation is 3 = ( + 8). Monitoring Progress 8 Chapter Writing Linear Functions Help in English and Spanish at BigIdeasMath.com Write an equation in point-slope form of the line that passes through the given point and has the given slope.. (3, ); m =. (, ); m = 3

12 ANOTHER WAY You can use either of the given points to write an equation of the line. Use m = and (3, ). ( ) = ( 3) + = + = + Writing Equations of Lines Given Two Points When ou are given two points on a line, ou can write an equation of the line using the following steps. Step Find the slope of the line. Step Use the slope and one of the points to write an equation of the line in point-slope form. Using Two Points to Write an Equation Write an equation in slope-intercept form of the line shown. Step Find the slope of the line. m = 3 =, or Step Use the slope m = and the point (, ) to write an equation of the line. = m( ) Write the point-slope form. = ( ) Substitute for m, for, and for. = + = + Distributive Propert Write in slope-intercept form. (, ) 3 5 (3, ) The equation is = +. Writing a Linear Function Write a linear function f with the values f() = and f(8) =. Note that ou can rewrite f() = as (, ) and f(8) = as (8, ). Step Find the slope of the line that passes through (, ) and (8, ). m = ( ) 8 =, or.5 Step Use the slope m =.5 and the point (8, ) to write an equation of the line. = m( ) Write the point-slope form. =.5( 8) Substitute.5 for m, 8 for, and for. =.5 =.5 8 A function is f() =.5 8. Distributive Propert Write in slope-intercept form. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Write an equation in slope-intercept form of the line that passes through the given points. 3. (, ), (3, ). (, ), (8, ) 5. Write a linear function g with the values g() = 3 and g() = 5. Section. Writing Equations in Point-Slope Form 83

13 Solving Real-Life Problems Modeling with Mathematics The student council is ordering customized foam hands to promote school spirit. The table shows the cost of ordering different numbers of foam hands. Can the situation be modeled b a linear equation? Eplain. If possible, write a linear model that represents the cost as a function of the number of foam hands. Number of foam hands 8 Cost (dollars) Understand the Problem You know five data pairs from the table. You are asked whether the data are linear. If so, write a linear model that represents the cost.. Make a Plan Find the rate of change for consecutive data pairs in the table. If the rate of change is constant, use the point-slope form to write an equation. Rewrite the equation in slope-intercept form so that the cost is a function of the number of foam hands. 3. Solve the Problem Step Find the rate of change for consecutive data pairs in the table. 3 =, 58 8 =, =, 8 7 = Because the rate of change is constant, the data are linear. So, use the pointslope form to write an equation that represents the data. Step Use the constant rate of change (slope) m = and the data pair (, 3) to write an equation. Let C be the cost (in dollars) and n be the number of foam hands. C C = m(n n ) Write the point-slope form. C 3 = (n ) Substitute for m, for n, and 3 for C. C 3 = n C = n + Distributive Propert Write in slope-intercept form. Because the cost increases at a constant rate, the situation can be modeled b a linear equation. The linear model is C = n +.. Look Back To check that our model is correct, verif that the other data pairs are solutions of the equation. = () + 58 = (8) + 7 = () + 8 = () + Number of months Total cost (dollars) Monitoring Progress Help in English and Spanish at BigIdeasMath.com. You pa an installation fee and a monthl fee for Internet service. The table shows the total cost for different numbers of months. Can the situation be modeled b a linear equation? Eplain. If possible, write a linear model that represents the total cost as a function of the number of months. 8 Chapter Writing Linear Functions

14 . Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. USING STRUCTURE Without simplifing, identif the slope of the line given b the equation 5 = ( + 5). Then identif one point on the line.. WRITING Eplain how ou can use the slope formula to write an equation of the line that passes through (3, ) and has a slope of. Monitoring Progress and Modeling with Mathematics In Eercises 3, write an equation in point-slope form of the line that passes through the given point and has the given slope. (See Eample.) 3. (, ); m =. (3, 5); m = 5. (7, ); m =. ( 8, ); m = 5 7. (9, ); m = 3 8. (, ); m = 9. (, ); m = 3. (5, ); m = 5 In Eercises, write an equation in slope-intercept form of the line shown. (See Eample.). 3 (, 3) (3, ) 3 5. (, ) (, 5) In Eercises, write a linear function f with the given values. (See Eample 3.). f() =, f() =. f(5) = 7, f( ) = 3. f( ) =, f() = 3. f( ) =, f( ) = 5. f( 3) =, f(3) = 5. f( 9) =, f( ) = In Eercises 7 3, tell whether the data in the table can be modeled b a linear equation. Eplain. If possible, write a linear equation that represents as a function of. (See Eample.) (, ) (, ). (8, ) (, ) In Eercises 5, write an equation in slope-intercept form of the line that passes through the given points. 5. (7, ), (, ). (, ), (, ) 7. (, ), (3, 7) 8. (, 5), (, 5) 9. (, 9), ( 3, 9). ( 5, 9), (5, 3) 3. ERROR ANALYSIS Describe and correct the error in writing a linear function g with the values g(5) = and g(3) =. m = 3 5 = = 3 = m = 3 5 = 3 A function is g() = 3. Section. Writing Equations in Point-Slope Form 85

15 3. ERROR ANALYSIS Describe and correct the error in writing an equation of the line that passes through the points (, ) and (, 3). m = 3 = 3 = ( ) MODELING WITH MATHEMATICS You are designing a sticker to advertise our band. A compan charges $5 for the first stickers and $8 for each additional stickers. a. Write an equation that represents the total cost (in dollars) of the stickers as a function of the number (in thousands) of stickers ordered. b. Find the total cost of 9 stickers. 3. MODELING WITH MATHEMATICS You pa a processing fee and a dail fee to rent a beach house. The table shows the total cost of renting the beach house for different numbers of das. Das 8 Total cost (dollars) a. Can the situation be modeled b a linear equation? Eplain. b. What is the processing fee? the dail fee? c. You can spend no more than $ on the beach house rental. What is the maimum number of das ou can rent the beach house? 35. WRITING Describe two was to graph the equation = 3 ( ). 3. THOUGHT PROVOKING The graph of a linear function passes through the point (, 5) and has a slope of. Represent this function in two other was HOW DO YOU SEE IT? The graph shows two points that lie on the graph of a linear function. 8 a. Does the -intercept of the graph of the linear function appear to be positive or negative? Eplain. b. Estimate the coordinates of the two points. How can ou use our estimates to confirm our answer in part (a)? 39. CONNECTION TO TRANSFORMATIONS Compare the graph of = to the graph of = ( + 3). Make a conjecture about the graphs of = m and k = m( h).. COMPARING FUNCTIONS Three siblings each receive mone for a holida and then spend it at a constant weekl rate. The graph describes Sibling A s spending, the table describes Sibling B s spending, and the equation = describes Sibling C s spending. The variable represents the amount of mone left after weeks. Mone left (dollars) Spending Mone 8 (, 5) (, ) 3 5 Week Week, Mone left, $ $75 3 $5 $5 37. REASONING You are writing an equation of the line that passes through two points that are not on the -ais. Would ou use slope-intercept form or point-slope form to write the equation? Eplain. Maintaining Mathematical Proficienc Write the reciprocal of the number. (Skills Review Handbook) Reviewing what ou learned in previous grades and lessons a. Which sibling received the most mone? the least mone? b. Which sibling spends mone at the fastest rate? the slowest rate? c. Which sibling runs out of mone first? last? 8 Chapter Writing Linear Functions

16 .3 Writing Equations of Parallel and Perpendicular Lines Essential Question How can ou recognize lines that are parallel or perpendicular? Recognizing Parallel Lines Work with a partner. Write each linear equation in slope-intercept form. Then use a graphing calculator to graph the three equations in the same square viewing window. (The graph of the first equation is shown.) Which two lines appear parallel? How can ou tell? a. 3 + = b. 5 + = 3 + = + = =.5 + = USING TOOLS STRATEGICALLY To be proficient in math, ou need to use a graphing calculator and other available technological tools, as appropriate, to help ou eplore relationships and deepen our understanding of concepts. 3 = = + 3 Recognizing Perpendicular Lines Work with a partner. Write each linear equation in slope-intercept form. Then use a graphing calculator to graph the three equations in the same square viewing window. (The graph of the first equation is shown.) Which two lines appear perpendicular? How can ou tell? a. 3 + = b. + 5 = 3 = + = 3 3 =.5 = = + 3 = + 5 Communicate Your Answer 3. How can ou recognize lines that are parallel or perpendicular?. Compare the slopes of the lines in Eploration. How can ou use slope to determine whether two lines are parallel? Eplain our reasoning. 5. Compare the slopes of the lines in Eploration. How can ou use slope to determine whether two lines are perpendicular? Eplain our reasoning. Section.3 Writing Equations of Parallel and Perpendicular Lines 87

