# 4 Writing Linear Functions

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Writing Linear Functions.1 Writing Equations in Slope-Intercept Form. Writing Equations in Point-Slope 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 problem-solving model that incorporates analzing given information, formulating a plan or strateg, determining a solution, justifing the solution, and evaluating the problem-solving process and the reasonableness of the solutions. (A.1.B) 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 \$3.19. 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 Slope-Intercept 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 Slope-Intercept Form 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 (, 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 real-life 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 Slope-Intercept Form 11

5 .1 Lesson Core Vocabular linear model, p. 1 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 characteristics. a. slope = 3; -intercept = 1 a. = m + b Write the slope-intercept 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 slope-intercept form. 5 = ( ) + b Substitute for m, for, and 5 for. 13 = b Solve for b. Write an equation. = m + b Write the slope-intercept 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 slope-intercept 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 slope-intercept 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 slope-intercept 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 Slope-Intercept Form 13

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 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 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 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 non-hdropower 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 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 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 slope-intercept 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 Slope-Intercept 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 slope-intercept form? Justif our answer. 3. THOUGHT PROVOKING Describe a real-life 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 slope-intercept 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 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 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 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 = 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 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 ( 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 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 \$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 Point-Slope Form 17

11 . Lesson Core Vocabular point-slope form, p. 18 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 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 point-slope form is 3 = 1 ( ). Core Concept Point-Slope Form Words A linear equation written in the form 1 = m( 1 ) is in point-slope 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 point-slope form of the line that passes through the point (8, 3) and has a slope of 1. 1 = m( 1 ) Using Point-Slope Form Identif the slope of the line + = 3( ). Then identif a point the line passes through. The equation is written in point-slope 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 point-slope 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 point-slope 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 point-slope form. Using Two Points to Write an Equation Write an equation in slope-intercept 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 point-slope form. = ( 1) Substitute for m, 1 for 1, and for 1. = + = + Distributive Propert 1 Write in slope-intercept 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 point-slope form. 1 = 1.5( 1) Substitute 1.5 for m, 1 for 1, and 1 for 1. = A function is f() = Write in slope-intercept form. Section. Writing Equations in Point-Slope Form 19

13 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.. (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 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 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 point-slope 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 point-slope form. C 3 = (n ) Substitute for m, for n 1, and 3 for C 1. C = n + 1 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 + 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 point-slope 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 slope-intercept form of the line shown. (See Eample 3.) (, ) (1, 3) (, ) (3, 1) (, ) (, 1) (8, ) (1, 5) 1 In Eercises 19, write an equation in slope-intercept 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 Point-Slope 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 slope-intercept form or point-slope 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 slope-intercept form and point-slope 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 slope-intercept form. Subtract 3 from each side. The equation in standard form is. b. = ( + ) The equation is in slope-intercept 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 point-slope form Write equations in standard form. Use linear equations to solve real-life 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 point-slope form. 1 = m( 1 ) Write the point-slope 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 slope-intercept form or point-slope 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 slope-intercept 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 Real-Life 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 whole-number 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 slope-intercept form or point-slope 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: slope-intercept, point-slope, 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 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 + = 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 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 = 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 point-slope 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 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. 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 slope-intercept 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 slope-intercept 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 point-slope form to write an equation of the perpendicular line that passes through ( 3, 1). 1 = m( 1 ) Write the point-slope form. 1 = [ ( 3)] Substitute for m, 3 for 1, and 1 for 1. 1 = = 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: + 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 point-slope form to write an equation of the shortest flight path that passes through (1, ). 1 = m( 1 ) Write the point-slope form. = 3 ( 1) Substitute 3 for m, 1 for 1, and for 1. = 3 1 Distributive Propert = 3 17 Write in slope-intercept 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

28 .1. What Did You Learn? Core Vocabular linear model, p. 1 point-slope form, p. 18 parallel lines, p. 18 perpendicular lines, p. 181 Core Concepts Section.1 Using Slope-Intercept Form, p. 1 Section. Using Point-Slope 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

