Overview of Section 2. Introduction The goal in this lesson is for students to create quadratic equations to represent the relationship between two quantities (HSA-CED.A.2). From the contet given, students are able to write the quadratic model in one of the equation forms previousl studied (verte or intercept). The first eploration reviews how to interpret the characteristics of a quadratic function in standard form. In the second eploration, students use the regression feature on their calculators to determine a quadratic model for a set of data. The eploration assumes students are familiar with making a scatter plot and performing regression. In the formal lesson, students eplore different techniques for writing quadratic equations. Just like with linear equations, different techniques are used depending on what information is known in the problem. The eamples presented demonstrate how quadratic functions can model real-world phenomena. This speaks to the usefulness of mathematics, and students should find the eamples interesting and engaging. Dnamic Teaching Tools Dnamic Assessment & Progress Monitoring Tool Lesson Planning Tool Interactive Whiteboard Lesson Librar Dnamic Classroom with Dnamic Investigations Resources Students need to use a graphing calculator in Eploration 2 and Eample in the formal lesson. In each case, the will use regression analsis to find a best-fitting quadratic model for data. Teaching Strateg When working with real-life data, it is ver possible that no function will be a perfect fit. The strateg is to find a function that best fits the data. The first step is to do a scatter plot of the data using a good viewing window for the data. Students should learn to look at the tendenc of the data as displaed in a scatter plot before deciding what tpe of model to fit to the data. 25 15 25 2 12 When deciding how to set the window, it is wise to fill as much of the viewing window as possible. If there is a curvature, it will be more noticeable in the window that is the tightest. Pacing Suggestion Use the Motivate and the two eplorations before beginning the formal lesson. Section 2. T-7
Common Core State Standards HSA-CED.A.2 Create equations in two variables to represent relationships between quantities; graph equations on coordinate aes with labels and scales. HSF-IF.B.6 Calculate and interpret the average rate of change of a function (presented smbolicall or as a table) over a specified interval. Estimate the rate of change from a graph. HSF-BF.A.1a Determine an eplicit epression, from a contet. HSS-ID.B.6a Fit a function to the data; use functions fitted to data to solve problems in the contet of the data. Eploration Motivate The first eample in the lesson is about a human cannonball. Search the Internet using human cannonball farthest distance for a video showing the Guinness World Record for this. Show the video. Ask, What are ou curious about? This is a ver open question, and the discussion needs to be focused on those questions that are related to math! How far the person traveled is noted in the video. Other questions might relate to the maimum height or wh the cannon was positioned at that angle. Eplain to students that in this lesson the will look at how quadratic models can be used to answer man real-life questions. Discuss Students should recall the characteristics of a quadratic function written in standard form. The verte represents the -value where the function takes on the maimum or minimum value. When the parabola opens upward, reading the function left to right, the function is decreasing and then increasing. When the parabola opens downward, reading the function left to right, the function is increasing and then decreasing. In the first eploration, ou will have a good assessment of how well students can interpret the characteristics of a quadratic in a contetual setting. Eploration 1 MP2 Reason Abstractl and Quantitativel: The profit equation is the general standard form. Students must use the constants a, b, and c in answering the questions. MP Model