17 .3 Lesson What You Will Learn Core Vocabular parallel lines, p. 88 perpendicular lines, p. 89 Previous reciprocal READING The phrase A if and onl if B is a wa of writing two conditional statements at once. It means that if A is true, then B is true. It also means that if B is true, then A is true. Identif and write equations of parallel lines. Identif and write equations of perpendicular lines. Use parallel and perpendicular lines in real-life problems. Identifing and Writing Equations of Parallel Lines Core Concept Parallel Lines and Slopes Two lines in the same plane that never intersect are parallel lines. Two distinct nonvertical lines are parallel if and onl if the have the same slope. All vertical lines are parallel. Identifing Parallel Lines Determine which of the lines are parallel. Find the slope of each line. Line a: m = 3 ( ) = 5 Line b: m = ( 3) = 5 ( ) Line c: m = ( 3) = 5 Lines a and c have the same slope, so the are parallel. Writing an Equation of a Parallel Line a b c (, 3) ( 3, ) 3 (, ) ( 3, ) (, 5) (, ) ANOTHER WAY You can also use the slope m = and the point-slope form to write an equation of the line that passes through (5, ). = m( ) ( ) = ( 5) = Write an equation of the line that passes through (5, ) and is parallel to the line = + 3. Step Find the slope of the parallel line. The graph of the given equation has a slope of. So, the parallel line that passes through (5, ) also has a slope of. Step Use the slope-intercept form to find the -intercept of the parallel line. = m + b Write the slope-intercept form. = (5) + b Substitute for m, 5 for, and for. = b Solve for b. Using m = and b =, an equation of the parallel line is =. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. Line a passes through ( 5, 3) and (, ). Line b passes through (3, ) and (, 7). Are the lines parallel? Eplain.. Write an equation of the line that passes through (, ) and is parallel to the line = Chapter Writing Linear Functions

18 REMEMBER The product of a nonzero number m and its negative reciprocal is : m ( m ) =. Identifing and Writing Equations of Perpendicular Lines Core Concept Perpendicular Lines and Slopes Two lines in the same plane that intersect to form right angles are perpendicular lines. Nonvertical lines are perpendicular if and onl if their slopes are negative reciprocals. Vertical lines are perpendicular to horizontal lines. = + = Identifing Parallel and Perpendicular Lines Determine which of the lines, if an, are parallel or perpendicular. Line a: = + Line b: + = 3 Line c: 8 = Write the equations in slope-intercept form. Then compare the slopes. Line a: = + Line b: = + 3 Line c: = Lines b and c have slopes of, so the are parallel. Line a has a slope of, the negative reciprocal of, so it is perpendicular to lines b and c. Writing an Equation of a Perpendicular Line ANOTHER WAY You can also use the slope m = and the slope-intercept form to write an equation of the line that passes through ( 3, ). = m + b = ( 3) + b 5 = b So, = 5. Write an equation of the line that passes through ( 3, ) and is perpendicular to the line = + 3. Step Find the slope of the perpendicular line. The graph of the given equation has a slope of. Because the slopes of perpendicular lines are negative reciprocals, the slope of the perpendicular line that passes through ( 3, ) is. Step Use the slope m = and the point-slope form to write an equation of the perpendicular line that passes through ( 3, ). = m( ) Write the point-slope form. = [ ( 3)] Substitute for m, 3 for, and for. = = 5 Simplif. Write in slope-intercept form. An equation of the perpendicular line is = 5. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 3. Determine which of the lines, if an, are parallel or perpendicular. Eplain. Line a: + = 3 Line b: = 3 8 Line c: + 8 = 9. Write an equation of the line that passes through ( 3, 5) and is perpendicular to the line = 3. Section.3 Writing Equations of Parallel and Perpendicular Lines 89

19 Writing Equations for Real-Life Problems Writing an Equation of a Perpendicular Line The position of a helicopter search and rescue crew is shown in the graph. The shortest flight path to the shoreline is one that is perpendicular to the shoreline. Write an equation that represents this path. water (, ) 8 shore. Understand the Problem You can see the line that represents the shoreline. You know the coordinates of the helicopter. You are asked to write an equation that represents the shortest flight path to the shoreline.. Make a Plan Find the slope of the line that represents the shoreline. Use the negative reciprocal of this slope, the coordinates of the helicopter, and the point-slope form to write an equation. 3. Solve the Problem Step Find the slope of the line that represents the shoreline. The line passes through points (, 3) and (, ). So, the slope is m = 3 = 3. Because the shoreline and shortest flight path are perpendicular, the slopes of their respective graphs are negative reciprocals. So, the slope of the graph of the shortest flight path is 3. Step Use the slope m = 3 and the point-slope form to write an equation of the shortest flight path that passes through (, ). = m( ) Write the point-slope form. = 3 ( ) Substitute 3 for m, for, and for. = 3 Distributive Propert = 3 7 Write in slope-intercept form. An equation that represents the shortest flight path is = Look Back To check that our equation is correct, verif that (, ) is a solution of the equation. = 3 () 7 Monitoring Progress Help in English and Spanish at BigIdeasMath.com 5. In Eample 5, a boat is traveling parallel to the shoreline and passes through (9, 3). Write an equation that represents the path of the boat. 9 Chapter Writing Linear Functions

20 .3 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. COMPLETE THE SENTENCE Two distinct nonvertical lines that have the same slope are.. VOCABULARY Two lines are perpendicular. The slope of one line is 5 7. What is the slope of the other line? Justif our answer. Monitoring Progress and Modeling with Mathematics In Eercises 3 8, determine which of the lines, if an, are parallel. Eplain. (See Eample.) 3.. (3, ) ( 3, ) (, 3) a (, ) 3 b c (3, ) (, 3) 3 5. Line a passes through (, ) and (, ). Line b passes through (, ) and (, ). Line c passes through (, ) and (, ). (, ) a (5, ). Line a passes through (, 3) and (, 9). Line b passes through (, ) and (, ). Line c passes through (3, 8) and (, ). b (, 5) (3, ) c (, ) (5, ) 5. Line a passes through (, ) and (, 3). Line b passes through (, ) and (, ). Line c passes through (, 3) and (, ).. Line a passes through (, ) and (, 3). Line b passes through (, 9) and (, ). Line c passes through (, ) and (, 9). 7. Line a: 3 = Line b: = 3 + Line c: + 3 = 8. Line a: = Line b: = Line c: + = In Eercises 9, write an equation of the line that passes through the given point and is perpendicular to the given line. (See Eample.) 9. (7, ); = 9. (, ); = 3 +. ( 3, 3); = 8. (8, ); + = 7. Line a: + = 8 Line b: + = Line c: = Line a: 3 = Line b: 3 = + 8 Line c: 3 = 9 In Eercises 3 and, write an equation of the line that passes through the given point and is (a) parallel and (b) perpendicular to the given line. In Eercises 9, write an equation of the line that passes through the given point and is parallel to the given line. (See Eample.) 9. (, 3); = +. (, ); = 5 +. (8, ); 3 =. (, 5); = 3 + In Eercises 3 8, determine which of the lines, if an, are parallel or perpendicular. Eplain. (See Eample 3.) 3.. ( 3, ) (, ) (, ) ( 5, ) b ( 3, ) (, ) a c (, 5) 3 (, ) c (, 5) (3, ) (, ) b 5 (, ) a ERROR ANALYSIS Describe and correct the error in writing an equation of the line that passes through (, 3) and is parallel to the line = +. (, 3) = m( ) 3 = ( ) 3 = + = (3, ) Section.3 Writing Equations of Parallel and Perpendicular Lines 9

21 . ERROR ANALYSIS Describe and correct the error in writing an equation of the line that passes through (, 5) and is perpendicular to the line = = m( ) ( 5) = 3( ) + 5 = 3 = MAKING AN ARGUMENT A hocke puck leaves the blade of a hocke stick, bounces off a wall, and travels in a new direction, as shown. Your friend claims the path of the puck forms a right angle. Is our friend correct? Eplain. (, ) (, 8) 8 (, ) 7. MODELING WITH MATHEMATICS A cit water department is proposing the construction of a new water pipe, as shown. The new pipe will be perpendicular to the old pipe. Write an equation that represents the new pipe. (See Eample 5.) Softball Academ (, ) 8 proposed water pipe recreation department is constructing a new bike path. The path will be parallel to the railroad tracks shown and pass through the parking area at the point (, 5). Write an equation that represents the path Student B Student A 3 Months of membership 8. MODELING WITH MATHEMATICS A parks and 8 Total cost (dollars) eisting (, 3) water pipe connector valve ( 3, ) students an initial registration fee plus a monthl fee. The graph shows the total amounts paid b two students over a -month period. The lines are parallel. 3. HOW DO YOU SEE IT? A softball academ charges a. Did one of the students pa a greater registration fee? Eplain. b. Did one of the students pa a greater monthl fee? Eplain. parking area (, 5) (, ) REASONING In Eercises 33 35, determine whether the statement is alwas, sometimes, or never true. Eplain our reasoning. 33. Two lines with positive slopes are perpendicular. (8, ) 9. MATHEMATICAL CONNECTIONS The vertices of a quadrilateral are A(, ), B(, ), C(8, ), and D(, 8). a. Is quadrilateral ABCD a parallelogram? Eplain. b. Is quadrilateral ABCD a rectangle? Eplain. 3. USING STRUCTURE For what value of a are the graphs of = + and = a 5 parallel? perpendicular? Maintaining Mathematical Proficienc 3. A vertical line is parallel to the -ais. 35. Two lines with the same -intercept are perpendicular. 3. THOUGHT PROVOKING You are designing a new logo for our math club. Your teacher asks ou to include at least one pair of parallel lines and at least one pair of perpendicular lines. Sketch our logo in a coordinate plane. Write the equations of the parallel and perpendicular lines. Reviewing what ou learned in previous grades and lessons Determine whether the relation is a function. Eplain. (Section 3.) 37. (3, ), (, 8), (5, ), (, 9), (7, ) 38. (, ), (, ), (, ), (, ), (, 5) 9 Chapter hsnb_alg_pe_3.indd 9 Writing Linear Functions //5 : PM