### Essential Question How can you describe the graph of the equation Ax + By = C? Number of adult tickets. adult

3. Graphing Linear Equations in Standard Form Essential Question How can ou describe the graph of the equation A + B = C? Using a Table to Plot Points Work with a partner. You sold a total of \$16 worth

### Essential Question How can you solve a system of linear equations? \$15 per night. Cost, C (in dollars) \$75 per Number of. Revenue, R (in dollars)

5.1 Solving Sstems of Linear Equations b Graphing Essential Question How can ou solve a sstem of linear equations? Writing a Sstem of Linear Equations Work with a partner. Your famil opens a bed-and-breakfast.

### Essential Question How can you graph a system of linear inequalities?

5.7 Sstems of Linear Inequalities Essential Question How can ou graph a sstem of linear inequalities? Graphing Linear Inequalities Work with a partner. Match each linear inequalit with its graph. Eplain

### Graphing Linear Equations in Slope-Intercept Form

4.4. Graphing Linear Equations in Slope-Intercept Form equation = m + b? How can ou describe the graph of the ACTIVITY: Analzing Graphs of Lines Work with a partner. Graph each equation. Find the slope

### Solving Systems of Linear Equations

5 Solving Sstems of Linear Equations 5. Solving Sstems of Linear Equations b Graphing 5. Solving Sstems of Linear Equations b Substitution 5.3 Solving Sstems of Linear Equations b Elimination 5. Solving

### Solving Special Systems of Linear Equations

5. Solving Special Sstems of Linear Equations Essential Question Can a sstem of linear equations have no solution or infinitel man solutions? Using a Table to Solve a Sstem Work with a partner. You invest

### Solving Systems of Linear Equations

5 Solving Sstems of Linear Equations 5. Solving Sstems of Linear Equations b Graphing 5. Solving Sstems of Linear Equations b Substitution 5.3 Solving Sstems of Linear Equations b Elimination 5. Solving

### 2.3 Writing Equations of Lines

. Writing Equations of Lines In this section ou will learn to use point-slope form to write an equation of a line use slope-intercept form to write an equation of a line graph linear equations using the

### The Graph of a Linear Equation

4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that

### A Library of Parent Functions. Linear and Squaring Functions. Writing a Linear Function. Write the linear function f for which f 1 3 and f 4 0.

0_006.qd 66 /7/0 Chapter.6 8:0 AM Page 66 Functions and Their Graphs A Librar of Parent Functions What ou should learn Identif and graph linear and squaring functions. Identif and graph cubic, square root,

### SLOPE OF A LINE 3.2. section. helpful. hint. Slope Using Coordinates to Find 6% GRADE 6 100 SLOW VEHICLES KEEP RIGHT

. Slope of a Line (-) 67. 600 68. 00. SLOPE OF A LINE In this section In Section. we saw some equations whose graphs were straight lines. In this section we look at graphs of straight lines in more detail

### Lesson 8.3 Exercises, pages

Lesson 8. Eercises, pages 57 5 A. For each function, write the equation of the corresponding reciprocal function. a) = 5 - b) = 5 c) = - d) =. Sketch broken lines to represent the vertical and horizontal

### Quadratic Functions and Models. The Graph of a Quadratic Function. These functions are examples of polynomial functions. Why you should learn it

0_00.qd 8 /7/05 Chapter. 9:0 AM Page 8 Polnomial and Rational Functions Quadratic Functions and Models What ou should learn Analze graphs of quadratic functions. Write quadratic functions in standard form

### Attributes and Transformations of Reciprocal Functions VOCABULARY

TEKS FOCUS - Attributes and Transformations of Reciprocal Functions VOCABULARY TEKS (6)(G) Analze the effect on the graphs of f () = when f () is replaced b af (), f (b), f ( - c), and f () + d for specific