with Mathematics and MP6 Attend to Precision: Students should note that the aes are labeled t (ears) and P (earl profit). Selective Responses: Circulate and read eplanations in part (d). Make notes so that ou ma elicit Selective Responses from students. MP2: Discuss student responses. Listen for accurate language and understanding. Eploration 2 COMMON ERROR The graph does not show the path of the wrench. Have students identif the independent and dependent variables. MP5 Use Appropriate Tools Strategicall: Partners using Turn and Talk should work together in using a graphing calculator. Pair students strategicall if students have forgotten how to make a scatter plot or perform regression on the calculator. MP2 and MP6: To get a good viewing window, students need to consider the known values in the domain and range and then set the window accordingl. Big Idea: The characteristics of a parabola can be interpreted and given meaning in a reallife application. In this eploration, one of the -intercepts has a meaning in the contet of the situation. Communicate Your Answer Question could be assigned as a homework eercise due in number of das. Connecting to Net Step Eplain that calculator regression was used in Eploration 2 to find the equation. In the formal lesson, other techniques for finding the equation will be eplored. T-75 Chapter 2
MODELING WITH MATHEMATICS 2. To be proficient in math, ou need to routinel interpret our results in the contet of the situation. Modeling with Quadratic Functions Essential Question How can ou use a quadratic function to model a real-life situation? Work with a partner. The graph shows a quadratic function of the form Modeling with a Quadratic Function P(t) = at 2 + bt + c P(t) = at 2 + bt + c which approimates the earl profits for a compan, where P(t) is the profit in ear t. a. Is the value of a positive, negative, or zero? Eplain. b. Write an epression in terms of a and b that represents the ear t when the compan t Year made the least profit. c. The compan made the same earl profits in 2 and 212. Estimate the ear in which the compan made the least profit. d. Assume that the model is still valid toda. Are the earl profits currentl increasing, decreasing, or constant? Eplain. Modeling with a Graphing Calculator Work with a partner. The table shows the heights h (in feet) of a wrench t seconds after it has been dropped from a building under construction. Time, t 1 2 3 Height, h 3 336 256 1 a. Use a graphing calculator to create a scatter plot of the data, as shown at the right. Eplain wh the data appear to fit a quadratic model. b. Use the quadratic regression feature to find a quadratic model for the data. c. Graph the quadratic function on the same screen as the scatter plot to verif that it fits the data. d. When does the wrench hit the ground? Eplain. Communicate Your Answer 3. How can ou use a quadratic function to model a real-life situation?. Use the Internet or some other reference to find eamples of real-life situations that can be modeled b quadratic functions. Yearl profit (dollars) P 5 Dnamic Teaching Tools Dnamic Assessment & Progress Monitoring Tool Lesson Planning Tool Interactive Whiteboard Lesson Librar Dnamic Classroom with Dnamic Investigations 1. a. positive; The parabola opens up. b. t = b 2a c. 2 d. increasing; The graph is increasing to the right of the ear of the minimum profit, so the profit is currentl increasing. 2. a. The height is not decreasing at a linear rate. b. h = 16t 2 + c. The function fits the scatter plot because the graph crosses through all 5 points. d. 5 sec; Substituting h = into the equation and solving for t results in t =±5. 3. A quadratic function can be used to model an real-life situation that involves a parabolic arc.. Sample answer: an object in free fall, or a person jumping off a diving board Section 2. Modeling with Quadratic Functions 75 Section 2. 75
