4 Writing Linear Functions


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1 Writing Linear Functions.1 Writing Equations in SlopeIntercept Form. Writing Equations in PointSlope Form.3 Writing Equations in Standard Form. Writing Equations of Parallel and Perpendicular Lines.5 Scatter Plots and Lines of Fit. Analzing Lines of Fit.7 Arithmetic Sequences Online Auction (p. 5) Old Faithful Geser (p. 19) Helicopter Rescue (p. 18) SEE the Big Idea School Spirit (p. 17) Renewable Energ (p. 1) Mathematical Thinking: Mathematicall proficient students can appl the mathematics the know to solve problems arising in everda life, societ, and the workplace.
2 Maintaining Mathematical Proficienc Using a Coordinate Plane (.11) Eample 1 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. 1. 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 (A.1.E) 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. = = 1 7. = = = 7 1. = 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. 159
3 Mathematical Thinking Mathematicall profi cient students use a problemsolving model that incorporates analzing given information, formulating a plan or strateg, determining a solution, justifing the solution, and evaluating the problemsolving process and the reasonableness of the solutions. (A.1.B) ProblemSolving Strategies Core Concept Solve a Simpler Problem When solving a reallife 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 ProblemSolving Strateg In the deli section of a grocer store, a half pound of sliced roast beef costs $3.19. You bu 1.81 pounds. How much do ou pa? Step 1 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. 1/ lb = $ 1 lb lb Rewrite $3 per 1 = $1 Simplif. In the simpler problem, ou pa $1. Step Appl the strateg to the original problem. 1. You work 37 1 hours and earn $35.5. What is our hourl wage? pound as $ per pound. Total cost = $ lb Use unit analsis to write a verbal model. 1/ lb = $.38 1 lb 1.81 lb Rewrite $3.19 per 1 = $11.55 Simplif. In the original problem, ou pa $ pound as $.38 per pound.. You drive 1.5 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? 1 Chapter Writing Linear Functions
4 .1 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..B A..C A.3.A Writing Equations in SlopeIntercept Form Essential Question Given the graph of a linear function, how can ou write an equation of the line? Work with a partner. Writing Equations in SlopeIntercept Form Find the slope and intercept of each line. Write an equation of each line in slopeintercept form. Use a graphing calculator to verif our equation. a. b. (, 3) (, ) 9 (, 1) 9 9 (, ) 9 c. d. EXPLAINING MATHEMATICAL IDEAS 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 reallife problems. 9 ( 3, 3) (3, 1) 9 9 Mathematical Modeling (, ) (, 1) 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.1 Writing Equations in SlopeIntercept Form 11
5 .1 Lesson Core Vocabular linear model, p. 1 Previous slopeintercept form function rate What You Will Learn Write equations in slopeintercept form. Use linear equations to solve reallife problems. Writing Equations in SlopeIntercept Form Using Slopes and Intercepts to Write Equations Write an equation of each line with the given characteristics. a. slope = 3; intercept = 1 a. = m + b Write the slopeintercept form. b. slope = ; passes through (, 5) = Substitute 3 for m and 1 for b. An equation is = b. Find the intercept. = m + b Write the slopeintercept form. 5 = ( ) + b Substitute for m, for, and 5 for. 13 = b Solve for b. Write an equation. = m + b Write the slopeintercept form. = + 13 Substitute for m and 13 for b. An equation is = Using a Graph to Write an Equation Write an equation of the line in slopeintercept form. (, 3) STUDY TIP You can use an two points on a line to find the slope. (, 3) Find the slope and intercept. Let ( 1, 1 ) = (, 3) and (, ) = (, 3). m = 1 = 3 ( 3) 1 =, or 3 Because the line crosses the ais at (, 3), the intercept is 3. 1 Chapter Writing Linear Functions So, the equation is = 3 3.