### The Slope-Intercept Form

7.1 The Slope-Intercept Form 7.1 OBJECTIVES 1. Find the slope and intercept from the equation of a line. Given the slope and intercept, write the equation of a line. Use the slope and intercept to graph

### LINEAR FUNCTIONS. Form Equation Note Standard Ax + By = C A and B are not 0. A > 0

LINEAR FUNCTIONS As previousl described, a linear equation can be defined as an equation in which the highest eponent of the equation variable is one. A linear function is a function of the form f ( )

### Solving Quadratic Equations

9 Solving Quadratic Equations 9.1 Properties of Radicals 9. Solving Quadratic Equations b Graphing 9. Solving Quadratic Equations Using Square Roots 9. Solving Quadratic Equations b Completing the Square

### GASOLINE The graph represents the cost of gasoline at \$3 per gallon.

9-6 Slope-Intercept Form MAIN IDEA Graph linear equations using the slope and -intercept. New Vocabular slope-intercept form -intercept Math Online glencoe.com Etra Eamples Personal Tutor Self-Check Quiz

### 1.5 Shifting, Reflecting, and Stretching Graphs

7_00.qd /7/0 0: AM Page 7. Shifting, Reflecting, and Stretching Graphs Section. Shifting, Reflecting, and Stretching Graphs 7 Summar of Graphs of Parent Functions One of the goals of this tet is to enable

### Filling in Coordinate Grid Planes

Filling in Coordinate Grid Planes A coordinate grid is a sstem that can be used to write an address for an point within the grid. The grid is formed b two number lines called and that intersect at the

### Let (x 1, y 1 ) (0, 1) and (x 2, y 2 ) (x, y). x 0. y 1. y 1 2. x x Multiply each side by x. y 1 x. y x 1 Add 1 to each side. Slope-Intercept Form

8 (-) Chapter Linear Equations in Two Variables and Their Graphs In this section Slope-Intercept Form Standard Form Using Slope-Intercept Form for Graphing Writing the Equation for a Line Applications

### 3 Quadratic Functions

3 Quadratic Functions 3.1 Transformations of Quadratic Functions 3. Characteristics of Quadratic Functions 3.3 Focus of a Parabola 3. Modeling with Quadratic Functions SEE the Big Idea Meteorologist (p.

### Exponential Functions

CHAPTER Eponential Functions 010 Carnegie Learning, Inc. Georgia has two nuclear power plants: the Hatch plant in Appling Count, and the Vogtle plant in Burke Count. Together, these plants suppl about

### 1.2 GRAPHS OF EQUATIONS

000_00.qd /5/05 : AM Page SECTION. Graphs of Equations. GRAPHS OF EQUATIONS Sketch graphs of equations b hand. Find the - and -intercepts of graphs of equations. Write the standard forms of equations of

### Solving Systems of Linear Equations by Graphing

. Solving Sstems of Linear Equations b Graphing How can ou solve a sstem of linear equations? ACTIVITY: Writing a Sstem of Linear Equations Work with a partner. Your famil starts a bed-and-breakfast. It

### 3.1 Quadratic Functions

33337_030.qp 252 2/27/06 Chapter 3 :20 PM Page 252 Polnomial and Rational Functions 3. Quadratic Functions The Graph of a Quadratic Function In this and the net section, ou will stud the graphs of polnomial

### 5.1. A Formula for Slope. Investigation: Points and Slope CONDENSED

CONDENSED L E S S O N 5.1 A Formula for Slope In this lesson ou will learn how to calculate the slope of a line given two points on the line determine whether a point lies on the same line as two given

### MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60

MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets

### EQUATIONS OF LINES IN SLOPE- INTERCEPT AND STANDARD FORM

. Equations of Lines in Slope-Intercept and Standard Form ( ) 8 In this Slope-Intercept Form Standard Form section Using Slope-Intercept Form for Graphing Writing the Equation for a Line Applications (0,

### Review of Essential Skills and Knowledge

Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope

### What are the x- and y-intercepts?

.: Graphing Linear Functions STARTER. What does it mean to INTERCEPT a pass in football? The path of the defender crosses the path of the thrown football. In algebra, what are - and -intercepts? What are

### Slope-Intercept Form and Point-Slope Form

Slope-Intercept Form and Point-Slope Form In this section we will be discussing Slope-Intercept Form and the Point-Slope Form of a line. We will also discuss how to graph using the Slope-Intercept Form.