22 ..3 What Did You Learn? Core Vocabular linear model, p. 78 point-slope form, p. 8 parallel lines, p. 88 perpendicular lines, p. 89 Core Concepts Section. Using Slope-Intercept Form, p. 7 Section. Using Point-Slope Form, p. 8 Section.3 Parallel Lines and Slopes, p. 88 Perpendicular Lines and Slopes, p. 89 Mathematical Practices. How can ou eplain to ourself the meaning of the graph in Eercise 3 on page 8?. How did ou use the structure of the equations in Eercise 39 on page 8 to make a conjecture? 3. How did ou use the diagram in Eercise 3 on page 9 to determine whether our friend was correct? Stud Skills Getting Activel Involved in Class If ou do not understand something at all and do not even know how to phrase a question, just ask for clarification. You might sa something like, Could ou please eplain the steps in this problem one more time? If our teacher asks for someone to go up to the board, volunteer. The student at the board often receives additional attention and instruction to complete the problem. 93

23 ..3 Quiz Write an equation of the line in slope-intercept form. (Section.). (, 3). (, 5) (3, ) 3. (, ) (, ) (, ) Write an equation in point-slope form of the line that passes through the given points. (Section.). (, 5), (, ) 5. ( 3, ), (, ). (, ), (, ) Write a linear function f with the given values. (Section. and Section.) 7. f() =, f(5) = 3 8. f( ) =, f() = 9. f( 3) =, f( ) = 3 Determine which of the lines, if an, are parallel or perpendicular. Eplain. (Section.3). Line a passes through (, ) and (, ).. Line a: + = Line b passes through (, 8) and (3, ). Line b: = 3 5 Line c passes through (, 3) and (, ). Line c: 3 = Write an equation of the line that passes through the given point and is (a) parallel and (b) perpendicular to the given line. (Section.3). (, ) (, ) 5 3 (, 3) 5. A website hosting compan charges an initial fee of $8 to set up a website. The compan charges $ per month to maintain the website. (Section.) a. Write a linear model that represents the total cost of setting up and maintaining a website as a function of the number of months it is maintained. b. Find the total cost of setting up a website and maintaining it for months. c. A different website hosting compan charges $ per month to maintain a website, but there is no initial set-up fee. You have $. At which compan can ou set up and maintain a website for the greatest amount of time? Eplain.. The table shows the amount of water remaining in a water tank as it drains. Can the situation be modeled b a linear equation? Eplain. If possible, write a linear model that represents the amount of water remaining in the tank as a function of time. (Section.) Time (minutes) 8 Water (gallons) Chapter Writing Linear Functions

24 . Scatter Plots and Lines of Fit Essential Question How can ou use a scatter plot and a line of fit to make conclusions about data? A scatter plot is a graph that shows the relationship between two data sets. The two data sets are graphed as ordered pairs in a coordinate plane. Finding a Line of Fit REASONING QUANTITATIVELY To be proficient in math, ou need to make sense of quantities and their relationships in problem situations. Work with a partner. A surve was taken of 79 married couples. Each person was asked his or her age. The scatter plot shows the results. a. Draw a line that approimates the data. Write an equation of the line. Eplain the method ou used. b. What conclusions can ou make from the equation ou wrote? Eplain our reasoning. Wife s age Ages of Married Couples Husband s age Work with a partner. The scatter plot shows the median ages of American women at their first marriage for selected ears from 9 through. a. Draw a line that approimates the data. Write an equation of the line. Let represent the number of ears since 9. Eplain the method ou used. b. What conclusions can ou make from the equation ou wrote? Finding a Line of Fit c. Use our equation to predict the median age of American women at their first marriage in the ear. Communicate Your Answer 3. How can ou use a scatter plot and a line of fit to make conclusions about data? Age Ages of American Women at First Marriage Year. Use the Internet or some other reference to find a scatter plot of real-life data that is different from those given above. Then draw a line that approimates the data and write an equation of the line. Eplain the method ou used. Section. Scatter Plots and Lines of Fit 95

25 . Lesson What You Will Learn Core Vocabular scatter plot, p. 9 correlation, p. 97 line of fit, p. 98 Interpret scatter plots. Identif correlations between data sets. Use lines of fit to model data. Interpreting Scatter Plots Core Concept Scatter Plot A scatter plot is a graph that shows the relationship between two data sets. The two data sets are graphed as ordered pairs in a coordinate plane. Scatter plots can show trends in the data. Interpreting a Scatter Plot Calories Smoothies Sugar (grams) The scatter plot shows the amounts (in grams) of sugar and the numbers of calories in smoothies. a. How man calories are in the smoothie that contains 5 grams of sugar? b. How man grams of sugar are in the smoothie that contains 3 calories? c. What tends to happen to the number of calories as the number of grams of sugar increases? a. Draw a horizontal line from the point that has an -value of 5. It crosses the -ais at 7. So, the smoothie has 7 calories. b. Draw a vertical line from the point that has a -value of 3. It crosses the -ais at 7. So, the smoothie has 7 grams of sugar. c. Looking at the graph, the plotted points go up from left to right. So, as the number of grams of sugar increases, the number of calories increases. Calories Smoothies Sugar (grams) Monitoring Progress Help in English and Spanish at BigIdeasMath.com. How man calories are in the smoothie that contains 5 grams of sugar?. How man grams of sugar are in the smoothie that contains 5 calories? 9 Chapter Writing Linear Functions

26 STUDY TIP You can think of a positive correlation as having a positive slope and a negative correlation as having a negative slope. Identifing Correlations between Data Sets A correlation is a relationship between data sets. You can use a scatter plot to describe the correlation between data. Positive Correlation Negative Correlation No Correlation As increases, increases. As increases, decreases. The points show no pattern. Identifing Correlations Tell whether the data show a positive, a negative, or no correlation. a. age and vehicles owned b. temperature and coat sales at a store Number of vehicles owned Age and Vehicles Owned Person s age (ears) Coats sold per da Temperature and Coat Sales Average dail temperature ( F) a. The points show no pattern. The number of vehicles owned does not depend on a person s age. So, the scatter plot shows no correlation. b. As the average temperature increases, the number of coats sold decreases. So, the scatter plot shows a negative correlation. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Make a scatter plot of the data. Tell whether the data show a positive, a negative, or no correlation. 3. Temperature ( F), Attendees (thousands), Age of a car (ears), Value (thousands), $ $ $9 $8 $5 $ $8 $7 Section. Scatter Plots and Lines of Fit 97

27 STUDY TIP A line of fit is also called a trend line. Using Lines of Fit to Model Data When data show a positive or negative correlation, ou can model the trend in the data using a line of fit. A line of fit is a line drawn on a scatter plot that is close to most of the data points. Core Concept Using a Line of Fit to Model Data Step Make a scatter plot of the data. Step Decide whether the data can be modeled b a line. Step 3 Draw a line that appears to fit the data closel. There should be approimatel as man points above the line as below it. Step Write an equation using two points on the line. The points do not have to represent actual data pairs, but the must lie on the line of fit. Finding a Line of Fit The table shows the weekl sales of a DVD and the number of weeks since its release. Write an equation that models the DVD sales as a function of the number of weeks since its release. Interpret the slope and -intercept of the line of fit. Week, Sales (millions), $9 $5 $3 $ $ $8 $7 $5 Sales (millions of dollars) 8 8 DVD Sales (5, ) (, 8) Week Step Make a scatter plot of the data. Step Decide whether the data can be modeled b a line. Because the scatter plot shows a negative correlation, ou can fit a line to the data. Step 3 Draw a line that appears to fit the data closel. Step Write an equation using two points on the line. Use (5, ) and (, 8). The slope of the line is m = 8 5 =. Use the slope m = and the point (, 8) to write an equation of the line. = m( ) Write the point-slope form. 8 = ( ) Substitute for m, for, and 8 for. = + Solve for. An equation of the line of fit is = +. The slope of the line is. This means the sales are decreasing b about $ million each week. The -intercept is. The -intercept has no meaning in this contet because there are no sales in week. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 98 Chapter Writing Linear Functions 5. The following data pairs show the monthl income (in dollars) and the monthl car pament (in dollars) of si people: (, ), (5, 35), (95, 5), (5, 95), (5, ), and (8, 375). Write an equation that models the monthl car pament as a function of the monthl income. Interpret the slope and -intercept of the line of fit.

