3. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..A A..E A..A A..B A..C 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 P(t) = at + 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 made the least profit. Modeling with a Quadratic Function Yearl profit (dollars) P P(t) = at + bt + c Year t c. The compan made the same earl profits in and 1. 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 3 Height, h 3 336 56 1 APPLYING MATHEMATICS To be proficient in math, ou need to routinel interpret our results in the contet of the situation. 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. Predict when the wrench will hit the ground. Eplain. Communicate Your Answer 5 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. Section 3. Modeling with Quadratic Functions 17
3. 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) Use verte form: = a( h) + k Given a point and -intercepts p and q Use intercept form: = a( p)( q) Given three points Write and solve a sstem of three equations in three variables. Writing an Equation Using a Verte and a Point 3 1 Human Cannonball (,15) (5, 35) 6 Horizontal distance (feet) 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) + k Verte form 15 = a( 5) + 35 Substitute for h, k,, and. = 5a Simplif.. = a Divide each side b 5. Because a =., h = 5, and k = 35, the path can be modeled b the equation =.( 5) + 35, where 9. Find the height when = 9. =.(9 5) + 35 Substitute 9 for. =.(16) + 35 Simplif. =. Simplif. So, the height of the net is about 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?. Write an equation of the parabola that passes through the point ( 1, ) and has verte (, 9). 1 Chapter 3 Quadratic Functions
Writing an Equation Using a Point and -Intercepts Temperature ( C) Temperature Forecast (, 9.6) 1 1 REMEMBER (, ) (, ) 3 9 15 Hours after midnight The average rate of change of a function f from 1 to is the slope of the line connecting ( 1, f( 1 )) and (, f( )): f( ) f( 1 ). 1 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 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( )( ) Substitute for p, q,, and. 9.6 = 96a Simplif..1 = a Divide each side b 96. Because a =.1, p =, and q =, the temperature over time can be modeled b f() =.1( )( ), where. The coldest temperature is the minimum value. So, find f() when = + = 1. f(1) =.1(1 )(1 ) Substitute 1 for. = 1 Simplif. So, the coldest temperature is 1 C at 1 hours after midnight, or 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 < <. Average rate of change Average rate of change over < < 1: over 1 < < : f(1) f() 1 9.6 = = 1. 1 1 f() f(1) 1 = ( 1) = 1 1 (, 9.6) 1 3 15 (, ) 1 (1, 1) Because 1. > 1, the average rate at which the temperature decreases from midnight to p.m. is greater than the average rate at which it increases from p.m. to midnight. Monitoring Progress 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 (, 5) and has -intercepts and. Section 3. Modeling with Quadratic Functions 19
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 fi rst differences. Quadratic data have constant second differences. The first and second differences of f() = are shown below. Equall-spaced -values 3 1 1 3 f() 9 1 1 9 first differences: 5 3 1 1 3 5 second differences: Writing a Quadratic Equation Using Three Points Time, t Height, h 1 6,9 15 9,5 3,6 5 31,65 3 3,1 35 3,5 31, 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. 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() h(5) h(3) h(35) h() 6,9 9,5 3,6 31,65 3,1 3,5 31, 15 1575 15 75 75 65 55 55 55 55 55 Because the second differences are constant, ou can model the data with a quadratic function. Step Write a quadratic function of the form h(t) = at + bt + c that models the data. Use an three points (t, h) from the table to write a sstem of equations. Use (1, 6,9): 1a + 1b + c = 6,9 Equation 1 Use (, 3,6): a + b + c = 3,6 Equation Use (3, 3,1): 9a + 3b + c = 3,1 Equation 3 Use the elimination method to solve the sstem. Subtract Equation 1 from Equation. Subtract Equation 1 from Equation 3. 3a + 1b = 37 New Equation 1 a + b = 5 New Equation a = Subtract times new Equation 1 from new Equation. a = 11 Solve for a. b = 7 Substitute into new Equation 1 to find b. c = 1, Substitute into Equation 1 to find c. The data can be modeled b the function h(t) = 11t + 7t + 1,. Step 3 Evaluate the function when t =.. h(.) = 11(.) + 7(.) + 1, = 3,.96 Passengers begin to eperience a weightless environment at about 3, feet. 13 Chapter 3 Quadratic Functions