English Language Learners Analzing Word Problems Guide students to highlight an word in a word problem that the do not understand. Pair each English language learner with an English speaking student and have the students use the contet of the problem to help them define each unknown word. Encourage pairs to draw diagrams to conve their understanding of each scenario. 2. Lesson What You Will Learn Core Vocabular Previous average rate of change sstem of three linear equations Write equations of quadratic functions using vertices, points, and -intercepts. Write quadratic equations to model data sets. Writing Quadratic Equations Core Concept Writing Quadratic Equations Given a point and the verte (h, k) Given a point and -intercepts p and q Use verte form: = a( h) 2 + k Use intercept form: = a( p)( q) Etra Eample 1 The graph shows the parabolic path of a performer who is shot out of a cannon, where is the height (in feet) and is the horizontal distance traveled (in feet). Write an equation of the parabola. The performer lands in a net 6 feet from the cannon. What is the height of the net to the nearest foot? Human Cannonball (25, 3) 3 2 (, 2) 1 2 6 Horizontal distance (feet) =.16( 25) 2 + 3, where 6; about 1 feet 3 Human Cannonball 2 (,15) 1 (5, 35) 2 6 Horizontal distance (feet) Given three points Write and solve a sstem of three equations in three variables. Writing an Equation Using a Verte and a Point The graph shows the parabolic path of a performer who is shot out of a cannon, where is the height (in feet) and is the horizontal distance traveled (in feet). Write an equation of the parabola. The performer lands in a net 9 feet from the cannon. What is the height of the net? SOLUTION From the graph, ou can see that the verte (h, k) is (5, 35) and the parabola passes through the point (, 15). Use the verte and the point to solve for a in verte form. = a( h) 2 + k Verte form 15 = a( 5) 2 + 35 Substitute for h, k,, and. 2 = 25a Simplif.. = a Divide each side b 25. Because a =., h = 5, and k = 35, the path can be modeled b the equation =.( 5) 2 + 35, where 9. Find the height when = 9. =.(9 5) 2 + 35 Substitute 9 for. =.(16) + 35 Simplif. = 22.2 Simplif. So, the height of the net is about 22 feet. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 1. WHAT IF? The verte of the parabola is (5, 37.5). What is the height of the net? 2. Write an equation of the parabola that passes through the point ( 1, 2) and has verte (, 9). MONITORING PROGRESS 1. about 23 ft 2. =.( ) 2 9 76 Chapter 2 Quadratic Functions Teacher Actions Review with students the different strategies the have used to determine the equation of a quadratic. Then write the Core Concept. Turn and Talk: Have students read the eample and view the graph. What information is known in the eample, and what question are the tring to answer? MP3 Construct Viable Arguments and Critique the Reasoning of Others: After partners have had a chance to discuss the eample, ask volunteers to share their thinking. Common Misconception: Although students recognize the verte and are comfortable with the first step of substituting for h and k in the equation, it is not alwas obvious that it is oka to substitute for an ordered pair (, ) as well. Substituting for,, h, and k means that the leading coefficient a can be solved for. Before solving for a, ask, What do ou know about a in this eample? Eplain. a must be negative because the parabola opens downward. 76 Chapter 2
Temperature ( C) Temperature Forecast (, 9.6) 1 (, ) (2, ) 3 9 15 1 REMEMBER Hours after midnight The average rate of change of a function f from 1 to 2 is the slope of the line connecting ( 1, f( 1 )) and ( 2, f( 2 )): f( 2 ) f( 1 ). 2 1 Writing an Equation Using a Point and -Intercepts A meteorologist creates a parabola to predict the temperature tomorrow, where is the number of hours after midnight and is the temperature (in degrees Celsius). a. Write a function f that models the temperature over time. What is the coldest temperature? b. What is the average rate of change in temperature over the interval in which the temperature is decreasing? increasing? Compare the average rates of change. SOLUTION a. The -intercepts are and 2 and the parabola passes through (, 9.6). Use the -intercepts and the point to solve for a in intercept form. = a( p)( q) Intercept form 9.6 = a( )( 2) Substitute for p, q,, and. 9.6 = 96a Simplif..1 = a Divide each side b 96. Because a =.1, p =, and q = 2, the temperature over time can be modeled b f() =.1( )( 2), where 2. The coldest temperature is the minimum value. So, find f() when = + 2 = 1. 