Chapter 3 Straight Lines and Linear Functions Math 1483

Transcription

1 Chapter 3 Straight Lines and Linear Functions Math 1483 In this chapter we are going to look at slope, rates of change, linear equations, linear data, and linear regression. Section 3.1: The Geometry of Lines The slope, or rate of change, of a line describes the steepness of the line. Ex. 1: A line has vertical intercept 4 and slope 2. What is its horizontal intercept? Ex. 2: A line has horizontal intercept 6 and slope 2. What is its vertical intercept? Ex. 3: A ladder leans against a wall so that its slope is 1.40 ft/ft. The top of the ladder is 7 vertical feet above the ground. What is the approximate horizontal distance from the base of the ladder to the wall? Assume that the positive direction points from the base of the ladder toward the wall.

2 Ex. 4: I am at the center of a circus tent, where the height is h = 27 feet. I am facing due west, which I take to be the positive direction. The slope of the tent line is 0.7 ft/ft. Assume that the roof of the circus tent extends in a straight line to the ground. How far from the center of the tent does the roof meet the ground? (Round your answer to one decimal place). Ex. 5: Trusses as shown in the figure below are to be constructed to support the roof of a building. The truss is to have a 16-foot horizontal base (joist) that spans from one wall to the opposite wall. The vertical center strut is 4 feet long. a. Vertical struts are located 3 horizontal feet inside each wall. How long are these vertical struts? b. The rafter extends 1.5 horizontal feet outside the wall. If the top of the wall is 8 feet above the floor, how high above the floor is the outside tip of the rafter?

3 Ex. 6: Twenty horizontal feet north of a hb = 50 foot building is a hw = 25 foot wall. A man 6 feet tall wishes to view the top of the building from the north side of the wall. How far north of the wall must he stand in order to view the top of the building? Round your answer to 2 decimal places. Ex. 7: Use the fact that for points (a1, b1) and (a2, b2) in the coordinate plane, we can calculate the slope of the line through these points using the following formula: y b2 b1 Slope x a a 2 1 Find the slope of the line through the points ( 4.4, 2.5) and (2.7, 1.1).

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5 Groupwork 3.1 Math 1483 Group Name: Group Members: You are at the center of a circus tent, where the height is h = 26 feet. You are facing due west, which you take to be the positive direction. The slope of the tent is 0.7 ft/ft. If you walk x = 5 feet west, how high is the tent? (Round your answer to one decimal place.)

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7 Section 3.2: Linear Functions Linear Functions have a constant rate of change. All linear functions can be described using their rate of change (slope) and their initial value (y-intercept). *** To help write your ordered pairs correctly, look for the following statement: is a function of. The first is the function (y-value). The second is the variable (x-value) Ex. 1: Suppose f is a linear function such that f(3) = 3 and f(6) = 18. a. What is the slope m of f? b. Find the equation for f. Ex. 2: The temperature T in degrees Fahrenheit is a linear function of the number C of cricket chirps per minute. 20 cricket chirps per minute corresponds to a temperature of 42 degrees Fahrenheit. Each additional chirp per minute corresponds to an increase of 0.25 degrees. a. Use a formula to express T as a linear function of C. b. What temperature is indicated by 120 cricket chirps per minute?

8 Ex. 3: Water freezes at 0 degrees Celsius, which is the same as 32 degrees Fahrenheit. Also water boils at 100 degrees Celsius, which is the same as 212 degrees Fahrenheit. a. Use the freezing and boiling points of water to find a formula expressing Celsius temperature C as a linear function of the Fahrenheit temperature F. b. What is the slope of the function? Explain its meaning in practical terms. Ex. 4: A study of average driver speed on rural highways found a linear relationship between average speed S, in miles per hour, and the amount of curvature D, in degrees, of the road. On a straight road (D = 0), the average speed was found to be mph. This was found to decrease by mph for each additional degree of curvature. a. Find a linear formula relating speed S to curvature D. b. Express using functional notation the speed for a road with curvature of 13 degrees. c. Calculate that value. (Round your answer to two decimal places).

9 Groupwork 3.2 Math 1483 Group Name: Group Members: A pilot reports that the temperature at 8 thousand feet is 30 o F and temperature is 20 o F at 18 thousand feet. Assume temperature T is a linear function of altitude A. a. What is the slope? Explain its meaning in practical terms. b. Write a formula for the temperature T as a linear function of the altitude A. c. How high is the aircraft if the temperature is 0 o F?

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11 Section 3.3: Modeling with Linear Functions Linear Functions have a constant rate of change. Ex. 1: Is the data in each table linear? a. t = time (in years) N = population b. t = time (in years) N = population Remember that the rate of change in a linear function is the slope of the line, ARC = m Also, the initial value (y-intercept) for a linear function is the value when the variable = 0, (0, b). y = mx + b Plotting Data Points on the Graphing Calculator FIRST TIME ONLY!! Press 2 nd Y= (stat plot) and 1 (plot 1) Make sure the curser is over ON and press ENTER. Move the curser over the picture of the scatter plot and press ENTER. Press STAT and 1 (edit). Enter the x data into L1 and the y data in L2 If there is already data in the lists, clear them by moving the curser onto the L1 or L2, press CLEAR, and then move the curser back down. Press ZOOM then 9 (zoom stat). This sets your window for you and creates your plot. To turn a plot on and off quickly: Press Y= At the top of the screen is Plot 1, Plot 2, Plot 3. If a plot is highlighted it means it is turned on, if it is not highlighted it means the plot is turned off. To either turn a plot on or off simply move the curser over plot 1 and press ENTER.