### Algebra I Semester 1 Practice Exam

. Find the product: 4 8 0 5 7 6 6 5. Which bo-and-whisker plot below represents the following set of data: {0, 4,, 8,, 4, 6, 40, 45, 46}? 4 6 0 40 56 48 48 40 4 0 40 56 6 6 5 4 6 0 5 7 6 6 5 0 0 0 0 40

### Rational Functions. 7.1 A Rational Existence. 7.2 A Rational Shift in Behavior. 7.3 A Rational Approach. 7.4 There s a Hole In My Function, Dear Liza

Rational Functions 7 The ozone laer protects Earth from harmful ultraviolet radiation. Each ear, this laer thins dramaticall over the poles, creating ozone holes which have stretched as far as Australia

### Words Algebra Graph. m 5 y 2 2 y 1. slope. Find slope in real life

. Find Slope and Rate of Change Before You graphed linear functions. Now You will find slopes of lines and rates of change. Wh? So ou can model growth rates, as in E. 6. Ke Vocabular slope parallel perpendicular

### BIG IDEAS MATH. Ron Larson Laurie Boswell. A Common Core Curriculum

BIG IDEAS MATH Ron Larson Laurie Boswell A Common Core Curriculum Introducing A New Common Core High School Series b Ron Larson and Laurie Boswell Big Ideas Math is pleased to introduce a new high school

### COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS

G COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS RECTANGULAR COORDINATE SYSTEMS Just as points on a coordinate line can be associated with real numbers, so points in a plane can be associated with pairs

### Identify a pattern and find the next three numbers in the pattern. 5. 5(2s 2 1) 2 3(s 1 2); s 5 4

Chapter 1 Test Do ou know HOW? Identif a pattern and find the net three numbers in the pattern. 1. 5, 1, 3, 7, c. 6, 3, 16, 8, c Each term is more than the previous Each term is half of the previous term;

### Transformations of Function Graphs

- - - 0 - - - - - - - Locker LESSON.3 Transformations of Function Graphs Teas Math Standards The student is epected to: A..C Analze the effect on the graphs of f () = when f () is replaced b af (), f (b),

### 2.7 Applications of Derivatives to Business

80 CHAPTER 2 Applications of the Derivative 2.7 Applications of Derivatives to Business and Economics Cost = C() In recent ears, economic decision making has become more and more mathematicall oriented.

### Linear Equations in Two Variables

Section. Sets of Numbers and Interval Notation 0 Linear Equations in Two Variables. The Rectangular Coordinate Sstem and Midpoint Formula. Linear Equations in Two Variables. Slope of a Line. Equations

### SECTION 2-2 Straight Lines

- Straight Lines 11 94. Engineering. The cross section of a rivet has a top that is an arc of a circle (see the figure). If the ends of the arc are 1 millimeters apart and the top is 4 millimeters above

### Exponential and Logarithmic Functions

Chapter 3 Eponential and Logarithmic Functions Section 3.1 Eponential Functions and Their Graphs Objective: In this lesson ou learned how to recognize, evaluate, and graph eponential functions. Course

### D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D0 APPENDIX D Precalculus Review APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b

### THE POINT-SLOPE FORM

. The Point-Slope Form (-) 67. THE POINT-SLOPE FORM In this section In Section. we wrote the equation of a line given its slope and -intercept. In this section ou will learn to write the equation of a

### Florida Algebra I EOC Online Practice Test

Florida Algebra I EOC Online Practice Test 1 Directions: This practice test contains 65 multiple-choice questions. Choose the best answer for each question. Detailed answer eplanations appear at the end