6 Using Points to Write Equations Write an equation of each line that passes through the given points. a. ( 3, 5), (, 1) 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 = 1 5 ( 3) = Because the line crosses the ais at (, 1), the intercept is 1. So, an equation is = 1. 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() = 1 and f() = 3. Step 1 Write f() = 1 as (, 1) and f() = 3 as (, 3). Step Find the slope of the line that passes through (, 1) and (, 3). 3 1 m = =, or Step 3 Write an equation of the line. Because the line crosses the ais at (, 1), the intercept is 1. = m + b Write the slopeintercept form. = + 1 Substitute for m and 1 for b. A function is f() = + 1. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Write an equation of the line with the given characteristics. 1. slope = 7; intercept =. slope = 1 ; passes through (, 1) 3 Write an equation of the line in slopeintercept form. 3.. (, 1) (, 3) (, 1) (5, 3) 5. Write an equation of the line that passes through (, ) and (, 1).. Write a linear function g with the values g() = 9 and g(8) = 7. Section.1 Writing Equations in SlopeIntercept Form 13
7 Solving RealLife Problems A linear model is a linear function that models a reallife 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 15 million megawatt hours of electricit in 7. B 1, the amount of electricit generated had increased to 19 million megawatt hours. Write a linear model that represents the number of megawatt hours generated b nonhdropower 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 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 1 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 1 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 1 corresponds to = 5. Let ( 1, 1 ) = (, 15) and (, ) = (5, 19). The initial value is the intercept b, which is 15. The rate of change is the slope m. Part 17 corresponds to = 1. m = 1 = = 11 5 =.8 Megawatt hours = Initial + Rate of Years (millions) value change since 7 = = Write the equation. = (1) Substitute 1 for. = 333 Simplif. The linear model is = The model predicts nonhdropower renewable energ sources will generate 333 million megawatt hours in 17.. Look Back To check that our model is correct, verif that (, 15) and (5, 19) 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 1. Write a linear model that represents the number of megawatt hours generated b hdropower as a function of the number of ears since 7. 1 Chapter Writing Linear Functions
8 .1 Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE A linear function that models a reallife situation is called a.. WRITING Eplain how ou can use slopeintercept 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 characteristics. (See Eample 1.) 3. slope:. slope: intercept: 9 passes through: ( 3, 5) 5. slope: 3. slope: 7 passes through: (, ) intercept: 1 7. slope: 3 8. slope: 3 intercept: 8 passes through: ( 8, ) In Eercises 9 1, write an equation of the line in slopeintercept form. (See Eample.) ( 3, ) (, ) (3, 3) In Eercises 13 18, write an equation of the line that passes through the given points. (See Eample 3.) 13. (3, 1), (, 1) 1. (, 7), (, 5) (, ) (, 3) (, ) (, ) (, ) In Eercises 19, write a linear function f with the given values. (See Eample.) 19. f() =, f() =. f() = 7, f(3) = 1 1. f() = 3, f() =. f(5) = 1, f() = 5 3. f( ) =, f() =. f() = 3, f( ) = 3 In Eercises 5 and, write a linear function f with the given values f() f() 1 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 = 1 5 = 3 5 = 3 5 (, ) (5, 1) 15. (, ), (, ) 1. (, ), (, ) = (, 5), ( 1.5, 1) 18. (, 3), ( 5,.5) Section.1 Writing Equations in SlopeIntercept Form 15
9 9. MODELING WITH MATHEMATICS In 19, the world record for the men s mile was 3.91 minutes. In 198, the record time was 3.81 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 19. 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 1 hours of recording time at the studio or a $75 music software program that ou can use to record on our own computer? Eplain. 31. WRITING A line passes through the points (, ) and (, 5). Is it possible to write an equation of the line in slopeintercept form? Justif our answer. 3. THOUGHT PROVOKING Describe a reallife situation involving a linear function whose graph passes through the points. 33. REASONING Recall that the standard form of a linear equation is A + B = C. Rewrite this equation in slopeintercept form. Use our answer to find the slope and intercept of the graph of the equation + 5 = 9. Maintaining Mathematical Proficienc Solve the equation. (Section 1.3) 8 (, ) (, 8) 3. MAKING AN ARGUMENT Your friend claims that given f() and an other value of a linear function f, ou can write an equation in slopeintercept 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 1, where = represents the ear. U.S. Bo Office Revenue 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 18? 