Essential Question How can you use a linear function to model and analyze a real-life situation?

Transcription

1 1.3 Modeling with Linear Functions Essential Question How can ou use a linear function to model and analze a real-life situation? Modeling with a Linear Function MODELING WITH MATHEMATICS To be proficient in math, ou need to routinel interpret our results in the context of the situation. Work with a partner. A compan purchases a copier for $1,. The spreadsheet shows how the copier depreciates over an -ear period. a. Write a linear function to represent the value V of the copier as a function of the number t of ears. b. Sketch a graph of the function. Explain wh this tpe of depreciation is called straight line depreciation. c. Interpret the slope of the graph in the context of the problem. Modeling with Linear Functions Work with a partner. Match each description of the situation with its corresponding graph. Explain our reasoning A Year, t B Value, V $1, $1,75 $9,5 $,5 $7, $5,75 $,5 $3,5 $, a. A person gives $ per week to a friend to repa a $ loan. b. An emploee receives $1.5 per hour plus $ for each unit produced per hour. c. A sales representative receives $3 per da for food plus $.55 for each mile driven. d. A computer that was purchased for $75 depreciates $1 per ear. A. B. 1 x x C. D. 1 x x Communicate Your Answer 3. How can ou use a linear function to model and analze a real-life situation?. Use the Internet or some other reference to find a real-life example of straight line depreciation. a. Use a spreadsheet to show the depreciation. b. Write a function that models the depreciation. c. Sketch a graph of the function. Section 1.3 Modeling with Linear Functions 1

2 1.3 Lesson Core Vocabular line of fit, p. line of best fit, p. 5 correlation coefficient, p. 5 Previous slope slope-intercept form point-slope form scatter plot What You Will Learn Write equations of linear functions using points and slopes. Find lines of fit and lines of best fit. Writing Linear Equations Core Concept Writing an Equation of a Line Given slope m and -intercept b Given slope m and a point (x 1, 1 ) Use slope-intercept form: = mx + b Use point-slope form: 1 = m(x x 1 ) Given points (x 1, 1 ) and (x, ) First use the slope formula to find m. Then use point-slope form with either given point. Distance (miles) Asteroid 1 DA1 1 (5, ) x Time (seconds) REMEMBER An equation of the form = mx indicates that x and are in a proportional relationship. Writing a Linear Equation from a Graph The graph shows the distance Asteroid 1 DA1 travels in x seconds. Write an equation of the line and interpret the slope. The asteroid came within 17, miles of Earth in Februar, 13. About how long does it take the asteroid to travel that distance? SOLUTION From the graph, ou can see the slope is m = =. and the -intercept is b =. 5 Use slope-intercept form to write an equation of the line. = mx + b Slope-intercept form =.x + Substitute. for m and for b. The equation is =.x. The slope indicates that the asteroid travels. miles per second. Use the equation to find how long it takes the asteroid to travel 17, miles. 17, =.x Substitute 17, for. 353 x Divide each side b.. Because there are 3 seconds in 1 hour, it takes the asteroid about 1 hour to travel 17, miles. Monitoring Progress 1. The graph shows the remaining balance on a car loan after making x monthl paments. Write an equation of the line and interpret the slope and -intercept. What is the remaining balance after 3 paments? Help in English and Spanish at BigIdeasMath.com Balance (thousands of dollars) 1 1 (1, 15) (, 1) Car Loan x Number of paments Chapter 1 Linear Functions

3 Modeling with Mathematics Lakeside Inn Number of students, x Total cost, 1 $15 15 $1 15 $1 175 $ $7 Two prom venues charge a rental fee plus a fee per student. The table shows the total costs for different numbers of students at Lakeside Inn. The total cost (in dollars) for x students at Sunview Resort is represented b the equation = 1x +. Which venue charges less per student? How man students must attend for the total costs to be the same? SOLUTION 1. Understand the Problem You are given an equation that represents the total cost at one venue and a table of values showing total costs at another venue. You need to compare the costs.. Make a Plan Write an equation that models the total cost at Lakeside Inn. Then compare the slopes to determine which venue charges less per student. Finall, equate the cost expressions and solve to determine the number of students for which the total costs are equal. 3. Solve the Problem First find the slope using an two points from the table. Use (x 1, 1 ) = (1, 15) and (x, ) = (15, 1). m = = x x = 3 5 = 1 Write an equation that represents the total cost at Lakeside Inn using the slope of 1 and a point from the table. Use (x 1, 1 ) = (1, 15). 1 = m(x x 1 ) Point-slope form 15 = 1(x 1) Substitute for m, x 1, and = 1x 1 Distributive Propert = 1x + 3 Add 15 to each side. Equate the cost expressions and solve. 1x + = 1x + 3 Set cost expressions equal. 3 = x Combine like terms. 15 = x Divide each side b. Comparing the slopes of the equations, Sunview Resort charges $1 per student, which is less than the $1 per student that Lakeside Inn charges. The total costs are the same for 15 students.. Look Back Notice that the table shows the total cost for 15 students at Lakeside Inn is $1. To check that our solution is correct, verif that the total cost at Sunview Resort is also $1 for 15 students. = 1(15) + Substitute 15 for x. = 1 Simplif. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. WHAT IF? Maple Ridge charges a rental fee plus a $1 fee per student. The total cost is $19 for 1 students. Describe the number of students that must attend for the total cost at Maple Ridge to be less than the total costs at the other two venues. Use a graph to justif our answer. Section 1.3 Modeling with Linear Functions 3

