Essential Question How can you use a quadratic function to model a real-life situation?

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2 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

3 Writing an Equation Using a Point and -Intercepts Temperature ( C) Temperature Forecast (, 9.6) 1 1 REMEMBER (, ) (, ) 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() = = f() f(1) 1 = ( 1) = 1 1 (, 9.6) (, ) 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

4 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 f() first differences: 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, 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

6 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? 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 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

8 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 Price decrease (dollars), Revenue (\$1s), Time (hours), 1 3, 6 Time (hours), Population (hundreds), 16 3 Time (das), 1 3, 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), , HOW DO YOU SEE IT? Use the graph to determine whether the average rate of change over each interval is positive, negative, or zero 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) 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 Maintaining Mathematical Proficienc Factor the trinomial. (Skills Review Handbook) Reviewing what ou learned in previous grades and lessons Chapter 3 Quadratic Functions

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