2 f(1) =.1(1 )(1 2) Substitute 1 for. = 1 Simplif. So, the coldest temperature is 1 C at 1 hours after midnight, or 2 p.m. b. The parabola opens up and the ais of smmetr is = 1. So, the function is decreasing over the interval < < 1 and increasing over the interval 1 < < 2. Average rate of change Average rate of change over < < 1: over 1 < < 2: f(1) f() 1 9.6 = = 1. 1 1 (, 9.6) 1 1 (2, ) 3 15 (1, 1) f(2) f(1) = ( 1) = 1 2 1 1 Because 1. > 1, the average rate at which the temperature decreases from midnight to 2 p.m. is greater than the average rate at which it increases from 2 p.m. to midnight. Monitoring Progress Section 2. Modeling with Quadratic Functions 77 Help in English and Spanish at BigIdeasMath.com 3. WHAT IF? The -intercept is.. How does this change our answers in parts (a) and (b)?. Write an equation of the parabola that passes through the point (2, 5) and has -intercepts 2 and. Teacher Actions Refer to the graph in the eample. Where in the United States and at what time of ear would the graph make sense? Listen for temperatures that are freezing (below C) at night. This eample is similar to Eample 1, ecept the intercept form is used. MP5: An alternate approach would be to use a graphing calculator to determine the minimum value once the equation has been found. Refer to the Remember note if students have forgotten what it means to find the average rate of change. MP: The graph is helpful in showing what was confirmed numericall, namel that the average rate at which the temperature decreases from midnight to 2 P.M. is greater than the average rate at which it increases from 2 P.M. to midnight. Monitoring Progress: Use whiteboards for a quick assessment of Question. Differentiated Instruction Kinesthetic Have students look at the graphs of different quadratic functions representing real-world scenarios. Ask real-world questions about the graphs, and have students point or trace with their fingers to show the part of the graph that gives the answers. The responses should require students to show where the graph is increasing or decreasing, the maimum or minimum, the intercepts, or a point on the parabola. Etra Eample 2 The meteorologist creates a parabola to predict the temperature the da after tomorrow, where is the number of hours after midnight and is the temperature (in degrees Celsius). Temperature ( C) Temperature Forecast (, 6.3) (6, ) (21, ) 12 Hours after midnight a. Write a function f that models the temperature over time. What is the coldest temperature? f() =.5( 6)( 21) for 2; The coldest temperature is about 2. C at 1:3 P.M. b. What is the average rate of change on the interval in which the temperature is decreasing? increasing? Compare the average rates of change. See Additional Answers. MONITORING PROGRESS 3. a. equation changes to =.5( )( 2); coldest temperature is 5 C b. average rates of change are cut in half. = 5 ( + 2)( ) Section 2. 77
Etra Eample 3 A former NASA emploee designs a model airplane that flies in a parabolic path. The table shows the heights h (in feet) of a plane t seconds after starting the flight path. Write and evaluate a function to approimate the height of the model airplane after 6.5 seconds. Time, t Height, h 1 19.1 2 2. 