28 . Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. COMPLETE THE SENTENCE When data show a positive correlation, the dependent variable tends to as the independent variable increases.. VOCABULARY What is a line of fit? Monitoring Progress and Modeling with Mathematics In Eercises 3, use the scatter plot to fill in the missing coordinate of the ordered pair. 3. (, ). (3, ) 5. (, ). (, 7) 7. INTERPRETING A SCATTER PLOT The scatter plot shows the hard drive capacities (in gigabtes) and the prices (in dollars) of laptops. (See Eample.) Price (dollars) 8 8 Laptops 8 8 Hard drive capacit (gigabtes) a. What is the price of the laptop with a hard drive capacit of 8 gigabtes? b. What is the hard drive capacit of the $ laptop? c. What tends to happen to the price as the hard drive capacit increases? 8. INTERPRETING A SCATTER PLOT The scatter plot shows the earned run averages and the winning percentages of eight pitchers on a baseball team. Winning percentage Pitchers 3 5 Earned run average a. What is the winning percentage of the pitcher with an earned run average of.? b. What is the earned run average of the pitcher with a winning percentage of.33? c. What tends to happen to the winning percentage as the earned run average increases? In Eercises 9, tell whether and show a positive, a negative, or no correlation. (See Eample.) Section. Scatter Plots and Lines of Fit 99

29 In Eercises 3 and, make a scatter plot of the data. Tell whether and show a positive, a negative, or no correlation MODELING WITH MATHEMATICS The table shows the world birth rates (number of births per people) ears since 9. (See Eample 3.) a. Write an equation that models the birthrate as a function of the number of ears since 9. b. Interpret the slope and -intercept of the line of fit.. MODELING WITH MATHEMATICS The table shows the total earnings (in dollars) of a food server who works hours a. Write an equation that models the server s earnings as a function of the number of hours the server works. b. Interpret the slope and -intercept of the line of fit. 7. OPEN-ENDED Give an eample of a real-life data set that shows a negative correlation. 8. MAKING AN ARGUMENT Your friend sas that the data in the table show a negative correlation because the dependent variable is decreasing. Is our friend correct? Eplain USING TOOLS Use a ruler or a ardstick to find the heights and arm spans of five people. a. Make a scatter plot using the data ou collected. Then draw a line of fit for the data. b. Interpret the slope and -intercept of the line of fit.. THOUGHT PROVOKING A line of fit for a scatter plot is given b the equation = 5 +. Describe a real-life data set that could be represented b the scatter plot.. WRITING When is data best displaed in a scatter plot, rather than another tpe of displa, such as a bar graph or circle graph?. HOW DO YOU SEE IT? The scatter plot shows part of a data set and a line of fit for the data set. Four data points are missing. Choose possible coordinates for these data points REASONING A data set has no correlation. Is it possible to find a line of fit for the data? Eplain.. ANALYZING RELATIONSHIPS Make a scatter plot of the data in the tables. Describe the relationship between the variables. Is it possible to fit a line to the data? If so, write an equation of the line. If not, eplain wh Maintaining Mathematical Proficienc Evaluate the function when = 3,, and. (Section 3.3) 5. g() =. h() = 7. f() = v() = 3 Reviewing what ou learned in previous grades and lessons Chapter Writing Linear Functions

30 .5 Analzing Lines of Fit Essential Question How can ou analticall find a line of best fit for a scatter plot? Finding a Line of Best Fit Work with a partner. The scatter plot shows the median ages of American women at their first marriage for selected ears from 9 through. In Eploration in Section., ou approimated a line of fit graphicall. To find the line of best fit, ou can use a computer, spreadsheet, or graphing calculator that has a linear regression feature. Age Ages of American Women at First Marriage Year a. The data from the scatter plot is shown in the table. Note that, 5,, and so on represent the numbers of ears since 9. What does the ordered pair (5, 3.3) represent? b. Use the linear regression feature to find an equation of the line of best fit. You should obtain results such as those shown below. LinReg =a+b a=.888 b= r = r= L L(55)= L L3 CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, ou need to reason inductivel about data. c. Write an equation of the line of best fit. Compare our result with the equation ou obtained in Eploration in Section.. Communicate Your Answer. How can ou analticall find a line of best fit for a scatter plot? 3. The data set relates the number of chirps per second for striped ground crickets and the outside temperature in degrees Fahrenheit. Make a scatter plot of the data. Then find an equation of the line of best fit. Use our result to estimate the outside temperature when there are 9 chirps per second. Chirps per second Temperature ( F) Chirps per second Temperature ( F) Section.5 Analzing Lines of Fit

31 .5 Lesson Core Vocabular residual, p. linear regression, p. 3 line of best fit, p. 3 correlation coefficient, p. 3 interpolation, p. 5 etrapolation, p. 5 causation, p. 5 What You Will Learn Use residuals to determine how well lines of fit model data. Use technolog to find lines of best fit. Distinguish between correlation and causation. Analzing Residuals One wa to determine how well a line of fit models a data set is to analze residuals. Core Concept Residuals A residual is the difference of the -value of a data point and the corresponding -value found using the line of fit. A residual can be positive, negative, or zero. A scatter plot of the residuals shows how well a model fits a data set. If the model is a good fit, then the absolute values of the residuals are relativel positive residual data point data point line of fit negative residual small, and the residual points will be more or less evenl dispersed about the horizontal ais. If the model is not a good fit, then the residual points will form some tpe of pattern that suggests the data are not linear. Wildl scattered residual points suggest that the data might have no correlation. Using Residuals Week, Sales (millions), $9 $5 3 $3 $ 5 $ $8 7 $7 8 $5 In Eample 3 in Section., the equation = + models the data in the table shown. Is the model a good fit? Step Calculate the residuals. Organize our results in a table. Step Use the points (, residual) to make a scatter plot. -Value from model Residual = 5 5 = = = 5 = = = = residual 8 The points are evenl dispersed about the horizontal ais. So, the equation = + is a good fit. Chapter Writing Linear Functions

32 Using Residuals The table shows the ages and salaries (in thousands of dollars) of eight emploees at a compan. The equation = models the data. Is the model a good fit? Age, Salar, Step Calculate the residuals. Organize our results in a table. Step Use the points (, residual) to make a scatter plot. -Value from model Residual = =. residual = = = = = =. The residual points form a -shaped pattern, which suggests the data are not linear. So, the equation = does not model the data well. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. The table shows the attendances (in thousands) at an amusement park from 5 to, where = represents the ear 5. The equation = models the data. Is the model a good fit? STUDY TIP You know how to use two points to find an equation of a line of fit. When finding an equation of the line of best fit, ever point in the data set is used. Year, Attendance, Finding Lines of Best Fit Graphing calculators use a method called linear regression to find a precise line of fit called a line of best fit. This line best models a set of data. A calculator often gives a value r, called the correlation coefficient. This value tells whether the correlation is positive or negative and how closel the equation models the data. Values of r range from to. When r is close to or, there is a strong correlation between the variables. As r, gets closer to, the correlation becomes weaker. r = r = r = strong negative correlation no correlation strong positive correlation Section.5 Analzing Lines of Fit 3

33 Finding a Line of Best Fit Using Technolog The table shows the durations (in minutes) of several eruptions of the geser Old Faithful and the times (in minutes) until the net eruption. (a) Use a graphing calculator to find an equation of the line of best fit. Then plot the data and graph the equation in the same viewing window. (b) Identif and interpret the correlation coefficient. (c) Interpret the slope and -intercept of the line of best fit. Duration, Time, a. Step Enter the data from the table into two lists. L L L L()= Step Use the linear regression feature. The values in the equation can be rounded to obtain = LinReg =a+b a=.9989 b= r = r= slope -intercept correlation coefficient PRECISION Be sure to analze the data values to select an appropriate viewing window for our graph. Step 3 Enter the equation = into the calculator. Then plot the data and graph the equation in the same viewing window. 5 b. The correlation coefficient is about.979. This means that the relationship between the durations and the times until the net eruption has a strong positive correlation and the equation closel models the data, as shown in the graph. c. The slope of the line is. This means the time until the net eruption increases b about minutes for each minute the duration increases. The -intercept is 35, but it has no meaning in this contet because the duration cannot be minutes. Monitoring Progress Chapter Writing Linear Functions Help in English and Spanish at BigIdeasMath.com. Use the data in Monitoring Progress Question. (a) Use a graphing calculator to find an equation of the line of best fit. Then plot the data and graph the equation in the same viewing window. (b) Identif and interpret the correlation coefficient. (c) Interpret the slope and -intercept of the line of best fit.

34 Using a graph or its equation to approimate a value between two known values is called interpolation. Using a graph or its equation to predict a value outside the range of known values is called etrapolation. In general, the farther removed a value is from the known values, the less confidence ou can have in the accurac of the prediction. STUDY TIP To approimate or predict an unknown value, ou can evaluate the model algebraicall or graph the model with a graphing calculator and use the trace feature. READING = + 35 X= Y= A causal relationship eists when one variable causes a change in another variable. Interpolating and Etrapolating Data Refer to Eample 3. Use the equation of the line of best fit. a. Approimate the duration before a time of 77 minutes. b. Predict the time after an eruption lasting 5. minutes. a. = Write the equation. 77 = Substitute 77 for. 3.5 = Solve for. An eruption lasts about 3.5 minutes before a time of 77 minutes. b. Use a graphing calculator to graph the equation. Use the trace feature to find the value of when 5., as shown. A time of about 95 minutes will follow an eruption of 5. minutes. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 3. Refer to Monitoring Progress Question. Use the equation of the line of best fit to predict the attendance at the amusement park in 7. Correlation and Causation When a change in one variable causes a change in another variable, it is called causation. Causation produces a strong correlation between the two variables. The converse is not true. In other words, correlation does not impl causation. Identifing Correlation and Causation Tell whether a correlation is likel in the situation. If so, tell whether there is a causal relationship. Eplain our reasoning. a. time spent eercising and the number of calories burned b. the number of banks and the population of a cit a. There is a positive correlation and a causal relationship because the more time ou spend eercising, the more calories ou burn. b. There ma be a positive correlation but no causal relationship. Building more banks will not cause the population to increase. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. Is there a correlation between time spent plaing video games and grade point average? If so, is there a causal relationship? Eplain our reasoning. Section.5 Analzing Lines of Fit 5