37. ABSTRACT REASONING Show that the equation of the line that passes through the points (, b) and (1, b + m) is = m + b. Eplain how ou can be sure that the point ( 1, b m) also lies on the line. Reviewing what ou learned in previous grades and lessons (, 1) (3, ) 38. 3( 15) = (3d + 3) = 7 + d. 5( 3n) = 1(n ) Determine whether and show direct variation. If so, identif the constant of variation. (Section 3.) 1. + =. + 5 = = 3 1 Chapter Writing Linear Functions
10 . TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..B A..C A.3.A Writing Equations in PointSlope 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 = 1 b. m = USING TECHNOLOGY To be proficient in math, ou need to understand the feasibilit, appropriateness, and limitations of the technological tools at our disposal. For instance, in reallife 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 ( 1, 1 ) 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 pointslope form of a linear equation. Writing an Equation Work with a partner. For four months, ou have saved $5 per month. You now have $175 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 ( 1, 1 ) 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 (, 175) (, ) t Time (months) Section. Writing Equations in PointSlope Form 17
11 . Lesson Core Vocabular pointslope form, p. 18 Previous slopeintercept 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 reallife problems. Writing Equations of Lines in PointSlope 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 1. Let (, ) be another point on the line where. You can write an equation relating and using the slope formula with ( 1, 1 ) = (, 3) and (, ) = (, ). m = 1 1 Write the slope formula. 1 = 3 Substitute values. 1 ( ) = 3 Multipl each side b ( ). The equation in pointslope form is 3 = 1 ( ). Core Concept PointSlope Form Words A linear equation written in the form 1 = m( 1 ) is in pointslope form. The line passes through the point ( 1, 1 ), and the slope of the line is m. (, ) 1 passes through ( 1, 1 ) ( 1, 1 ) Algebra 1 = m( 1 ) slope 1 18 Chapter Writing Linear Functions Using a Slope and a Point to Write an Equation Write an equation in pointslope form of the line that passes through the point (8, 3) and has a slope of 1. 1 = m( 1 ) Using PointSlope Form Identif the slope of the line + = 3( ). Then identif a point the line passes through. The equation is written in pointslope form, 1 = m( 1 ), where m = 3, 1 =, and 1 =. So, the slope of the line is 3, and the line passes through the point (, ). Write the pointslope form. 3 = 1 ( 8) Substitute 1 for m, 8 for 1, and 3 for 1. The equation is 3 = 1 ( 8).
12 Monitoring Progress Help in English and Spanish at BigIdeasMath.com 1. Identif the slope of the line 1 = ( 1 ). Then identif a point the line passes through. Write an equation in pointslope form of the line that passes through the given point and has the given slope.. (3, 1); m = 3. (, ); m = 3 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 1 Find the slope of the line. Step Use the slope and one of the points to write an equation of the line in pointslope form. Using Two Points to Write an Equation Write an equation in slopeintercept form of the line shown. Step 1 Find the slope of the line. m = 3 1 =, or Step Use the slope m = and the point (1, ) to write an equation of the line. 1 = m( 1 ) Write the pointslope form. = ( 1) Substitute for m, 1 for 1, and for 1. = + = + Distributive Propert 1 Write in slopeintercept form. 1 (1, ) 3 5 (3, ) The equation is = +. Writing a Linear Function Write a linear function f with the values f() = and f(1) = 1. Note that ou can rewrite f() = as (, ) and f(1) = 1 as (1, 1). Step 1 Find the slope of the line that passes through (, ) and (1, 1). 1 ( ) m = 1 = 1, or Step Use the slope m = 1.5 and the point (1, 1) to write an equation of the line. 1 = m( 1 ) Write the pointslope form. 1 = 1.5( 1) Substitute 1.5 for m, 1 for 1, and 1 for 1. = A function is f() = Write in slopeintercept form. Section. Writing Equations in PointSlope Form 19