3 19.9 17.6 5 13.5 h(t) =.9t 2 + t + 16; After 6.5 seconds, the model plane is at a height of about feet. Time, t Height, h 1 26,9 15 29,25 2 3,6 25 31,625 3 32,1 35 32,25 31, Writing Equations to Model Data When data have equall-spaced inputs, ou can analze patterns in the differences of the outputs to determine what tpe of function can be used to model the data. Linear data have constant first differences. Quadratic data have constant second differences. The first and second differences of f() = 2 are shown below. Equall-spaced -values 3 2 1 1 2 3 f() 9 1 1 9 first differences: 5 3 1 1 3 5 second differences: 2 2 2 2 2 Writing a Quadratic Equation Using Three Points NASA can create a weightless environment b fling a plane in parabolic paths. The table shows heights h (in feet) of a plane t seconds after starting the flight path. After about 2. seconds, passengers begin to eperience a weightless environment. Write and evaluate a function to approimate the height at which this occurs. SOLUTION Step 1 The input values are equall spaced. So, analze the differences in the outputs to determine what tpe of function ou can use to model the data. h(1) h(15) h(2) h(25) h(3) h(35) h() 26,9 29,25 3,6 31,625 32,1 32,25 31, 2125 1575 125 75 75 625 55 55 55 55 55 Because the second differences are constant, ou can model the data with a quadratic function. Step 2 Write a quadratic function of the form h(t) = at 2 + bt + c that models the data. Use an three points (t, h) from the table to write a sstem of equations. Use (1, 26,9): 1a + 1b + c = 26,9 Equation 1 Use (2, 3,6): a + 2b + c = 3,6 Equation 2 Use (3, 32,1): 9a + 3b + c = 32,1 Equation 3 Use the elimination method to solve the sstem. Subtract Equation 1 from Equation 2. Subtract Equation 1 from Equation 3. 3a + 1b = 37 New Equation 1 a + 2b = 52 New Equation 2 2a = 22 Subtract 2 times new Equation 1 from new Equation 2. a = 11 Solve for a. b = 7 Substitute into new Equation 1 to find b. c = 21, Substitute into Equation 1 to find c. The data can be modeled b the function h(t) = 11t 2 + 7t + 21,. Step 3 Evaluate the function when t = 2.. h(2.) = 11(2.) 2 + 7(2.) + 21, = 3,.96 Passengers begin to eperience a weightless environment at about 3, feet. 7 Chapter 2 Quadratic Functions Teacher Actions Students should be familiar with identifing a linear function from a table of values where the constant rate of change was found in the first differences. Introduce the idea that when second differences are equal, the data is quadratic. This can be quickl demonstrated using familiar data from = 2. Students ma ask wh regression is not used for this eample, which it clearl could be. The technique used here is a review of a skill from Algebra 1. There are still was in which the equation can be found without technolog. Students should recall using the elimination method in solving a sstem of two equations in two unknowns. In this eample, there are three equations and three unknowns, but the technique is the same, onl longer. Turn and Talk: Finish the eample and have partners review the solution. What part of the solution is unclear? Are there steps that do not make sense? Etension: If time permits, use three different points and repeat the process. The equation will be the same. 7 Chapter 2
Miles per hour, STUDY TIP Miles per gallon, 2 1.5 2 17.5 3 21.2 36 23.7 25.2 5 25. 5 25. 56 25.1 6 2. 7 19.5 The coefficient of determination R 2 shows how well an equation fits a set of data. The closer R 2 is to 1, the better the fit. Real-life data that show a quadratic relationship usuall do not have constant second differences because the data are not eactl quadratic. Relationships that are approimatel quadratic have second differences that are relativel close in value. Man technolog tools have a quadratic regression feature that ou can use to find a quadratic function that best models a set of data. Using Quadratic Regression The table shows fuel efficiencies of a vehicle at different speeds. Write a function that models the data. Use the model to approimate the optimal driving speed. SOLUTION Because the -values are not equall spaced, ou cannot analze the differences in the outputs. Use a graphing calculator to find a function that models the data. Step 1 Enter the data in a graphing calculator using two lists and create a scatter plot. The data show a quadratic relationship. 