35 .5 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. VOCABULARY When is a residual positive? When is it negative?. WRITING Eplain how ou can use residuals to determine how well a line of fit models a data set. 3. VOCABULARY Compare interpolation and etrapolation.. WHICH ONE DOESN T BELONG? Which correlation coefficient does not belong with the other three? Eplain our reasoning. r =.98 r =.9 r =.9 r =.97 Monitoring Progress and Modeling with Mathematics In Eercises 5 8, use residuals to determine whether the model is a good fit for the data in the table. Eplain. (See Eamples and.) 5. = = = ANALYZING RESIDUALS The table shows the approimate numbers (in thousands) of movie tickets sold from Januar to June for a theater. In the table, = represents Januar. The equation = models the data. Is the model a good fit? Eplain. Month, Ticket sales, In Eercises, use a graphing calculator to find an equation of the line of best fit for the data. Identif and interpret the correlation coefficient = ANALYZING RESIDUALS The table shows the growth (in inches) of an elk s antlers during week. The equation = models the data. Is the model a good fit? Eplain. Week, 3 5 Growth, Chapter Writing Linear Functions

36 ERROR ANALYSIS In Eercises 5 and, describe and correct the error in interpreting the graphing calculator displa. 5.. LinReg =a+b a=-.7 b=3. r = r= An equation of the line of best fit is = The data have a strong positive correlation. 7. MODELING WITH MATHEMATICS The table shows the total numbers of people who reported an earthquake minutes after it ended. (See Eample 3.) a. Use a graphing calculator to find an equation of the line of best fit. Then plot the data and graph the equation in the same viewing window. b. Identif and interpret the correlation coefficient. Minutes, People, c. Interpret the slope and -intercept of the line of best fit. 8. MODELING WITH MATHEMATICS The table shows the numbers of people who volunteer at an animal shelter on each da. Da, People, a. Use a graphing calculator to find an equation of the line of best fit. Then plot the data and graph the equation in the same viewing window. b. Identif and interpret the correlation coefficient. c. Interpret the slope and -intercept of the line of best fit. 9. MODELING WITH MATHEMATICS The table shows Dnamic the Solutions mileages available (in thousands at BigIdeasMath.com of miles) and the selling prices (in thousands of dollars) of several used automobiles of the same ear and model. (See Eample.) Mileage, Price, a. Use a graphing calculator to find an equation of the line of best fit. b. Identif and interpret the correlation coefficient. c. Interpret the slope and -intercept of the line of best fit. d. Approimate the mileage of an automobile that costs $5,5. e. Predict the price of an automobile with miles.. MODELING WITH MATHEMATICS The table shows the lengths and costs of several sailboats. a. Use a graphing calculator to find an equation of the line of best fit. b. Identif and interpret the correlation coefficient. c. Interpret the slope and -intercept of the line of best fit. d. Approimate the cost of a sailboat that is feet long. Length (feet), e. Predict the length of a sailboat that costs $7,. Cost (thousands of dollars), In Eercises, tell whether a correlation is likel in the situation. If so, tell whether there is a causal relationship. Eplain our reasoning. (See Eample 5.). the amount of time spent talking on a cell phone and the remaining batter life. the height of a toddler and the size of the toddler s vocabular 3. the number of hats ou own and the size of our head. the weight of a dog and the length of its tail Section.5 Analzing Lines of Fit 7

37 5. OPEN-ENDED Describe a data set that has a strong correlation but does not have a causal relationship.. HOW DO YOU SEE IT? Match each graph with its correlation coefficient. Eplain our reasoning. a. b. 5 5 c. 5 d MAKING AN ARGUMENT A student spends hours watching television each week and has a grade point average of.. Your friend sas including this information in the data set in Eercise 7 will weaken the correlation. Is our friend correct? Eplain. 9. USING MODELS Refer to Eercise 7. a. Predict the total numbers of people who reported an earthquake 9 minutes and 5 minutes after it ended. b. The table shows the actual data. Describe the accurac of our etrapolations in part (a) Minutes, 9 5 People, 75 3 A. r = B. r =.98 C. r =.97 D. r =.9 7. ANALYZING RELATIONSHIPS The table shows the grade point averages of several students and the numbers of hours the spend watching television each week. a. Use a graphing calculator to find an equation of the line of best fit. Identif and interpret the correlation coefficient. b. Interpret the slope and -intercept of the line of best fit. c. Another student watches about hours of television each week. Approimate the student s grade point average. d. Do ou think there is a causal relationship between time spent watching television and grade point average? Eplain. Maintaining Mathematical Proficienc Determine whether the table represents a linear or nonlinear function. Eplain. (Section 3.) Hours, Grade point average, THOUGHT PROVOKING A data set consists of the numbers of people at Beach and the numbers of people at Beach recorded dail for week. Sketch a possible graph of the data set. Describe the situation shown in the graph and give a possible correlation coefficient. Determine whether there is a causal relationship. Eplain. 3. COMPARING METHODS The table shows the numbers (in billions) of tet messages sent each ear in a five-ear period, where = represents the first ear in the five-ear period. Year, 3 5 Tet messages (billions), Reviewing what ou learned in previous grades and lessons a. Use a graphing calculator to find an equation of the line of best fit. Identif and interpret the correlation coefficient. b. Is there a causal relationship? Eplain our reasoning. c. Calculate the residuals. Then make a scatter plot of the residuals and interpret the results. d. Compare the methods ou used in parts (a) and (c) to determine whether the model is a good fit. Which method do ou prefer? Eplain. 8 Chapter Writing Linear Functions

38 . Arithmetic Sequences Essential Question How can ou use an arithmetic sequence to describe a pattern? An arithmetic sequence is an ordered list of numbers in which the difference between each pair of consecutive terms, or numbers in the list, is the same. Describing a Pattern Work with a partner. Use the figures to complete the table. Plot the points given b our completed table. Describe the pattern of the -values. a. n = n = n = 3 n = n = 5 LOOKING FOR A PATTERN To be proficient in math, ou need to look closel to discern patterns and structure. Number of stars, n 3 5 Number of sides, n b. n = n = n = 3 n = n = n 3 5 Number of circles, 3 5 n c. n = n = n = 3 n = n = 5 8 Number of rows, n n Number of dots, Communicate Your Answer. How can ou use an arithmetic sequence to describe a pattern? Give an eample from real life. 3. In chemistr, water is called H O because each molecule of water has two hdrogen atoms and one ogen atom. Describe the pattern shown below. Use the pattern to determine the number of atoms in 3 molecules. n = n = n = 3 n = n = 5 Section. Arithmetic Sequences 9

39 . Lesson What You Will Learn Core Vocabular sequence, p. term, p. arithmetic sequence, p. common difference, p. Previous point-slope form function notation READING An ellipsis (...) is a series of dots that indicates an intentional omission of information. In mathematics, the... notation means and so forth. The ellipsis indicates that there are more terms in the sequence that are not shown. Write the terms of arithmetic sequences. Graph arithmetic sequences. Write arithmetic sequences as functions. Writing the Terms of Arithmetic Sequences A sequence is an ordered list of numbers. Each number in a sequence is called a term. Each term a n has a specific position n in the sequence. Core Concept 5,, 5,, 5,..., a n,... Etending an Arithmetic Sequence Write the net three terms of the arithmetic sequence. 7,,, 8,... st position 3rd position nth position Arithmetic Sequence In an arithmetic sequence, the difference between each pair of consecutive terms is the same. This difference is called the common difference. Each term is found b adding the common difference to the previous term. 5,, 5,,... Terms of an arithmetic sequence common difference Use a table to organize the terms and find the pattern. Position 3 Term 7 8 +( 7) +( 7) +( 7) Add 7 to a term to find the net term. Each term is 7 less than the previous term. So, the common difference is 7. Position Term ( 7) +( 7) +( 7) The net three terms are 35,, and 9. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Write the net three terms of the arithmetic sequence..,,,,.....,.,,.,... 3., 3 3, 3, 3,... Chapter Writing Linear Functions

40 Graphing Arithmetic Sequences To graph a sequence, let a term s position number n in the sequence be the -value. The term a n is the corresponding -value. Plot the ordered pairs (n, a n ). Graphing an Arithmetic Sequence Graph the arithmetic sequence, 8,,,.... What do ou notice? Make a table. Then plot the ordered pairs (n, a n ). Position, n Term, a n 8 3 The points lie on a line. a n 8 (, ) (3, ) (, 8) (, ) n Identifing an Arithmetic Sequence from a Graph Does the graph represent an arithmetic sequence? Eplain. Make a table to organize the ordered pairs. Then determine whether there is a common difference. a n 8 (, 5) (, ) (3, 9) (, ) n Position, n 3 Term, a n 5 9 +( 3) +( 3) +( 3) Each term is 3 less than the previous term. So, the common difference is 3. a n 8 (, ) (3, 7) (, ) (, ) n Consecutive terms have a common difference of 3. So, the graph represents the arithmetic sequence 5,, 9,,.... Monitoring Progress Graph the arithmetic sequence. What do ou notice? Help in English and Spanish at BigIdeasMath.com. 3,, 9,,... 5.,,,,....,.8,.,., Does the graph shown represent an arithmetic sequence? Eplain. Section. Arithmetic Sequences