13 Monitoring Progress Help in English and Spanish at BigIdeasMath.com Write an equation in slopeintercept form of the line that passes through the given points.. (1, ), (3, 1) 5. (, 1), (8, ) Write a linear function g with the given values.. g() = 3, g() = 5 7. g( 1) = 8, g() = 1 Solving RealLife 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 Cost (dollars) Step 1 Find the rate of change for consecutive data pairs in the table. 3 =, 58 8 =, =, = 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 1 = m(n n 1 ) Write the pointslope form. C 3 = (n ) Substitute for m, for n 1, and 3 for C 1. C = n + 1 Write in slopeintercept form. Because the cost increases at a constant rate, the situation can be modeled b a linear equation. The linear model is C = n + 1. Number of months Total cost (dollars) Monitoring Progress Help in English and Spanish at BigIdeasMath.com 8. 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. 17 Chapter Writing Linear Functions
14 . Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check 1. 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, identif the slope of the line. Then identif a point the line passes through. (See Eample 1.) 3. = ( 9). 1 = ( 3) = 8 ( + 1 ). + = 3 5 In Eercises 7 1, write an equation in pointslope form of the line that passes through the given point and has the given slope. (See Eample.) 7. (, 1); m = 8. (3, 5); m = 1 9. (7, ); m = 1. ( 8, ); m = (9, ); m = 3 1. (, ); m = 13. (, ); m = 3 1. (5, 1); m = 5 In Eercises 15 18, write an equation in slopeintercept form of the line shown. (See Eample 3.) (, ) (1, 3) (, ) (3, 1) (, ) (, 1) (8, ) (1, 5) 1 In Eercises 19, write an equation in slopeintercept form of the line that passes through the given points. 19. (7, ), (, 1). (, ), (1, 1) 1. (, 1), (3, 7). (, 5), (, 5) 3. (1, 9), ( 3, 9). ( 5, 19), (5, 13) In Eercises 5 3, write a linear function f with the given values. (See Eample.) 5. f() =, f(1) = 1. f(5) = 7, f( ) = 7. f( ) =, f() = 3 8. f( 1) =, f( ) = 9. f( 3) = 1, f(13) = 5 3. f( 9) = 1, f( 1) = In Eercises 31 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 5.) ERROR ANALYSIS Describe and correct the error in writing an equation of the line that passes through the point (1, 5) and has a slope of. 1 = m( 1 ) 5 = ( 1) Section. Writing Equations in PointSlope Form 171
15 3. ERROR ANALYSIS Describe and correct the error in writing an equation of the line that passes through the points (1, ) and (, 3). m = 3 1 = 1 3 = 1 ( ) MODELING WITH MATHEMATICS You are designing a sticker to advertise our band. A compan charges $5 for the first 1 stickers and $8 for each additional 1 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. 38. 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 $1 on the beach house rental. What is the maimum number of das ou can rent the beach house? 39. WRITING Describe two was to graph the equation 1 = 3 ( ).. THOUGHT PROVOKING The graph of a linear function passes through the point (1, 5) and has a slope of. Represent this function in two other was. 5. 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)? 3. CONNECTION TO TRANSFORMATIONS Compare the graph of = to the graph of 1 = ( + 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) (, ) Week Week, Mone left, 1 $1 $75 3 $5 $5 1. REASONING You are writing an equation of the line that passes through two points that are not on the ais. Would ou use slopeintercept form or pointslope form to write the equation? Eplain. Maintaining Mathematical Proficienc Use intercepts to graph the linear equation. (Section 3.) 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? Reviewing what ou learned in previous grades and lessons 5. + = = = 8. 7 = 1 17 Chapter Writing Linear Functions
16 .3 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..B A..C A.3.A Writing Equations in Standard Form Essential Question How can ou write the equation of a line in standard form? Writing Equations in Standard Form Work with a partner. So far ou have written equations of lines in slopeintercept form and pointslope form. Linear equations can also be written in standard form, A + B = C. Write each equation in standard form. a. = 3 5 The equation is in slopeintercept form. Subtract 3 from each side. The equation in standard form is. b. = ( + ) The equation is in slopeintercept form. Distributive Propert Add to each side. Add to each side. The equation in standard form is. c. = + d. = 7 e. 1 = 3( + ) f. + 8 = ( 3) Finding the Slope and Intercept Work with a partner. The slope and intercept of a line are not eplicitl known from a linear equation written in standard form. Find the slope and intercept of the line represented b each equation. a. + 3 = 15 The equation is in standard form. Add to each side. Divide each side b 3. The slope is, and the intercept is. MAKING MATHEMATICAL ARGUMENTS To be proficient in math, ou need to justif our conclusions and communicate them to others. b. 5 1 = c. 8 = 5 Communicate Your Answer 3. How can ou write the equation of a line in standard form?. How can ou find the slope and intercept of a line given the equation of the line in standard form? 5. Consider the graph of A + B = C. a. Does changing the value of A change the slope? Does changing the value of B change the slope? Eplain our reasoning. b. Does changing the value of A change the intercept? Does changing the value of B change the intercept? Eplain our reasoning. Section.3 Writing Equations in Standard Form 173