35 Step 3 Graph the regression equation with the scatter plot. In this contet, the optimal driving speed is the speed at which the mileage per gallon is maimized. Using the maimum feature, ou can see that the maimum mileage per gallon is about 26. miles per gallon when driving about.9 miles per hour. So, the optimal driving speed is about 9 miles per hour. Monitoring Progress 75 Step 2 Use the quadratic regression feature. A quadratic model that represents the data is =.1 2 + 1.37 7.1. QuadReg =a 2 +b+c a=-.19739 b=1.3662167 c=-7.15213 R 2 =.9992752 Help in English and Spanish at BigIdeasMath.com 5. Write an equation of the parabola that passes through the points ( 1, ), (, 1), and (2, 7). 6. The table shows the estimated profits (in dollars) for a concert when the charge is dollars per ticket. Write and evaluate a function to determine what the charge per ticket should be to maimize the profit. Ticket price, 2 5 11 1 17 Profit, 26 65 6 9 7 1 Maimum X=.92565 Y=26.1671 75 7. The table shows the results of an eperiment testing the maimum weights (in tons) supported b ice inches thick. Write a function that models the data. How much weight can be supported b ice that is 22 inches thick? Ice thickness, 12 1 15 1 2 2 27 Maimum weight, 3. 7.6 1. 1.3 25..6 5.3 35 Etra Eample The table shows fuel efficiencies of a vehicle at different speeds. Write a function that models the data. Use the model to approimate the optimal driving speed. Miles per hour, Miles per gallon, 23 17.1 3 23. 2 27.5 7 2.6 5 29.6 61 26.2 72 22. =.15 2 + 1.55 1.9; about 52 miles per hour MONITORING PROGRESS 5. = 2 2 + 1 6. = 1 2 + 2 1; $1 7. =.1 2.5 5; about 32. tons Section 2. Modeling with Quadratic Functions 79 Teacher Actions Discuss with students that real-life data can be mess; it ma not fit a quadratic model eactl. Using quadratic regression ou can find a function that best models the set of data. If Eploration 2 was completed, students should be familiar with the calculator techniques. Viewing the scatter plot and observing the behavior of the data is an important first step. There should be visual evidence that the data could be modeled b a quadratic. Some graphing calculators will displa the coefficient of determination R 2 when the regression equation is generated. The Stud Tip eplains that the closer R 2 is to 1, the better the fit. Monitoring Progress: In Question 7, note that both linear and quadratic models are good fits. Point this out to students, but show how the linear model gives a higher value when is 22. Discuss the real-life implications of this difference. Closure Is it possible to write the equation of a quadratic function given two ordered pairs? Eplain. es; It is possible if one of the ordered pairs is the verte of the parabola. Section 2. 79
Assignment Guide and Homework Check ASSIGNMENT Basic: 1 2, 3 13 odd, 17 21 odd, 2, 36, 3 1 Average: 1 2, 1 even, 1, 2, 21, 2 2 even, 36, 3 1 Advanced: 1 2, 2 even, 21, 22 36 even, 3 1 HOMEWORK CHECK Basic: 5, 13, 17, 19, 21 Average: 6, 1, 1, 2, 2 Advanced:, 1, 1, 2, 26 1. A quadratic model is appropriate when the second differences are constant. 2. What is the distance from f () to f (2)? units, or about 2. units; 1 3. = 3( + 2) 2 + 6. =.25( ) 2 1 5. =.6( 3) 2 + 2 6. = 6( + 5) 2 + 9 7. = 1 3 ( + 6)2 12. = 3 7 ( + 1)2 + 1 9. = ( 2)( ) 1. = ( + 1)( 2) 11. = 1 ( 12)( + 6) 1 12. = 2( 9)( 1) 13. = 2.25( + 16)( + 2) 1. =.1( + 7)( + 3) 15. If given the -intercepts, it is easier to write the equation in intercept form. If given the verte, it is easier to write the equation in verte form. 16. A and C 17. = 16( 3) 2 + 15 1. = 16 2 + 1 2. Eercises Vocabular and Core Concept Check Chapter 2 Quadratic Functions Dnamic Solutions available at BigIdeasMath.com Monitoring Progress and Modeling with Mathematics In Eercises 3, write an equation of the parabola in verte form. (See Eample 1.) 3. ( 2, 6) ( 1, 3). (, 3) (, 1) 5. passes through (13, ) and has verte (3, 2) 6. passes through ( 7, 15) and has verte ( 5, 9) 7. passes through (, 2) and has verte ( 6, 12). passes through (6, 35) and has verte ( 1, 1) In Eercises 9 1, write an equation of the parabola in intercept form. (See Eample 2.) 9. 