12 Ex. 2: As a diver descends into the ocean, pressure increases linearly with depth, as given in the table below. d = depth (feet) P = pressure (pounds per square inch) a. Show that this data can be modeled by a linear function. b. If P is the pressure and d is the depth below the surface, write an equation that expresses P as a function of d. c. Plot the data points and add the graph of the linear formula you found in part (b). d. How deep can a scuba diver go if the safe pressure for his equipment and experience is 40 pounds per square inch?

13 Ex. 3: Coronary risk increases with the cholesterol level, according to the following table. l = Cholesterol Level (mg/dl) R = Coronary Risk (%) a. Show that this data can be modeled by a linear function and find a formula for R as a function of l. b. What is the slope for the linear function? Explain its meaning in practical terms. c. Plot the data points and add the graph of the linear formula you found in part (a). d. What is the coronary risk for a cholesterol level of 260?

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15 Groupwork 3.3 Math 148 Group Name: Group Members: Merck & Co., Inc., is the world s largest pharmaceutical company. Net income values for Merck & Co., Inc. are given below. Year Net income, in billions of dollars $1.2 $1.5 $1.8 $2.1 $2.4 a. Show that the data in the table can be modeled by a linear function. b. Find an equation that represents the net income N as a linear function of the time t since c. Graph the data in the table and add the linear function found in part (b) to the graph. d. Based on the data in the table, what would you expect Merck s net income to be in the year 1995?

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17 Section 3.4: Linear Regression Ex. 1: Determine whether the data below are linear. Then plot the data on your calculator. t = time (in years) N = population Finding a Linear Regression Equation Using a Graphing Calculator Enter the data into L1 and L2 (section 3.3) Plot the data (section 3.3) Press STAT, arrow over to CALC, then press 4 (LinReg(ax+b)). Press ENTER. To graph the linear regression equation press Y= and enter the equation. Then press GRAPH.

18 Ex. 2: The following table shows the length (in meters) of the winning long jump in the Olympic Games or the indicated year. Year Length a. Find the equation of the regression line that gives the length as a function of time. Let t be the number of years since 1900 and L the length of the winning long jump, in meters. Round the regression parameters to three decimal places). b. Explain in practical terms the meaning of the slope of the regression line. c. Plot the data points and the regression line. d. Would you expect the regression line formula to be a good model of the winning length over a long period of time? e. There were no Olympic Games in 1916 because of WWI, but the winning long jump in the 1920 Olympic Games was 7.15 meters. Compare this with the value that the regression line model gives. Is the result consistent with your answer to part (d)?

19 Ex. 3: A scientist collected the following data on the speed, in centimeters, at which ants ran at the given ambient temperature, in degrees Celsius. Temperature Speed a. Find the equation of the regression line, giving speed S as a function of the temperature t. (Round the regression line parameters to four decimal places) b. Explain in practical terms the meaning of the slope of the regression line. c. Express, using functional notation, the speed at which the ants run when the ambient temperature is 27 degrees Celsius. Then estimate that value. (Round your answer to two decimal places) d. The scientist observes the ants running at a speed of 2.42 centimeters per second. What is the ambient temperature? (Round your answer to two decimal places)

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21 Groupwork 3.4 Math 1483 Group Name: Group Members: The table shows the U.S. Bureau of Census population data for St. Louis, Missouri. y = year P = population 856, , , , ,215 a. Plot the data points. b. Find the equation of the regression line and add its graph to your data plot. c. Explain in practical terms the meaning of the slope of the regression line. d. Estimate the population of St. Louis in the year e. How many years will it take until the population of St. Louis drops to 200,000?

22 Section 3.5: Systems of Equations A system of linear equations is a set of two or more linear equations. The solution to a system of two linear equations Algebraically, it s the point (x, y) that satisfies both equations. Graphically, it s the point (x, y) where the lines intersect. Substitution Method o Solve one of the equations for one of the variables. o Substitute this into the other equation. o Solve for the remaining variable. o Substitute again to solve for the other variable. Ex. 1: 2x 3y 3 x y 4 Elimination Method o o o o Multiply one or both equations by a number so that one of the variables will be eliminated when the equations are added together. Add the equations together. Solve for the remaining variable. Substitute to solve for the other variable. Ex. 2: 2x 3y 3 x y 4

23 Crossing Graphs Method o Solve each equation for y. o Type each equation into the Y= screen on your calculator. o Find a window that shows the point of intersection. o Use CALC 5 (intersect) to find the point of intersection. Ex. 3: 2x 3y 3 x y 4 Write a system of equations and then solve using one of the above methods. Ex. 4: A bag contains 44 coins, consisting of nickels and dimes. The total value is $3.50. How many nickels and how many dimes are in the bag? Ex. 5: Two competing rental car companies charge different rates. ABC Rentals charges $29 per day and 6 cents for each mile driven. XYZ Rental charges $35 per day and 5 cents for each mile driven. If you rent the car for only one day, how many miles must be driven for the costs to be the same?