17 .3 Lesson What You Will Learn Core Vocabular Previous standard form equivalent equation pointslope form Write equations in standard form. Use linear equations to solve reallife problems. Writing Equations in Standard Form Recall that the linear equation A + B = C is in standard form, where A, B, and C are real numbers and A and B are not both zero. All linear equations can be written in standard form. Writing Equivalent Equations in Standard Form REMEMBER You can produce an equivalent equation b multipling or dividing each side of an equation b the same nonzero number. Write two equations in standard form that are equivalent to =. To write one equivalent equation, multipl each side of the original equation b. ( ) = () 1 = 8 To write another equivalent equation, divide each side of the original equation b. = 3 = Using Two Points to Write an Equation ANOTHER WAY You can use either of the given points to write an equation of the line. Use m = 3 and (, ). ( ) = 3( ) + = = 3 + = Write an equation in standard form of the line shown. Step 1 Find the slope of the line. m = 1 ( ) 1 = 3, or 3 1 Step Use the slope m = 3 and the point (1, 1) to write an equation in pointslope form. 1 = m( 1 ) Write the pointslope form. 1 = 3( 1) Substitute 3 for m, 1 for 1, and 1 for 1. Step 3 Write the equation in standard form. 1 = 3( 1) 1 = = = Write the equation. Distributive Propert Add 3 to each side. Add 1 to each side. (1, 1) (, ) An equation is 3 + =. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 1. Write two equations in standard form that are equivalent to = 3.. Write an equation in standard form of the line that passes through (3, 1) and (, 3). 17 Chapter Writing Linear Functions
18 Recall that equations of horizontal lines have the form = b and equations of vertical lines have the form = a. You cannot write an equation of a vertical line in slopeintercept form or pointslope form because the slope of a vertical line is undefined. However, ou can write an equation of a vertical line in standard form. Horizontal and Vertical Lines ANOTHER WAY Using the slopeintercept form to find an equation of the horizontal line gives ou =, or =. Write an equation of the specified line. a. blue line b. red line a. The coordinate of the given point on the blue line is. This means that all points on the line have a coordinate of. (, 1) (, ) So, an equation of the line is =. b. The coordinate of the given point on the red line is. This means that all points on the line have an coordinate of. So, an equation of the line is =. Completing an Equation in Standard Form Find the missing coefficient in the equation of the line shown. Write the completed equation. ( 1, ) A + 3 = Step 1 Find the value of A. Substitute the coordinates of the given point for and in the equation. Then solve for A. A + 3 = Write the equation. A( 1) + 3() = Substitute 1 for and for. A = Simplif. A = Divide each side b 1. Step Complete the equation. + 3 = Substitute for A. An equation is + 3 =. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Write equations of the horizontal and vertical lines that pass through the given point. 3. ( 8, 9). (13, 5) Find the missing coefficient in the equation of the line that passes through the given point. Write the completed equation B = 7; ( 1, 1). A + = 3; (, 11) Section.3 Writing Equations in Standard Form 175
19 Solving RealLife Problems Modeling with Mathematics Your class is taking a trip to the public librar. You can travel in small and large vans. A small van holds 8 people, and a large van holds 1 people. Your class can fill 15 small vans and large vans. a. Write an equation in standard form that models the possible combinations of small and large vans that our class can fill. b. Graph the equation from part (a). c. Find four possible combinations. a. Write a verbal model. Then write an equation. Number of Capacit of + small vans large van 8 s + 1 Capacit of small van Number of large vans = People on trip = p Because our class can fill 15 small vans and large vans, use (15, ) to find the value of p. 8s + 1 = p Write the equation. 