1. WRITING Eplain when it is appropriate to use a quadratic model for a set of data. 2. DIFFERENT WORDS, SAME QUESTION Which is different? Find both answers. What is the average rate of change over 2? What is the slope of the line segment? (3, ) (, ) (2, ) 1. 2 ( 1, ) (2, ) (1, 2) 11. -intercepts of 12 and 6; passes through (1, ) 12. -intercepts of 9 and 1; passes through (, 1) What is the distance from f() to f(2)? f(2) f() What is? 2 13. -intercepts of 16 and 2; passes through ( 1, 72) 1. -intercepts of 7 and 3; passes through ( 2,.5) 1 15. WRITING Eplain when to use intercept form and when to use verte form when writing an equation of a parabola. 16. ANALYZING EQUATIONS Which of the following equations represent the parabola? 2 ( 1, ) A = 2( 2)( + 1) B = 2( +.5) 2.5 C = 2(.5) 2.5 D = 2( + 2)( 1) (2, ) (.5,.5) In Eercises 17 2, write an equation of the parabola in verte form or intercept form. 17. 1. 16 2 1 Flare Signal (3, 15) (1, 6) 2 6 Time (seconds) 1 2 3 5 f 16 New Ride (, 1) (1, 16) 2 Time (seconds) Chapter 2
19. Human Jump 2. (3, 2.25) 2 (, ) (, ) 2 Distance (feet) Frog Jump 1..5 (3, 1) ( 1, 5 9 ). 2 Distance (feet) 2. MODELING WITH MATHEMATICS A baseball is thrown up in the air. The table shows the heights (in feet) of the baseball after seconds. Write an equation for the path of the baseball. Find the height of the baseball after 5 seconds. Time, 2 6 Baseball height, 6 22 22 6 Dnamic Teaching Tools Dnamic Assessment & Progress Monitoring Tool Interactive Whiteboard Lesson Librar Dnamic Classroom with Dnamic Investigations 21. ERROR ANALYSIS Describe and correct the error in writing an equation of the parabola. 2 ( 1, ) (3, ) 2 (2, ) = a( p)( q) = a(3 1)(3 + 2) a = 2 5 = 2 ( 1)( + 2) 5 22. MATHEMATICAL CONNECTIONS The area of a rectangle is modeled b the graph where is the area (in square meters) and is the width (in meters). Write an equation of the parabola. Find the dimensions and corresponding area of one possible rectangle. What dimensions result in the maimum area? Area (square meters) Rectangles 12 (1, 6) (, ) (7, ) Width (meters) 23. MODELING WITH MATHEMATICS Ever rope has a safe working load. A rope should not be used to lift a weight greater than its safe working load. The table shows the safe working loads S (in pounds) for ropes with circumference C (in inches). Write an equation for the safe working load for a rope. Find the safe working load for a rope that has a circumference of 1 inches. (See Eample 3.) Circumference, C 1 2 3 Safe working load, S 1 72 162 25. COMPARING METHODS You use a sstem with three variables to find the equation of a parabola that passes through the points (, ), (2, 2), and (1, ). Your friend uses intercept form to find the equation. Whose method is easier? Justif our answer. 26. MODELING WITH MATHEMATICS The table shows the distances a motorcclist is from home after hours. Time (hours), 1 2 3 Distance (miles), 5 9 135 a. Determine what tpe of function ou can use to model the data. Eplain our reasoning. b. Write and evaluate a function to determine the distance the motorcclist is from home after 6 hours. 27. USING TOOLS The table shows the heights h (in feet) of a sponge t seconds after it was dropped b a window cleaner on top of a skscraper. (See Eample.) Time, t 1 1.5 2.5 3 Height, h 2 26 2 1 136 a. Use a graphing calculator to create a scatter plot. Which better represents the data, a line or a parabola? Eplain. b. Use the regression feature of our calculator to find the model that best fits the data. c. Use the model in part (b) to predict when the sponge will hit the ground. d. Identif and interpret the domain and range in this situation. 2. MAKING AN ARGUMENT Your friend states that quadratic functions with the same -intercepts have the same equations, verte, and ais of smmetr. Is our friend correct? Eplain our reasoning. Section 2. Modeling with Quadratic Functions 1 19. =.75( ) 1 2. = 9 ( 3)2 + 1 21. The -intercepts were substituted incorrectl. = a( p)( q) = a(3 + 1)(3 2) a = 1 = ( + 1)( 2) 22. = 2 + 7; Sample answer: A rectangle 1 meter b 6 meters would result in an area of 6 square meters; A rectangle 3.5 meters b 3.5 meters would result in a maimum area of 12.25 square meters. 