4 Finding Lines of Fit and Lines of Best Fit Data do not alwas show an exact linear relationship. When the data in a scatter plot show an approximatel linear relationship, ou can model the data with a line of fit. Core Concept Finding a Line of Fit Step 1 Create a scatter plot of the data. Step Sketch the line that most closel appears to follow the trend given b the data points. There should be about as man points above the line as below it. Step 3 Choose two points on the line and estimate the coordinates of each point. These points do not have to be original data points. Step Write an equation of the line that passes through the two points from Step 3. This equation is a model for the data. Finding a Line of Fit Femur length, x Height, The table shows the femur lengths (in centimeters) and heights (in centimeters) of several people. Do the data show a linear relationship? If so, write an equation of a line of fit and use it to estimate the height of a person whose femur is 35 centimeters long. SOLUTION Step 1 Create a scatter plot of the data. The data show a linear relationship. Step Sketch the line that most closel appears to fit the data. One possibilit is shown. Step 3 Choose two points on the line. For the line shown, ou might choose (, 17) and (5, 195). Step Write an equation of the line. First, find the slope. m = = x x 1 5 = 5 1 =.5 Use point-slope form to write an equation. Use (x 1, 1 ) = (, 17). 1 = m(x x 1 ) Point-slope form 17 =.5(x ) Substitute for m, x 1, and =.5x 1 =.5x + 7 Use the equation to estimate the height of the person. Height (centimeters) Distributive Propert Add 17 to each side. Human Skeleton 1 (, 17) 3 (5, 195) 5 x Femur length (centimeters) =.5(35) + 7 Substitute 35 for x. = Simplif. The approximate height of a person with a 35-centimeter femur is centimeters. Chapter 1 Linear Functions

5 The line of best fit is the line that lies as close as possible to all of the data points. Man technolog tools have a linear regression feature that ou can use to find the line of best fit for a set of data. The correlation coefficient, denoted b r, is a number from 1 to 1 that measures how well a line fits a set of data pairs (x, ). When r is near 1, the points lie close to a line with a positive slope. When r is near 1, the points lie close to a line with a negative slope. When r is near, the points do not lie close to an line. Using a Graphing Calculator humerus femur Use the linear regression feature on a graphing calculator to find an equation of the line of best fit for the data in Example 3. Estimate the height of a person whose femur is 35 centimeters long. Compare this height to our estimate in Example 3. SOLUTION Step 1 Enter the data into two lists. L1 L L1(1)= L3 Step Use the linear regression feature. The line of best fit is =.x + 5. The value of r is close to 1. LinReg =ax+b a= b=.997 r = r= Step 3 Graph the regression equation with the scatter plot. 1 Step Use the trace feature to find the value of when x = =.x + 5 ATTENDING TO PRECISION Be sure to analze the data values to help ou select an appropriate viewing window for our graph X=35 Y=15 1 The approximate height of a person with a 35-centimeter femur is 15 centimeters. This is less than the estimate found in Example 3. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 3. The table shows the humerus lengths (in centimeters) and heights (in centimeters) of several females. 55 Humerus length, x Height, a. Do the data show a linear relationship? If so, write an equation of a line of fit and use it to estimate the height of a female whose humerus is centimeters long. b. Use the linear regression feature on a graphing calculator to find an equation of the line of best fit for the data. Estimate the height of a female whose humerus is centimeters long. Compare this height to our estimate in part (a). Section 1.3 Modeling with Linear Functions 5