8(15) + 1() = p Substitute 15 for s and for. 1 = p Simplif. So, the equation 8s + 1 = 1 models the possible combinations. ANOTHER WAY Another wa to find possible combinations is to substitute values for s or in the equation and solve for the other variable. b. Use intercepts to graph the equation. Find the intercepts. Substitute for s. Substitute for. 8() + 1 = 1 8s + 1() = 1 = 1 s = 18 Plot the points (, 1) and (18, ). Connect them with a line segment. For this problem, onl wholenumber values of s and make sense c. The graph passes through (, 1), (, 8), (1, ), and (18, ). So, four possible combinations are small and 1 large, small and 8 large, 1 small and large, and 18 small and large. 1 8 (, 1) 1 (18, ) Monitoring Progress Help in English and Spanish at BigIdeasMath.com 7. WHAT IF? Eight students decide to not go on the class trip. Write an equation in standard form that models the possible combinations of small and large vans that our class can fill. Find four possible combinations. 17 Chapter Writing Linear Functions
20 .3 Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check 1. WRITING Eplain how to write an equation in standard form of a line when two points on the line are given.. WHICH ONE DOESN'T BELONG? Which equation does not belong with the other three? Eplain our reasoning = 1 = 5 9 = 1 + = Monitoring Progress and Modeling with Mathematics In Eercises 3 8, write two equations in standard form that are equivalent to the given equation. (See Eample 1.) 3. + = = = = = = 5 In Eercises 9 1, write an equation in standard form of the line that passes through the given point and has the given slope. 9. ( 3, ); m = 1 1. (, 1); m = (, 5); m = 1. ( 8, ); m = 13. (, ); m = 3 1. (, 1); m = 1 In Eercises 15 18, write an equation in standard form of the line shown. (See Eample.) (, 3) ( 5, ) (, ) ( 3, ) (, 5) (1, 1) (, ) (, ) In Eercises 19, write equations of the horizontal and vertical lines that pass through the given point. (See Eample 3.) 19. (, 3). ( 5, ) 1. (8, 1). (, ) In Eercises 3, find the missing coefficient in the equation of the line shown. Write the completed equation. (See Eample.) 3. A + 3 = 5 (, 1). 5.. (, ) + B = 1 A = 1 (, 1) ( 5, ) 8 + B = 7. ERROR ANALYSIS Describe and correct the error in finding the value of A for the equation A 3 = 5, when the graph of the equation passes through the point (1, ). A( ) 3(1) = 5 A = 8 A = Section.3 Writing Equations in Standard Form 177
21 8. MAKING AN ARGUMENT Your friend sas that ou can write an equation of a horizontal line in standard form but not in slopeintercept form or pointslope form. Is our friend correct? Eplain. 9. MODELING WITH MATHEMATICS The diagram shows the prices of two tpes of ground cover plants. A gardener can afford to bu 15 vinca plants and phlo plants. (See Eample 5.) 3. HOW DO YOU SEE IT? A dog kennel charges $5 per night to board our dog. The kennel also sells dog treats for $5 each. The graph shows the possible combinations of nights at the kennel and treats that ou can bu for $1. Number of nights Dog Kennel = Number of treats a. List two possible combinations. a. Write an equation in standard form that models the possible combinations of vinca and phlo plants the gardener can afford to bu. b. Graph the equation from part (a). c. Find four possible combinations. 3. MODELING WITH MATHEMATICS One bus ride costs $.75. One subwa ride costs $1. A monthl pass for unlimited bus and subwa rides costs the same as 3 bus rides plus 3 subwa rides. a. Write an equation in standard form that models the possible combinations of bus and subwa rides with the same total cost as the pass. b. Interpret the intercepts of the graph. 33. ABSTRACT REASONING Write an equation in standard form of the line that passes through (a, ) and (, b), where a and b. 3. THOUGHT PROVOKING Use the graph shown. A + B = C b. Graph the equation from part (a). c. You ride the bus times in one month. How man times must ou ride the subwa for the total cost of the rides to equal the cost of the pass? Eplain our reasoning. 31. WRITING There are three forms of an equation of a line: slopeintercept, pointslope, and standard form. Which form would ou prefer to use to do each of the following? Eplain. a. Graph the equation. b. Find the intercept of the graph of the equation. c. Write an equation of the line given two points on the line. a. What are the signs of B and C when A is positive? when A is negative? b. Eplain how to change the equation so that the graph is reflected in the ais. c. Eplain how to change the equation so that the graph is translated horizontall. 35. MATHEMATICAL CONNECTIONS Write an equation in standard form that models the possible lengths and widths (in feet) of a rectangle with the same perimeter as a rectangle that is 1 feet wide and feet long. Make a table that shows five possible lengths and widths of the rectangle. Maintaining Mathematical Proficienc Reviewing what ou learned in previous grades and lessons Write the reciprocal of the number. (Skills Review Handbook) Chapter Writing Linear Functions