41 Writing Arithmetic Sequences as Functions Because consecutive terms of an arithmetic sequence have a common difference, the sequence has a constant rate of change. So, the points represented b an arithmetic sequence lie on a line. You can use the first term and the common difference to write a linear function that describes an arithmetic sequence. Let a = and d = 3. ANOTHER WAY An arithmetic sequence is a linear function whose domain is the set of positive integers. You can think of d as the slope and (, a ) as a point on the graph of the function. An equation in point-slope form for the function is a n a = d(n ). This equation can be rewritten as a n = a + (n )d. Position, n Term, a n Written using a and d Numbers first term, a a second term, a a + d + 3 = 7 3 third term, a 3 a + d + (3) = fourth term, a a + 3d + 3(3) = 3 n nth term, a n a + (n )d + (n )(3) Core Concept Equation for an Arithmetic Sequence Let a n be the nth term of an arithmetic sequence with first term a and common difference d. The nth term is given b a n = a + (n )d. Finding the nth Term of an Arithmetic Sequence STUDY TIP Notice that the equation in Eample is of the form = m + b, where is replaced b a n and is replaced b n. Write an equation for the nth term of the arithmetic sequence,, 8, 5,.... Then find a 5. The first term is, and the common difference is 3. a n = a + (n )d Equation for an arithmetic sequence a n = + (n )( 3) Substitute for a and 3 for d. a n = 3n + 7 Simplif. Use the equation to find the 5th term. a n = 3n + 7 Write the equation. a 5 = 3(5) + 7 Substitute 5 for n. = 33 Simplif. The 5th term of the arithmetic sequence is 33. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Write an equation for the nth term of the arithmetic sequence. Then find a 5. 8., 5,, 7, ,,, 3,....,,,,... Chapter Writing Linear Functions

42 You can rewrite the equation for an arithmetic sequence with first term a and common difference d in function notation b replacing a n with f(n). f(n) = a + (n )d The domain of the function is the set of positive integers. Writing Real-Life Functions Online bidding for a purse increases b $5 for each bid after the $ initial bid. Bid number 3 Bid amount $ $5 $7 $75 a. Write a function that represents the arithmetic sequence. b. Graph the function. c. The winning bid is $5. How man bids were there? a. The first term is, and the common difference is 5. f(n) = a + (n )d Function for an arithmetic sequence f(n) = + (n )5 Substitute for a and 5 for d. f(n) = 5n + 55 Simplif. REMEMBER The domain is the set of positive integers. The function f(n) = 5n + 55 represents the arithmetic sequence. b. Make a table. Then plot the ordered pairs (n, a n ). Bid Bid number, n amount, a n Bid amount (dollars) c. Use the function to find the value of n for which f(n) = 5. f(n) = 5n + 55 Write the function. Bidding on a Purse a n = 5n + 55 Substitute 5 for f(n). = n Solve for n. (, 75) (3, 7) (, 5) (, ) 3 5 Bid number n There were bids. Games Total cost $7 $9 3 $ $3 Monitoring Progress Help in English and Spanish at BigIdeasMath.com. A carnival charges $ for each game after ou pa a $5 entr fee. a. Write a function that represents the arithmetic sequence. b. Graph the function. c. How man games can ou pla when ou take $9 to the carnival? Section. Arithmetic Sequences 3

43 . Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. WRITING Describe the graph of an arithmetic sequence.. DIFFERENT WORDS, SAME QUESTION Consider the arithmetic sequence represented b the graph. Which is different? Find both answers. Find the slope of the linear function. Find the difference between the terms a and a. Find the difference between consecutive terms of the arithmetic sequence. Find the common difference of the arithmetic sequence. a n (5, 9) (, ) (3, 3) (, ) n Monitoring Progress and Modeling with Mathematics In Eercises 3 and, write the net three terms of the arithmetic sequence. 3. First term: Common difference: 3. First term: 8 Common difference: In Eercises 5, find the common difference of the arithmetic sequence. 5. 3, 8, 3, 8, , 5, 5,,... 7.,, 8,,... 8., 3 3, 3 3, 3, , 5, 3.5,,...., 7,,,... In Eercises, write the net three terms of the arithmetic sequence. (See Eample.). 9,, 5, 8,....,, 3, 3,... 3.,,, 3,...., 3,, 3, ,,.7,.,.... 5, 3,, 3,... In Eercises 7, graph the arithmetic sequence. (See Eample.) 7.,,, 8, ,, 5, 3,... 9., 3, 5, 7,...., 9, 3, 53,... In Eercises 3, determine whether the graph represents an arithmetic sequence. Eplain. (See Eample 3.) a n (, ) (, ) (, ) (3, ) n a n 7 (, 7) 5 (, 55) (3, ) 3 (, 5) n.. a n 8 8 a n (, 5) (3, 9) (, ) (, ) n (, ) (, ) (, ) (3, ) n In Eercises 7 3, determine whether the sequence is arithmetic. If so, find the common difference. 7. 3,, 39, 5, , 9,,, ,,,, , 8, 75, 9, FINDING A PATTERN Write a sequence that represents the number of smile faces in each group. Is the sequence arithmetic? Eplain..,, 9, 3,...., 5.5,.5, 3.75,... Chapter Writing Linear Functions

44 3. FINDING A PATTERN Write a sequence that represents the sum of the numbers in each roll. Is the sequence arithmetic? Eplain. Roll Roll Roll 3 Roll REPEATED REASONING In Eercises 3 and, (a) draw the net three figures in the sequence and (b) describe the th figure in the sequence. 3.. In Eercises 33 38, write an equation for the nth term of the arithmetic sequence. Then find a. (See Eample.) 33. 5,, 3,,... 3., 9,, 5, ,,,,... 3.,,, 3, ,,,, , 7, 5 7, 7, ERROR ANALYSIS Describe and correct the error in finding the common difference of the arithmetic sequence.,,,,... The common difference is.. ERROR ANALYSIS Describe and correct the error in writing an equation for the nth term of the arithmetic sequence.,, 3, 38,... a n = a + nd a n = + 8n. NUMBER SENSE The first term of an arithmetic sequence is 3. The common difference of the sequence is.5 times the first term. Write the net three terms of the sequence. Then graph the sequence.. NUMBER SENSE The first row of a dominoes displa has dominoes. Each row after the first has two more dominoes than the row before it. Write the first five terms of the sequence that represents the number of dominoes in each row. Then graph the sequence. 5. MODELING WITH MATHEMATICS The total number of babies born in a countr each minute after midnight Januar st can be estimated b the sequence shown in the table. (See Eample 5.) Minutes after midnight Januar st 3 Total babies born 5 5 a. Write a function that represents the arithmetic sequence. b. Graph the function. c. Estimate how man minutes after midnight Januar st it takes for babies to be born.. MODELING WITH MATHEMATICS The amount of mone a movie earns each week after its release can be approimated b the sequence shown in the graph. Earnings (millions of dollars) Movie Earnings a n (, 5) 5 3 (, 8) (3, ) (, 3) 3 5 Week a. Write a function that represents the arithmetic sequence. b. In what week does the movie earn $ million? c. How much mone does the movie earn overall? n Section. Arithmetic Sequences 5

45 MATHEMATICAL CONNECTIONS In Eercises 7 and 8, each small square represents square inch. Determine whether the areas of the figures form an arithmetic sequence. If so, write a function f that represents the arithmetic sequence and find f(3) REPEATED REASONING Firewood is stacked in a pile. The bottom row has logs, and the top row has logs. Each row has one more log than the row above it. How man logs are in the pile? 5. HOW DO YOU SEE IT? The bar graph shows the costs of advertising in a magazine. Magazine Advertisement 8. Cost (dollars) 7,, 5,, 3,,, 3 Size of advertisement (pages) 9. REASONING Is the domain of an arithmetic sequence discrete or continuous? Is the range of an arithmetic sequence discrete or continuous? 5. MAKING AN ARGUMENT Your friend sas that the range of a function that represents an arithmetic sequence alwas contains onl positive numbers or onl negative numbers. Your friend claims this is true because the domain is the set of positive integers and the output values either constantl increase or constantl decrease. Is our friend correct? Eplain. 5. OPEN-ENDED Write the first four terms of two different arithmetic sequences with a common difference of 3. Write an equation for the nth term of each sequence. 5. THOUGHT PROVOKING Describe an arithmetic sequence that models the numbers of people in a real-life situation. a. Does the graph represent an arithmetic sequence? Eplain. b. Eplain how ou would estimate the cost of a si-page advertisement in the magazine. 55. REASONING Write a function f that represents the arithmetic sequence shown in the mapping diagram PROBLEM SOLVING A train stops at a station ever minutes starting at : a.m. You arrive at the station at 7:9 a.m. How long must ou wait for the train? 57. ABSTRACT REASONING Let be a constant. Determine whether each sequence is an arithmetic sequence. Eplain. a. +, 3 +, 5 +, 7 +,... b. +, 3 +, 9 +, 7 +,... Maintaining Mathematical Proficienc Reviewing what ou learned in previous grades and lessons Solve the inequalit. Graph the solution. (Section.) < b. t < Graph the function. Compare the graph to the graph of f() =. Describe the domain and range. (Section 3.7). h() = 3 3. v() = 5. g() = + 5. r() = Chapter Writing Linear Functions

46 .7 Piecewise Functions Essential Question How can ou describe a function that is represented b more than one equation? Work with a partner. Writing Equations for a Function a. Does the graph represent as a function of? Justif our conclusion. CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, ou need to justif our conclusions and communicate them to others. b. What is the value of the function when =? How can ou tell? c. Write an equation that represents the values of the function when. f() =, if d. Write an equation that represents the values of the function when >. f() =, if > e. Combine the results of parts (c) and (d) to write a single description of the function. f() =, if, if > Writing Equations for a Function Work with a partner. a. Does the graph represent as a function of? Justif our conclusion. b. Describe the values of the function for the following intervals. f() =, if < 3, if 3 <, if < 3, if 3 < Communicate Your Answer 3. How can ou describe a function that is represented b more than one equation?. Use two equations to describe the function represented b the graph. Section.7 Piecewise Functions 7