6 1.3 Exercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE The linear equation = 1 x + 3 is written in form.. VOCABULARY A line of best fit has a correlation coefficient of.9. What can ou conclude about the slope of the line? Monitoring Progress and Modeling with Mathematics In Exercises 3, use the graph to write an equation of the line and interpret the slope. (See Example 1.) 3. Tipping. Gasoline Tank Tip (dollars) (1, ) 1 x Cost of meal (dollars) Fuel (gallons) 9 (9, 9) 3 1 x Distance (miles) 9. MODELING WITH MATHEMATICS Two newspapers charge a fee for placing an advertisement in their paper plus a fee based on the number of lines in the advertisement. The table shows the total costs for different length advertisements at the Dail Times. The total cost (in dollars) for an advertisement that is x lines long at the Greenville Journal is represented b the equation = x +. Which newspaper charges less per line? How man lines must be in an advertisement for the total costs to be the same? (See Example.) Dail Times 5. Savings Account Balance (dollars) (, 3) 1 x Time (weeks) 7. Tping Speed Words tped (1, 55) (3, 15) x Time (minutes). Tree Growth. Tree height (feet) Volume (cubic feet) x Age (ears) Swimming Pool (3, 3) (5, 1) x Time (hours) Number of lines, x Total cost, PROBLEM SOLVING While on vacation in Canada, ou notice that temperatures are reported in degrees Celsius. You know there is a linear relationship between Fahrenheit and Celsius, but ou forget the formula. From science class, ou remember the freezing point of water is C or 3 F, and its boiling point is 1 C or 1 F. a. Write an equation that represents degrees Fahrenheit in terms of degrees Celsius. b. The temperature outside is C. What is this temperature in degrees Fahrenheit? c. Rewrite our equation in part (a) to represent degrees Celsius in terms of degrees Fahrenheit. d. The temperature of the hotel pool water is 3 F. What is this temperature in degrees Celsius? Chapter 1 Linear Functions

7 ERROR ANALYSIS In Exercises 11 and 1, describe and correct the error in interpreting the slope in the context of the situation. 11. Balance (dollars) Savings Account (, 1) 11 (, 1) x Year The slope of the line is 1, so after 7 ears, the balance is $ MODELING WITH MATHEMATICS The data pairs (x, ) represent the average annual tuition (in dollars) for public colleges in the United States x ears after 5. Use the linear regression feature on a graphing calculator to find an equation of the line of best fit. Estimate the average annual tuition in. Interpret the slope and -intercept in this situation. (See Example.) (, 11,3), (1, 11,731), (, 11,) (3, 1,375), (, 1,), (5, 13,97) 1. MODELING WITH MATHEMATICS The table shows the numbers of tickets sold for a concert when different prices are charged. Write an equation of a line of fit for the data. Does it seem reasonable to use our model to predict the number of tickets sold when the ticket price is $5? Explain. 1. Income (dollars) Earnings (3, 33) (, ) x Hours The slope is 3, so the income is $3 per hour. Ticket price (dollars), x 17 Tickets sold, USING TOOLS In Exercises 19, use the linear regression feature on a graphing calculator to find an equation of the line of best fit for the data. Find and interpret the correlation coefficient. 19. x. x In Exercises 13 1, determine whether the data show a linear relationship. If so, write an equation of a line of fit. Estimate when x = 15 and explain its meaning in the context of the situation. (See Example 3.) Minutes walking, x Calories burned, Months, x Hair length (in.), Hours, x Batter life (%), 1 5 Shoe size, x Heart rate (bpm), x x.. x x 5. OPEN-ENDED Give two real-life quantities that have (a) a positive correlation, (b) a negative correlation, and (c) approximatel no correlation. Explain. Section 1.3 Modeling with Linear Functions 7

8 . HOW DO YOU SEE IT? You secure an interest-free loan to purchase a boat. You agree to make equal monthl paments for the next two ears. The graph shows the amount of mone ou still owe. Loan balance (hundreds of dollars) 3 1 Boat Loan 1 x Time (months) a. What is the slope of the line? What does the slope represent? b. What is the domain and range of the function? What does each represent? c. How much do ou still owe after making paments for 1 months? 3. MATHEMATICAL CONNECTIONS Which equation has a graph that is a line passing through the point (, 5) and is perpendicular to the graph of = x + 1? A = 1 x 5 B = x + 7 C = 1 x 7 D = 1 x PROBLEM SOLVING You are participating in an orienteering competition. The diagram shows the position of a river that cuts through the woods. You are currentl miles east and 1 mile north of our starting point, the origin. What is the shortest distance ou must travel to reach the river? North = 3x + East 1 3 x 7. MAKING AN ARGUMENT A set of data pairs has a correlation coefficient r =.3. Your friend sas that because the correlation coefficient is positive, it is logical to use the line of best fit to make predictions. Is our friend correct? Explain our reasoning.. THOUGHT PROVOKING Points A and B lie on the line = x +. Choose coordinates for points A, B, and C where point C is the same distance from point A as it is from point B. Write equations for the lines connecting points A and C and points B and C. 9. ABSTRACT REASONING If x and have a positive correlation, and and z have a negative correlation, then what can ou conclude about the correlation between x and z? Explain. 3. ANALYZING RELATIONSHIPS Data from North American countries show a positive correlation between the number of personal computers per capita and the average life expectanc in the countr. a. Does a positive correlation make sense in this situation? Explain. b. Is it reasonable to conclude that giving residents of a countr personal computers will lengthen their lives? Explain. Maintaining Mathematical Proficienc Solve the sstem of linear equations in two variables b elimination or substitution. (Skills Review Handbook) 33. 3x + = 7 3. x + 3 = 35. x + = 3 x = 9 x 3 = 1 x = 1 1 Reviewing what ou learned in previous grades and lessons 3. = 1 + x 37. x + = 3. = x x + = x = 1 x + = Chapter 1 Linear Functions