24 Ex. 6: One evening 1500 concert tickets were sold for the Fairmont Summer Jazz Festival. Tickets cost $25 for covered pavilion seats and $15 for lawn seats. Total receipts were $28,500. How many of each type of ticket were sold?

25 Groupwork 3.5 Math 1483 Group Name: Group Members: 1. Solve the system of equations: 4x 3y 6 5x 3y 3 2. You have $90 to spend on refreshments for a party. Large bags of chips cost $5.00 and sodas cost $0.50. You need to buy exactly five times as many sodas as bags of chips. How many bags of chips and how many sodas can you buy?

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27 Exam 3 Review Section 3.1 A resort has a ski mountain that is at an elevation of 6000 feet at the bottom of the ski run. The mountain has an elevation of 8500 feet when a skier is 2000 horizontal feet from the bottom of the mountain. The top of the mountain is at an elevation of 12,900 feet. 12,900 ft 8500 ft 6000 ft 2000 ft 1. Assuming the side of the mountain slopes upward in a straight line, what is the slope of the mountain? 2. How far, in horizontal distance, is the top of the mountain from the bottom of the ski run? 3. At a distance of 3500 horizontal feet from the bottom of the ski run, what is the elevation of the mountain? Section 3.2 A spring will stretch when a weight is attached to it. The length L of the spring is a linear function of the amount of weight W attached to it. The length of the spring is 4 cm when a weight of 100 grams is attached and it is 6 cm when a weight of 200 grams is attached to it. 4. What is the slope? Explain its meaning in practical terms. 5. What is the initial value? Explain its meaning in practical terms. 6. Write the equation of the length L as a linear function of the weight W. 7. If the length of the spring is 5.6 cm, how much weight is attached to the spring?

28 Section 3.3 Your maximum heart rate during aerobic exercise decreases as you age, as shown in the table. t = age in years H = heart rate in beats per minute Determine if the data in the table are linear. 9. Calculate the slope for heart rate as a function of age. Explain in practical terms the meaning of the slope. 10. Find the y-intercept. Explain in practical terms the meaning of the y-intercept. 11. Write the equation for heart rate as a linear function of age. 12. What is the heart rate of a person who is 85 years old? 13. What is the age of a person with a heart rate of 203 beats per minute? Section 3.4 The average life expectancy for men of all races in the U.S. is given in the table below. This is the number of years of life expected at birth. B = year of birth (since 1900) M = Male life expectancy (years) Plot the data points. 15. Write the equation of the regression line. 16. Graph the regression line and add it to your plot in # Explain the meaning of the slope in practical terms. 18. Explain the meaning of the y-intercept in practical terms. 19. Use the regression model to predict the life expectancy of a man born in the year In what year would a man be born if he had a life expectancy of 75 years at birth? Section 3.5 You are planning to spend $248 for dogwood trees and redbud trees for your yard. Dogwoods cost $16 each and redbuds cost $10 each. You would like to purchase 20 trees. 21. Write a system of equations that represent this situation. Let D represent the number of dogwood trees and R represent the number of redbud trees. 22. How many trees of each type should you buy?

29 Exam 3 Review ANSWERS 1. m = = 4.25 feet/foot =, run = The top of the mountain is ft from the bottom of the ski run =, elevation is 20,875 The elevation of the mountain 3500 horizontal feet from the bottom of the ski run is 20, m = =.02 cm/g The spring stretches.02 cm for each gram of weight attached =.02(100) + b, b = 2cm The spring is 2cm long before attaching any weight. 6. L =.02W =.02W + 2, W = 180 grams If the length of the spring is 5.6 cm, then there is 180 grams attached to the spring. 8. = -1.1 = -1.1 = -1.1 = -1.1 There is a constant rate of change, so the data are linear. 9. m = -1.1 beats per minute/year. Your maximum heart rate during aerobic exercise decreases by 1.1 beats per minute every year that you live = -1.1(20) + b, b = 222 When you were born, your maximum heart rate was 222 beats per minute. 11. H = -1.1t H = -1.1(85) + 222, H = The maximum heart rate of an 85 year old would be beats per minute = -1.1t + 222, t = A person with a maximum heart rate of 203 beats per minute would be years old. 14. Use your calculator to make the plot and copy what you see x axis is B year of birth since 1900, y axis M male life expectancy in years. Stat, edit, 1, enter data, zoom 9

30 15. M = B Use the calculator to graph the line on the plot copy what you see. Y = enter equation, graph 17. The life expectancy of a male increases by year for each additional year. 18. If a male was born in 1900 the life expectancy would be years. 19. M = (82) M = A man born in 1982 would have a life expectancy of years = (B) B = A man with a life expectancy of 75 years, would have been born in D + R = 20 16D + 10R = dogwood trees 12 redbud trees