22 . TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..C A..E A..F A..G A.3.A 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 slopeintercept 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 + = 1 + = = = SELECTING TOOLS 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 slopeintercept 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 = 1 3 = 1 + = 3 3 = 1.5 = = + 3 = + 5 Communicate Your Answer 3. How can ou recognize lines that are parallel or perpendicular?. Compare the slopes of the lines in Eploration 1. 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. Writing Equations of Parallel and Perpendicular Lines 179
23 . Lesson What You Will Learn Core Vocabular parallel lines, p. 18 perpendicular lines, p. 181 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. 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. 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 1 ( ) = 1 5 Line b: m = 1 1 ( 3) = 1 5 ( ) Line c: m = ( 3) = 1 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 1 (1, ) ( 3, ) (, 5) (1, 1) ANOTHER WAY You can also use the slope m = and the pointslope form to write an equation of the line that passes through (5, ). 1 = m( 1 ) ( ) = ( 5) = 1 Write an equation of the line that passes through (5, ) and is parallel to the line = + 3. Step 1 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 slopeintercept form to find the intercept of the parallel line. = m + b Write the slopeintercept form. = (5) + b Substitute for m, 5 for, and for. 1 = b Solve for b. Using m = and b = 1, an equation of the parallel line is = 1. Monitoring Progress 18 Chapter Writing Linear Functions Help in English and Spanish at BigIdeasMath.com 1. Line a passes through ( 5, 3) and (, 1). 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 =
24 REMEMBER The product of a nonzero number m and its negative reciprocal is 1: m ( 1 m ) = 1. 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. = + 1 = 1 Identifing Parallel and Perpendicular Lines Determine which of the lines, if an, are parallel or perpendicular. Line a: = + Line b: + = 3 Line c: 8 = 1 Write the equations in slopeintercept form. Then compare the slopes. Line a: = + Line b: = Line c: = 1 Lines b and c have slopes of 1, so the are parallel. Line a has a slope of, the negative reciprocal of 1, 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 slopeintercept form to write an equation of the line that passes through ( 3, 1). = m + b 1 = ( 3) + b 5 = b So, = 5. Write an equation of the line that passes through ( 3, 1) and is perpendicular to the line = Step 1 Find the slope of the perpendicular line. The graph of the given equation has a slope of 1. Because the slopes of perpendicular lines are negative reciprocals, the slope of the perpendicular line that passes through ( 3, 1) is. Step Use the slope m = and the pointslope form to write an equation of the perpendicular line that passes through ( 3, 1). 1 = m( 1 ) Write the pointslope form. 1 = [ ( 3)] Substitute for m, 3 for 1, and 1 for 1. 1 = = 5 Simplif. Write in slopeintercept 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: + 18 = 9. Write an equation of the line that passes through ( 3, 5) and is perpendicular to the line = 3 1. Section. Writing Equations of Parallel and Perpendicular Lines 181
25 REMEMBER The slope of a horizontal line is. The slope of a vertical line is undefined. Horizontal and Vertical Lines Write an equation of a line that is (a) parallel to the ais and (b) perpendicular to the ais. What is the slope of each line? a. The ais (the line = ) is a horizontal line and has a slope of. All horizontal lines are parallel. Equations of horizontal lines have the form = b, where b is a constant. Let b equal an number other than, such as 5. An equation of a line parallel to the ais is = 5. The slope of the line is. b. The ais is a horizontal line. Vertical lines are perpendicular to horizontal lines. The slope of a vertical line is undefined. Equations of vertical lines have the form = a, where a is a constant. Let a equal an number, such as. An equation of a line perpendicular to the ais is =. The slope of the line is undefined. 