### P1. Plot the following points on the real. P2. Determine which of the following are solutions

Section 1.5 Rectangular Coordinates and Graphs of Equations 9 PART II: LINEAR EQUATIONS AND INEQUALITIES IN TWO VARIABLES 1.5 Rectangular Coordinates and Graphs of Equations OBJECTIVES 1 Plot Points in

### COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS

a p p e n d i f COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS RECTANGULAR COORDINATE SYSTEMS Just as points on a coordinate line can be associated with real numbers, so points in a plane can be associated

### Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form

SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving

### Name Class Date. Additional Vocabulary Support

- Additional Vocabular Support Rate of Change and Slope Concept List negative slope positive slope rate of change rise run slope slope formula slope of horizontal line slope of vertical line Choose the

1.3 LINEAR EQUATIONS IN TWO VARIABLES Copyright Cengage Learning. All rights reserved. What You Should Learn Use slope to graph linear equations in two variables. Find the slope of a line given two points

### 5. Equations of Lines: slope intercept & point slope

5. Equations of Lines: slope intercept & point slope Slope of the line m rise run Slope-Intercept Form m + b m is slope; b is -intercept Point-Slope Form m( + or m( Slope of parallel lines m m (slopes

### Algebra 2 Honors: Quadratic Functions. Student Focus

Resources: SpringBoard- Algebra Online Resources: Algebra Springboard Tet Algebra Honors: Quadratic Functions Semester 1, Unit : Activit 10 Unit Overview In this unit, students write the equations of quadratic

### Q (x 1, y 1 ) m = y 1 y 0

. Linear Functions We now begin the stud of families of functions. Our first famil, linear functions, are old friends as we shall soon see. Recall from Geometr that two distinct points in the plane determine

### ANSWERS The width of the box is u inches. The length of the box is u inches. 2. What is the problem asking you to determine?

page page - ELL Support Relations and Functions - Think About a Plan Relations and Functions Complete the vocabular chart b filling in the missing information. Word or Word Phrase relation Definition A

### Math 152, Intermediate Algebra Practice Problems #1

Math 152, Intermediate Algebra Practice Problems 1 Instructions: These problems are intended to give ou practice with the tpes Joseph Krause and level of problems that I epect ou to be able to do. Work

### Rational Exponents and Radical Functions

Rational Eponents and Radical Functions.1 nth Roots and Rational Eponents. Properties of Rational Eponents and Radicals. Graphing Radical Functions. Solving Radical Equations and Inequalities. Performing

### How can you construct and interpret a scatter plot? ACTIVITY: Constructing a Scatter Plot

9. Scatter Plots How can ou construct and interpret a scatter plot? ACTIVITY: Constructing a Scatter Plot Work with a partner. The weights (in ounces) and circumferences C (in inches) of several sports

### Linear Equations and Arithmetic Sequences

CONDENSED LESSON 3.1 Linear Equations and Arithmetic Sequences In this lesson ou will write eplicit formulas for arithmetic sequences write linear equations in intercept form You learned about recursive

### 6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions:

Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(-, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions - 4, use the graph at the right.. Eplain wh the graph

### Essential Question: What are two ways to solve an absolute value inequality? A2.6.F Solve absolute value linear inequalities.

Locker LESSON.3 Solving Absolute Value Inequalities Name Class Date.3 Solving Absolute Value Inequalities Teas Math Standards The student is epected to: A.6.F Essential Question: What are two was to solve

### Why should we learn this? One real-world connection is to find the rate of change in an airplane s altitude. The Slope of a Line VOCABULARY

Wh should we learn this? The Slope of a Line Objectives: To find slope of a line given two points, and to graph a line using the slope and the -intercept. One real-world connection is to find the rate

### Coordinate Geometry. Positive gradients: Negative gradients:

8 Coordinate Geometr Negative gradients: m < 0 Positive gradients: m > 0 Chapter Contents 8:0 The distance between two points 8:0 The midpoint of an interval 8:0 The gradient of a line 8:0 Graphing straight

### Reteaching Masters. To jump to a location in this book. 1. Click a bookmark on the left. To print a part of the book. 1. Click the Print button.