47 .7 Lesson What You Will Learn Core Vocabular piecewise function, p. 8 step function, p. Previous absolute value function verte form verte Evaluate piecewise functions. Graph and write piecewise functions. Graph and write step functions. Write absolute value functions. Evaluating Piecewise Functions Core Concept Piecewise Function A piecewise function is a function defined b two or more equations. Each piece of the function applies to a different part of its domain. An eample is shown below. f() = {, if +, if > The epression represents the value of f when is less than f() = +, > or equal to. The epression + represents the value of f when is greater than. f() =, Evaluating a Piecewise Function Evaluate the function f above when (a) = and (b) =. a. f() = Because, use the first equation. f() = Substitute for. f() = Simplif. The value of f is when =. b. f() = + Because >, use the second equation. f() = () + Substitute for. f() = 9 Simplif. The value of f is 9 when =. Monitoring Progress Evaluate the function. Help in English and Spanish at BigIdeasMath.com f() = { 3, +,, if < if 5 if > 5. f( 8). f( ) 3. f(). f(3) 5. f(5). f() 8 Chapter Writing Linear Functions

48 Graphing and Writing Piecewise Functions, Graph = {, Graphing a Piecewise Function if <. Describe the domain and range. if Step Graph = for <. Because is not equal to, use an open circle at (, ). Step Graph = for. Because is greater than or equal to, use a closed circle at (, ). The domain is all real numbers. The range is >. =, < =, Monitoring Progress Graph the function. Describe the domain and range. 7. = { +, if 8. =, if > {,, Help in English and Spanish at BigIdeasMath.com if < if Writing a Piecewise Function Write a piecewise function for the graph. Each piece of the function is linear. Left Piece When <, the graph is the line given b = + 3. Right Piece When, the graph is the line given b =. So, a piecewise function for the graph is f() = { + 3,, if < if. Monitoring Progress Write a piecewise function for the graph. Help in English and Spanish at BigIdeasMath.com Section.7 Piecewise Functions 9

49 STUDY TIP The graph of a step function looks like a staircase. Graphing and Writing Step Functions A step function is a piecewise function defined b a constant value over each part of its domain. The graph of a step function consists of a series of line segments. 8 f() =, if < 3, if <, if < 5, if < 8, if 8 < 7, if < Graphing and Writing a Step Function You rent a karaoke machine for 5 das. The rental compan charges $5 for the first da and $5 for each additional da. Write and graph a step function that represents the relationship between the number of das and the total cost (in dollars) of renting the karaoke machine. Step Use a table to organize the information. Number of das Total cost (dollars) < 5 < 75 < 3 3 < 5 < 5 5 Step Write the step function. { 5, if < 75, if < f() =, if < 3 5, if 3 < 5, if < 5 Step 3 Graph the step function. Karaoke Machine Rental Total cost (dollars) Number of das Monitoring Progress Help in English and Spanish at BigIdeasMath.com. A landscaper rents a wood chipper for das. The rental compan charges $ for the first da and $5 for each additional da. Write and graph a step function that represents the relationship between the number of das and the total cost (in dollars) of renting the chipper. Chapter Writing Linear Functions

50 REMEMBER The verte form of an absolute value function is g() = a h + k, where a. The verte of the graph of g is (h, k). STUDY TIP Recall that the graph of an absolute value function is smmetric about the line = h. So, it makes sense that the piecewise definition splits the function at = 5. Writing Absolute Value Functions The absolute value function f() = can be written as a piecewise function. f() = {,, if < if Similarl, the verte form of an absolute value function g() = a h + k can be written as a piecewise function. a[ ( h)] + k, g() = { a( h) + k, if h < if h Writing an Absolute Value Function In holograph, light from a laser beam is split into two beams, a reference beam and an object beam. Light from the object beam reflects off an object and is recombined with the reference beam to form images on film that can be used to create three-dimensional images. a. Write an absolute value function that represents the path of the reference beam. b. Write the function in part (a) as a piecewise function. a. The verte of the path of the reference beam is (5, 8). So, the function has the form g() = a Substitute the coordinates of the point (, ) into the equation and solve for a. g() = a Verte form of the function = a Substitute for and for g().. = a Solve for a. So, the function g() = represents the path of the reference beam. b. Write g() = as a piecewise function..[ ( 5)] + 8, if 5 < g() = {.( 5) + 8, if 5 Simplif each epression and solve the inequalities. So, a piecewise function for g() = is g() = {.,. +, if < 5 if 5. 8 (5, 8) mirror reference reference beam object beam beam splitter (, ) laser object beam mirror 8 film plate Monitoring Progress Help in English and Spanish at BigIdeasMath.com. WHAT IF? The reference beam originates at (3, ) and reflects off a mirror at (5, ). a. Write an absolute value function that represents the path of the reference beam. b. Write the function in part (a) as a piecewise function. Section.7 Piecewise Functions

51 .7 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. VOCABULARY Compare piecewise functions and step functions.. WRITING Use a graph to eplain wh ou can write the absolute value function = as a piecewise function. Monitoring Progress and Modeling with Mathematics In Eercises 3, evaluate the function. (See Eample.) 5, if < f() = { + 3, if { +, g() = 3, 5, if if < < if 3. f( 3). f( ) 5. f(). f(5) 7. g( ) 8. g( ) 9. g(). g(). g(). g(5) 3. MODELING WITH MATHEMATICS On a trip, the total distance (in miles) ou travel in hours is represented b the piecewise function d() = { 55, 5, if if < 5. How far do ou travel in hours?. MODELING WITH MATHEMATICS The total cost (in dollars) of ordering custom shirts is represented b the piecewise function { 7 +, if < 5 c() = 5.8 +, if 5 < 5. +, if 5 Determine the total cost of ordering shirts. In Eercises 5, graph the function. Describe the domain and range. (See Eample.) 5. = {, if <, if. = {,, 3, 7. = { +, 8. = { + 8,, 9. = {,, +, { +,. = +, 3, if 3 if > 3 if if > if < if if < 3 if 3 3 if > 3 if if < < if. ERROR ANALYSIS Describe and correct the error in 3, if < 5 finding f(5) when f() = { + 8, if 5. f(5) = (5) 3 = 7. ERROR ANALYSIS Describe and correct the error in graphing = { +, if, if >. 5 3 Chapter Writing Linear Functions

52 In Eercises 3 3, write a piecewise function for the graph. (See Eample 3.) MODELING WITH MATHEMATICS The cost to join an intramural sports league is $8 per team and includes the first five team members. For each additional team member, there is a $3 fee. You plan to have nine people on our team. Write and graph a step function that represents the relationship between the number p of people on our team and the total cost of joining the league. (See Eample.) 3. MODELING WITH MATHEMATICS The rates for a parking garage are shown. Write and graph a step function that represents the relationship between the number of hours a car is parked in the garage and the total cost of parking in the garage for da In Eercises 37, write the absolute value function as a piecewise function = = =. = + 5. = + 3. = 3. = 5 8. = 3 + In Eercises 3 3, graph the step function. Describe the domain and range. 3. f() = { 3. f() = { 33. f() = { 3,, 5,,,, 8,, 9,, 5,, if < if < if < if < 8 if < if < 3 if 3 < if < 5 if < if < if < 9 if 9 < 5. = 3 +. = MODELING WITH MATHEMATICS You are sitting on a boat on a lake. You can get a sunburn from the sunlight that hits ou directl and also from the sunlight that reflects off the water. (See Eample 5.) f() = {,,,, if < 5 if 5 < 3 if 3 < if < a. Write an absolute value function that represents the path of the sunlight that reflects off the water. b. Write the function in part (a) as a piecewise function. Section.7 Piecewise Functions 3

53 8. MODELING WITH MATHEMATICS You are tring to 5. HOW DO YOU SEE IT? The graph shows the total cost C of making photocopies at a cop shop. make a hole in one on the miniature golf green. Making Photocopies C Total cost (dollars) 5 a. Write an absolute value function that represents the path of the golf ball. a. Does it cost more mone to make photocopies or photocopies? Eplain. 9. REASONING The piecewise function f consists of two linear pieces. The graph of f is shown. b. You have $ to make photocopies. Can ou bu more than 5 photocopies? Eplain. 53. USING STRUCTURE The output of the greatest integer function is the greatest integer less than or equal to the input value. This function is written as f() =. Graph the function for <. Is it a piecewise function? a step function? Eplain. a. What is the value of f( )? b. What is the value of f(8)? 5. THOUGHT PROVOKING Eplain wh = 5. CRITICAL THINKING Describe how the graph of each piecewise function changes when < is replaced with and is replaced with >. Do the domain and range change? Eplain. a. f() = b. f() = { { 3 +, if < + 3, if, if 3 3, if MAKING AN ARGUMENT During a 9-hour snowstorm, it snows at a rate of inch per hour for the first hours, inches per hour for the net hours, and inch per hour for the final hour. a. Write and graph a piecewise function that represents the depth of the snow during the snowstorm. 5. USING STRUCTURE Graph = { does not represent a function. How can ou redefine so that it does represent a function? +, if <, if Number of copies b. Write the function in part (a) as a piecewise function. 3 5 { +, if., if > b. Your friend sas inches of snow accumulated during the storm. Is our friend correct? Eplain. Describe the domain and range. Maintaining Mathematical Proficienc Reviewing what ou learned in previous grades and lessons Write the sentence as an inequalit. Graph the inequalit. (Section.5) 5. A number r is greater than and no more than A number t is less than or equal to or no less than 8. Graph f and h. Describe the transformations from the graph of f to the graph of h. (Section 3.) 58. f() = ; h() = + 3 Chapter hsnb_alg_pe_7.indd 59. f() = ; h() = 8. f() = ; h() = + 5 Writing Linear Functions //5 : PM