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 (1, ) shore Step 1 Find the slope of the line that represents the shoreline. The line passes through points (1, 3) and (, 1). So, the slope is m = = 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 pointslope form to write an equation of the shortest flight path that passes through (1, ). 1 = m( 1 ) Write the pointslope form. = 3 ( 1) Substitute 3 for m, 1 for 1, and for 1. = 3 1 Distributive Propert = 3 17 Write in slopeintercept form. An equation that represents the shortest flight path is = Monitoring Progress Help in English and Spanish at BigIdeasMath.com 5. Write an equation of a line that is (a) parallel to the line = 3 and (b) perpendicular to the line = 3. What is the slope of each line?. WHAT IF? In Eample, a boat is traveling perpendicular to the shoreline and passes through (9, 3). Write an equation that represents the path of the boat. 18 Chapter Writing Linear Functions
26 . Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE Nonvertical lines have the same slope.. 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 1.) 3.. (3, ) ( 3, 1) (, 3) a (, ) 3 b c 1 (3, ) (, 3) 3 5. Line a passes through ( 1, ) and (1, ). Line b passes through (, ) and (, ). Line c passes through (, ) and ( 1, 1).. Line a passes through ( 1, 3) and (1, 9). Line b passes through (, 1) and ( 1, 1). Line c passes through (3, 8) and (, 1). 7. Line a: + = 8 Line b: + = Line c: = Line a: 3 = Line b: 3 = + 18 Line c: 3 = 9 In Eercises 9 1, write an equation of the line that passes through the given point and is parallel to the given line. (See Eample.) 9. ( 1, 3); = + 1. (1, ); = (18, ); 3 = 1 1. (, 5); = In Eercises 13 18, determine which of the lines, if an, are parallel or perpendicular. Eplain. (See Eample 3.) ( 3, 1) (, 1) (, ) ( 5, ) b ( 3, ) (, ) a c b (, 5) (, 5) (3, ) c (, ) (5, ) (, ) 1 a (5, ) 3 ( 1, 1) c (, 5) (3, ) (, ) b 5 (, ) a 15. Line a passes through (, 1) and (, 3). Line b passes through (, 1) and (, ). Line c passes through (1, 3) and (, 1). 1. Line a passes through (, 1) and (, 13). Line b passes through (, 9) and (, 1). Line c passes through (, 1) and (, 9). 17. Line a: 3 = Line b: = 3 + Line c: + 3 = 18. Line a: = Line b: = Line c: + = 1 In Eercises 19, write an equation of the line that passes through the given point and is perpendicular to the given line. (See Eample.) 19. (7, 1); = 1 9. (, 1); = ( 3, 3); = 8. (8, 1); + = 1 In Eercises 3, write an equation of a line that is (a) parallel to the given line and (b) perpendicular to the given line. (See Eample 5.) 3. the ais. = 5. =. = 7 7. ERROR ANALYSIS Describe and correct the error in writing an equation of the line that passes through (1, 3) and is parallel to the line = = m( 1 ) 3 = ( 1) 3 = + = + 7 Section. Writing Equations of Parallel and Perpendicular Lines 183
27 8. 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( 1) ( 5) = 3( ) + 5 = 3 1 = 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. (, 1) 1 (, 8) 1 8 (, ) 9. 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.) Softball Academ (, ) 8 proposed water pipe Total cost (dollars) eisting (, 3) water pipe connector valve 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, 1) 3. HOW DO YOU SEE IT? A softball academ charges Student B Student A 1 3 Months of membership 3. MODELING WITH MATHEMATICS A parks and 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. 8 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) (11, ) REASONING In Eercises 35 37, determine whether the statement is alwas, sometimes, or never true. Eplain our reasoning. 35. Two lines with positive slopes are perpendicular. (8, ) MATHEMATICAL CONNECTIONS The vertices of a quadrilateral are A(, ), B(, ), C(8, 1), 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. 37. Two lines with the same intercept are perpendicular. 38. 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 3.1) 39. (3, ), (, 8), (5, 1), (, 9), (7, 1). ( 1, ), (1, ), ( 1, ), (1, ), ( 1, 5) 18 Chapter Writing Linear Functions
28 .1. What Did You Learn? Core Vocabular linear model, p. 1 pointslope form, p. 18 parallel lines, p. 18 perpendicular lines, p. 181 Core Concepts Section.1 Using SlopeIntercept Form, p. 1 Section. Using PointSlope Form, p. 18 Section.3 Writing Equations in Standard Form, p. 17 Section. Parallel Lines and Slopes, p. 18 Perpendicular Lines and Slopes, p. 181 Mathematical Thinking 1. How can ou eplain to ourself the meaning of the graph in Eercise 3 on page 1?. How did ou use the structure of the equations in Eercise 3 on page 17 to make a conjecture? 3. How did ou use the diagram in Eercise 33 on page 18 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. 185
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