Reteaching Masters To jump to a location in this book. Click a bookmark on the left. To print a part of the book. Click the Print button.. When the Print window opens, tpe in a range of pages to print.

### The Rectangular Coordinate System

3.2 The Rectangular Coordinate Sstem 3.2 OBJECTIVES 1. Graph a set of ordered pairs 2. Identif plotted points 3. Scale the aes NOTE In the eighteenth centur, René Descartes, a French philosopher and mathematician,

### D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its

### Lesson 6: Linear Functions and their Slope

Lesson 6: Linear Functions and their Slope A linear function is represented b a line when graph, and represented in an where the variables have no whole number eponent higher than. Forms of a Linear Equation

### How can you write an equation of a line when you are given the slope and the y-intercept of the line? ACTIVITY: Writing Equations of Lines

. Writing Equations in Slope-Intercept Form How can ou write an equation of a line when ou are given the slope and the -intercept of the line? ACTIVITY: Writing Equations of Lines Work with a partner.

### The Point-Slope Form

7. The Point-Slope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope

### 1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model

. Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses

### C1: Coordinate geometry of straight lines

B_Chap0_08-05.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the

### 2.1 Equations of Lines

Section 2.1 Equations of Lines 1 2.1 Equations of Lines The Slope-Intercept Form Recall the formula for the slope of a line. Let s assume that the dependent variable is and the independent variable is

### Math 40 Chapter 3 Lecture Notes. Professor Miguel Ornelas

Math 0 Chapter Lecture Notes Professor Miguel Ornelas M. Ornelas Math 0 Lecture Notes Section. Section. The Rectangular Coordinate Sstem Plot each ordered pair on a Rectangular Coordinate Sstem and name

### INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1

Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.

### Use order of operations to simplify. Show all steps in the space provided below each problem. INTEGER OPERATIONS

ORDER OF OPERATIONS In the following order: 1) Work inside the grouping smbols such as parenthesis and brackets. ) Evaluate the powers. 3) Do the multiplication and/or division in order from left to right.

### Essential Question How can you use completing the square to solve a quadratic equation?

9.4 Solving Quadratic Equations Completing the Square Essential Question How can ou use completing the square to solve a quadratic equation? Work with a partner. a. Write the equation modeled the algera

### Alex and Morgan were asked to graph the equation y = 2x + 1

Which is better? Ale and Morgan were asked to graph the equation = 2 + 1 Ale s make a table of values wa Morgan s use the slope and -intercept wa First, I made a table. I chose some -values, then plugged

### 2 Solving Systems of. Equations and Inequalities

Solving Sstems of Equations and Inequalities. Solving Linear Sstems Using Substitution. Solving Linear Sstems Using Elimination.3 Solving Linear Sstems Using Technolog.4 Solving Sstems of Linear Inequalities

### SLOPES AND EQUATIONS OF LINES CHAPTER

CHAPTER 90 8 CHAPTER TABLE OF CONTENTS 8- The Slope of a Line 8- The Equation of a Line 8-3 Midpoint of a Line Segment 8-4 The Slopes of Perpendicular Lines 8-5 Coordinate Proof 8-6 Concurrence of the

### Answers for the lesson Write Linear Equations in Slope-Intercept Form

LESSON 4.1 Answers for the lesson Write Linear Equations in Slope-Intercept Form Skill Practice 1. slope. You can substitute the slope for m and the y-intercept for b to get the equation of the line..