23. S(C) = 1C 2 ; 1, lbs 2. = 2 2 + 12 + 6; 16 ft 25. intercept form; The three points can be substituted into the intercept form of a quadratic equation to solve for a, and then the equation can be written. This method is much shorter than writing and solving a sstem of three equations, although it can onl be used when given the intercepts. 26. a. linear; and change at a constant rate. b. = 5; 27 mi 27. a. parabola; not a constant rate of change b. h = 16t 2 + 2 c. about.1 sec d. The domain is t.1 and represents the time the sponge was in the air. The range is h 2 and represents the height of the sponge. 2. no; Because the ais of smmetr is alwas directl between the -intercepts, the ais of smmetr will be the same. The vertices could have different -coordinates, and therefore different equations. Section 2. 1
29. quadratic; The second differences are constant; = 2 2 + 2 + 7 3. linear; The first differences are constant; = 2 + 31 1. See Additional Answers. Mini-Assessment 1. A clown who is shot out of a cannon lands in a net 5 feet awa. Write an equation of the parabola. What is the height of the net? =.( 3) 2 + 5; 3 feet In Eercises 29 32, analze the differences in the outputs to determine whether the data are linear, quadratic, or neither. Eplain. If linear or quadratic, write an equation that fits the data. 29. 3. 31. 32. Price decrease (dollars), Revenue ($1s), 5 1 15 2 7 63 69 65 51 Time (hours), 1 2 3, 2 6 Time (hours), 1 2 3 5 Population (hundreds), 2 16 32 Time (das), 1 2 3, 32 33 25 173 6 3. THOUGHT PROVOKING Describe a real-life situation that can be modeled b a quadratic equation. Justif our answer. 35. PROBLEM SOLVING The table shows the heights of a competitive water-skier seconds after jumping off a ramp. Write a function that models the height of the water-skier over time. When is the water-skier 5 feet above the water? How long is the skier in the air? Time (seconds),.25.75 1 1.1, 22 22.5 17.5 12 9.2 36. HOW DO YOU SEE IT? Use the graph to determine whether the average rate of change over each interval is positive, negative, or zero. 6 Human Cannonball 6 (3, 5) 2 (, 1) 2 Horizontal distance (feet) 2. A meteorologist creates a parabola to predict the temperature the net da, where is the number of hours after midnight and is the temperature (in degrees Celsius). At midnight, the temperature is 9.6 C. At 5 A.M. and P.M., the temperature is C. Write a function f that models the temperature over time. What is the coldest temperature? f() =.96( 5)( 2) for 2; 5. C at 12:3 P.M. 3. The table shows the height of a ball over time after it was kicked. Write a function that models the data. Approimate the height of the ball after 6.7 seconds. 33. PROBLEM SOLVING The graph shows the number of students absent from school due to the flu each da. Number of students 16 12 (, 1) 2 Flu Epidemic (6, 19) 6 1 12 Das a. Interpret the meaning of the verte in this situation. b. Write an equation for the parabola to predict the number of students absent on da 1. c. Compare the average rates of change in the students with the flu from to 6 das and 6 to 11 das. Maintaining Mathematical Proficienc Factor the trinomial. (Skills Review Handbook) 3. 2 + + 3 39. 2 3 + 2. 3 2 15 + 12 1. 5 2 + 5 3 2 Chapter 2 Quadratic Functions If students need help... 2 6 a. 2 b. 2 5 c. 2 d. Reviewing what ou learned in previous grades and lessons 2 37. REPEATED REASONING The table shows the number of tiles in each figure. Verif that the data show a quadratic relationship. Predict the number of tiles in the 12th figure. Figure 1 Figure 2 Figure 3 Figure If students got it... Figure 1 2 3 Number of Tiles 1 5 11 19 Time, Height of ball, 1 12.7 2 15.2 3 1.9 13.1 5 1. =.5 2 +.3 + 9.32; about. foot Resources b Chapter Practice A and Practice B Puzzle Time Student Journal Practice Differentiating the Lesson Skills Review Handbook Resources b Chapter Enrichment and Etension Cumulative Review Start the net Section 2 Chapter 2