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 Miles per hour, STUDY TIP Miles per gallon, 1.5 17.5 3 1. 36 3.7 5. 5 5. 5 5. 56 5.1 6. 7 19.5 The coefficient of determination R shows how well an equation fits a set of data. The closer R is to 1, the better the fit. 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 6. miles per gallon when driving about.9 miles per hour. So, the optimal driving speed is about 9 miles per hour. 75 Step Use the quadratic regression feature. A quadratic model that represents the data is =.1 + 1.37 7.1. QuadReg =a +b+c a=-.19739 b=1.366167 c=-7.1513 R =.99975 35 Maimum X=.9565 Y=6.1671 75 Monitoring Progress Help in English and Spanish at BigIdeasMath.com 5. Write an equation of the parabola that passes through the points ( 1, ), (, 1), and (, 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, 5 11 1 17 Profit, 6 65 6 9 7 1 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 inches thick? Ice thickness, 1 1 15 1 7 Maimum weight, 3. 7.6 1. 1.3 5..6 5.3 Section 3. Modeling with Quadratic Functions 131
3. Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. WRITING Eplain when it is appropriate to use a quadratic model for a set of data.. DIFFERENT WORDS, SAME QUESTION Which is different? Find both answers. What is the average rate of change over? What is the distance from f() to f()? f What is the slope of the line segment? f() f() What is? 1 1 1 3 5 Monitoring Progress and Modeling with Mathematics In Eercises 3, write an equation of the parabola in verte form. (See Eample 1.) 3. (, 6) ( 1, 3). (, 3) (, 1) 5. passes through (13, ) and has verte (3, ) 6. passes through ( 7, 15) and has verte ( 5, 9) 7. passes through (, ) and has verte ( 6, 1). passes through (6, 35) and has verte ( 1, 1) In Eercises 9 1, write an equation of the parabola in intercept form. (See Eample.) 9. (3, ) (, ) (, ) 1. ( 1, ) (, ) (1, ) 11. -intercepts of 1 and 6; passes through (1, ) 1. -intercepts of 9 and 1; passes through (, 1) 13. -intercepts of 16 and ; passes through ( 1, 7) 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? ( 1, ) A = ( )( + 1) B = ( +.5).5 C = (.5).5 D = ( + )( 1) (, ) (.5,.5) In Eercises 17, write an equation of the parabola in verte form or intercept form. 17. 1. 16 Flare Signal (3, 15) (1, 6) 6 Time (seconds) 16 New Ride (, 1) (1, 16) Time (seconds) 1. -intercepts of 7 and 3; passes through (,.5) 13 Chapter 3 Quadratic Functions
19. Human Jump. (3,.5) (, ) (, ) Distance (feet) Frog Jump 1..5 (3, 1) ( 1, 5 9 ). Distance (feet). 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, 6 Baseball height, 6 6 1. ERROR ANALYSIS Describe and correct the error in writing an equation of the parabola. ( 1, ) (3, ) (, ) = a( p)( q) = a(3 1)(3 + ) a = 5 = ( 1)( + ) 5. 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 1 (1, 6) (, ) (7, ) Width (meters) 3. 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 3 Safe working load, S 1 7 16 5. COMPARING METHODS You use a sstem with three variables to find the equation of a parabola that passes through the points (, ), (, ), and (1, ). Your friend uses intercept form to find the equation. Whose method is easier? Justif our answer. 6. MODELING WITH MATHEMATICS The table shows the distances a motorcclist is from home after hours. Time (hours), 1 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. 7. 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 1.5.5 3 Height, h 6 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.. 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 3. Modeling with Quadratic Functions 133
In Eercises 9 3, 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. 3. THOUGHT PROVOKING Describe a real-life situation that can be modeled b a quadratic equation. Justif our answer. 9. 3. 31. 3. Price decrease (dollars), Revenue ($1s), 5 1 15 7 63 69 65 51 Time (hours), 1 3, 6 Time (hours), 1 3 5 Population (hundreds), 16 3 Time (das), 1 3, 3 33 5 173 6 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),.5.75 1 1.1,.5 17.5 1 9. 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 33. PROBLEM SOLVING The graph shows the number of students absent from school due to the flu each da. Number of students 16 1 (, 1) Flu Epidemic (6, 19) 6 1 1 Das 6 a. b. 5 c. d. 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 1th figure. 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. Figure 1 Figure Figure 3 Figure Figure 1 3 Number of tiles 1 5 11 19 Maintaining Mathematical Proficienc Factor the trinomial. (Skills Review Handbook) Reviewing what ou learned in previous grades and lessons 3. + + 3 39. 3 +. 3 15 + 1 1. 5 + 5 3 13 Chapter 3 Quadratic Functions