### Graphing Linear Equations

6.3 Graphing Linear Equations 6.3 OBJECTIVES 1. Graph a linear equation b plotting points 2. Graph a linear equation b the intercept method 3. Graph a linear equation b solving the equation for We are

### 2-5. The Graph of y = kx 2. Vocabulary. Rates of Change. Lesson. Mental Math

Chapter 2 Lesson 2-5 The Graph of = k 2 BIG IDEA The graph of the set of points (, ) satisfing = k 2, with k constant, is a parabola with verte at the origin and containing the point (1, k). Vocabular

### Solving Systems Using Tables and Graphs

- Think About a Plan Solving Sstems Using Tables and Graphs Sports You can choose between two tennis courts at two universit campuses to learn how to pla tennis. One campus charges \$ per hour. The other

### Chapter 3 & 8.1-8.3. Determine whether the pair of equations represents parallel lines. Work must be shown. 2) 3x - 4y = 10 16x + 8y = 10

Chapter 3 & 8.1-8.3 These are meant for practice. The actual test is different. Determine whether the pair of equations represents parallel lines. 1) 9 + 3 = 12 27 + 9 = 39 1) Determine whether the pair

### 17.1 Connecting Intercepts and Zeros

Locker LESSON 7. Connecting Intercepts and Zeros Teas Math Standards The student is epected to: A.7.A Graph quadratic functions on the coordinate plane and use the graph to identif ke attributes, if possible,

### NAME DATE PERIOD. 11. Is the relation (year, percent of women) a function? Explain. Yes; each year is

- NAME DATE PERID Functions Determine whether each relation is a function. Eplain.. {(, ), (0, 9), (, 0), (7, 0)} Yes; each value is paired with onl one value.. {(, ), (, ), (, ), (, ), (, )}. No; in the

### Polynomial and Rational Functions

Chapter Section.1 Quadratic Functions Polnomial and Rational Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Course Number Instructor Date Important

### Algebra 2 Unit 1 Practice

Algebra Unit Practice Lesson - Use this information for Items. Aaron has \$ to rent a bike in the cit. It costs \$ per hour to rent a bike. The additional fee for a helmet is \$ for the entire ride.. Write

### 3.1 Graphically Solving Systems of Two Equations

3.1 Graphicall Solving Sstems of Two Equations (Page 1 of 24) 3.1 Graphicall Solving Sstems of Two Equations Definitions The plot of all points that satisf an equation forms the graph of the equation.

### 4-1. Quadratic Functions and Transformations. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary

4-1 Quadratic Functions and Transformations Vocabular Review 1. Circle the verte of each absolute value graph. Vocabular Builder parabola (noun) puh RAB uh luh Related Words: verte, ais of smmetr, quadratic

### Systems of Equations. from Campus to Careers Fashion Designer

Sstems of Equations from Campus to Careers Fashion Designer Radius Images/Alam. Solving Sstems of Equations b Graphing. Solving Sstems of Equations Algebraicall. Problem Solving Using Sstems of Two Equations.

### 5.3 Graphing Cubic Functions

Name Class Date 5.3 Graphing Cubic Functions Essential Question: How are the graphs of f () = a ( - h) 3 + k and f () = ( 1_ related to the graph of f () = 3? b ( - h) 3 ) + k Resource Locker Eplore 1

### Zeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system.

_.qd /7/ 9:6 AM Page 69 Section. Zeros of Polnomial Functions 69. Zeros of Polnomial Functions What ou should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polnomial

### Summer Math Exercises. For students who are entering. Pre-Calculus

Summer Math Eercises For students who are entering Pre-Calculus It has been discovered that idle students lose learning over the summer months. To help you succeed net fall and perhaps to help you learn

### Course 2 Answer Key. 1.1 Rational & Irrational Numbers. Defining Real Numbers Student Logbook. The Square Root Function Student Logbook

Course Answer Ke. Rational & Irrational Numbers Defining Real Numbers. integers; 0. terminates; repeats 3. two; number 4. ratio; integers 5. terminating; repeating 6. rational; irrational 7. real 8. root

### Section 1.1 Linear Equations: Slope and Equations of Lines

Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of