54 ..7 What Did You Learn? Core Vocabular scatter plot, p. 9 correlation, p. 97 line of fit, p. 98 residual, p. linear regression, p. 3 line of best fit, p. 3 correlation coefficient, p. 3 interpolation, p. 5 etrapolation, p. 5 causation, p. 5 sequence, p. term, p. arithmetic sequence, p. common difference, p. piecewise function, p. 8 step function, p. Core Concepts Section. Scatter Plot, p. 9 Identifing Correlations, p. 97 Section.5 Residuals, p. Lines of Best Fit, p. 3 Using a Line of Fit to Model Data, p. 98 Correlation and Causation, p. 5 Section. Arithmetic Sequence, p. Equation for an Arithmetic Sequence, p. Section.7 Piecewise Function, p. 8 Step Function, p. Writing Absolute Value Functions, p. Mathematical Practices. What resources can ou use to help ou answer Eercise 7 on page?. What calculations are repeated in Eercises on page? When finding a term such as a 5, is there a general method or shortcut ou can use instead of repeating calculations? 3. Describe the definitions ou used when ou eplained our answer in Eercise 53 on page. An Beginning With so man was to represent a linear relationship, where do ou start? Use what ou know to move between equations, graphs, tables, and contets. To eplore the answers to this question and more, go to BigIdeasMath.com. Performance Task 5

55 Chapter Review Dnamic Solutions available at BigIdeasMath.com. Writing Equations in Slope-Intercept Form (pp. 75 8) Write an equation of the line in slope-intercept form. (, 3) (3, 5) Find the slope and -intercept. Let (, ) = (, 3) and (, ) = (3, 5). m = = = 3 Because the line crosses the -ais at (, 3), the -intercept is 3. So, the equation is = Write an equation of the line in slope-intercept form. (, ) (, ). Writing Equations in Point-Slope Form (pp. 8 8) Write an equation in point-slope form of the line that passes through the point (, 8) and has a slope of 3. = m( ) Write the point-slope form. ( 8) = 3[ ( )] Substitute 3 for m, for, and 8 for. + 8 = 3( + ) The equation is + 8 = 3( + ). Simplif.. Write an equation in point-slope form of the line that passes through the point (, 7) and has a slope of. Write a linear function f with the given values. 3. f() = 5, f() = 3. f(3) =, f(5) = 5. f() = 8, f(9) = 3.3 Writing Equations of Parallel and Perpendicular Lines (pp. 87 9) Determine which of the lines, if an, are parallel or perpendicular. Line a: = + 3 Line b: + = 5 Line c: 8 = Write the equations in slope-intercept form. Then compare the slopes. Line a: = + 3 Line b: = + 5 Line c: = Lines a and c have slopes of, so the are parallel. Line b has a slope of, the negative reciprocal of, so it is perpendicular to lines a and c. Chapter Writing Linear Functions

56 Determine which of the lines, if an, are parallel or perpendicular. Eplain.. Line a passes through (, ) and (, 3). 7. Line a: 7 = Line b passes through (, ) and (, ). Line b: = 7 8 Line c passes through (, ) and (, ). Line c: + 7 = 8. Write an equation of the line that passes through (, 5) and is parallel to the line = Write an equation of the line that passes through (, 3) and is perpendicular to the line = 3.. Scatter Plots and Lines of Fit (pp. 95 ) The scatter plot shows the roasting times (in hours) and weights (in pounds) of seven turkes. Tell whether the data show a positive, a negative, or no correlation. As the weight of a turke increases, the roasting time increases. So, the scatter plot shows a positive correlation. Use the scatter plot in the eample.. What is the roasting time for a -pound turke?. Write an equation that models the roasting time as a function of the weight of a turke. Interpret the slope and -intercept of the line of fit. Roasting time (hours) Roasting Turkes Weight (pounds).5 Analzing Lines of Fit (pp. 8) The table shows the heights (in inches) and shoe sizes of several students. Use a graphing calculator to find an equation of the line of best fit. Identif and interpret the correlation coefficient. Step Enter the data from the table into two lists. Step Use the linear regression feature. LinReg =a+b a= b= r = r= An equation of the line of best fit is = The correlation coefficient is about.97. This means that the relationship between the heights and the shoe sizes has a strong positive correlation and the equation closel models the data.. Make a scatter plot of the residuals to verif that the model in the eample is a good fit. 3. Use the data in the eample. (a) Approimate the height of a student whose shoe size is 9. (b) Predict the shoe size of a student whose height is inches.. Is there a causal relationship in the data in the eample? Eplain. Height, Shoe size, Chapter Chapter Review 7

57 . Arithmetic Sequences (pp. 9 ) Write an equation for the nth term of the arithmetic sequence 3, 5, 7, 9,.... Then find a. The first term is 3, and the common difference is. a n = a + (n )d Equation for an arithmetic sequence a n = 3 + (n )( ) Substitute 3 for a and for d. a n = n Use the equation to find the th term. Simplif. a = () Substitute for n. = Simplif. The th term of the arithmetic sequence is. Write an equation for the nth term of the arithmetic sequence. Then find a 3. 5.,, 9, 8,....,, 8,, ,, 3,,....7 Piecewise Functions (pp. 7 ) 3 Graph = { + 3, if. Describe the domain and range., if > Step Graph = for. Because is less than or equal to, use a closed circle at (, 3). Step Graph = for >. Because is not equal to, use an open circle at (, ). The domain is all real numbers. The range is Evaluate the function in the eample when (a) = and (b) = 5. 3 = + 3, 3 =, > Graph the function. Describe the domain and range. 9. = { +, 3, if if > +,. = {, Write the absolute value function as a piecewise function. if < if. = + 5. = = + 3. You are organizing a school fair and rent a popcorn machine for 3 das. The rental compan charges $5 for the first da and $35 for each additional da. Write and graph a step function that represents the relationship between the number of das and the total cost (in dollars) of renting the popcorn machine. 8 Chapter Writing Linear Functions

58 Chapter Test Graph the function. Describe the domain and range. +, if. = { 3, if > {,,. =,, if < 3 if 3 < if < 9 if 9 < Write an equation in slope-intercept form of the line with the given characteristics. 3. slope = ; -intercept = 7 5. passes through (, ) and (3, 3) 5. parallel to the line = 3 ; passes through (, 8). perpendicular to the line = 9; passes through (, ) Write an equation in point-slope form of the line with the given characteristics. 7. slope = ; passes through (, ) 8. passes through ( 3, ) and (, ) 9. The first row of an auditorium has seats. Each row after the first has three more seats than the row before it. a. Find the number of seats in Row 5. b. Which row has 9 seats?. The table shows the amount (in dollars) spent on advertising for a neighborhood festival and the attendance of the festival for several ears. a. Make a scatter plot of the data. Describe the correlation. b. Write an equation that models the attendance as a function of the amount spent on advertising. c. Interpret the slope and -intercept of the line of fit.. Consider the data in the table in Eercise. a. Use a graphing calculator to find an equation of the line of best fit. b. Identif and interpret the correlation coefficient. c. What would ou epect the scatter plot of the residuals to look like? d. Is there a causal relationship in the data? Eplain our reasoning. e. Predict the amount that must be spent on advertising to get people to attend the festival. Advertising (dollars), Yearl attendance, Let a, b, c, and d be constants. Determine which of the lines, if an, are parallel or perpendicular. Eplain. Line : c = a Line : a = b Line 3: a + = d 3. Write a piecewise function defined b three equations that has a domain of all real numbers and a range of 3 <. Chapter Chapter Test 9

59 Cumulative Assessment. Which function represents the arithmetic sequence shown in the graph? A B C D f(n) = 5 + 3n f(n) = 3n f(n) = 7 3n f(n) = 3n a n 8 8 (, ) (, ) (3, 8) (, 5) n. Which of the inequalities are equivalent? Complete the table for the four situations below. Eplain our reasoning. a. the price of a pair of pants and the number sold b. the number of cell phones and the number of tais in a cit c. a person s IQ and the time it takes the person to run 5 meters d. the amount of time spent studing and the score earned Situation a. b. c. d. Correlation Causation Yes No Yes No. Consider the function f() =. Select the functions that are shown in the graph. Eplain our reasoning. g() = f( + ) h() = f(3) k() = f() + p() = f( ) r() = 3f() q() = f() 3 Chapter Writing Linear Functions

60 5. Use the numbers to fill in values for m and b in the equation = m + b so that its graph passes through the points (, ) and (, 3) Fill in the piecewise function with, +, <,, >, or so that the function is represented b the graph. 3, if = { 3, if 7. You claim that ou can create a relation that is a function, and our friend claims that she can create a relation that is not a function. Using the given numbers, create a relation of five ordered pairs that supports our claim. What relation of five ordered pairs can our friend use to support her claim? You have two coupons ou can use at a restaurant. Write and solve an equation to determine how much our total bill must be for both coupons to save ou the same amount of mone. % OFF our entire purchase $5 Off an meal of $ or more 9. The table shows the dail high temperatures (in degrees Fahrenheit) and the numbers of frozen fruit bars sold on eight randoml selected das. The equation = 3 5 models the data. Temperature ( F), Frozen fruit bars, a. Select the points that appear on a scatter plot of the residuals. (9, ) (78, 9) (, ) (8, 58) (98, ) (7, 9) (5, 7) (9, 78) (, ) (8, ) b. Determine whether the model is a good fit for the data. Eplain our reasoning. Chapter Cumulative Assessment 3

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