7.1 Connecting Intercepts and Zeros

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1 Locker LESSON 7. Connecting Intercepts and Zeros Common Core Math Standards The student is epected to: F-IF.7a Graph linear and quadratic functions and show intercepts, maima, and minima. Also A-REI., A-APR.3, A-REI., F-LE.6 Mathematical Practices MP.5 Using Tools PAGE 95 Name Class Date 7. Connecting Intercepts and Zeros Essential Question: How can ou use the graph of a quadratic function to solve its related quadratic equation? Eplore Graphing Quadratic Functions in Standard Form A parabola can be graphed using its verte and ais of smmetr. Use these characteristics, the -intercept, and smmetr to graph a quadratic function. Graph b completing the steps. Resource Locker Language Objective Given a quadratic function modeling a real-world situation, eplain to a partner what the zeros of the function represent. ENGAGE Essential Question: How can ou use the graph of a quadratic function to solve its related quadratic equation? You can write the equation with one side equal to 0, graph the related function, and find the zeros of the function. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss what tpe of path is made b a diver diving into the water when her initial height is the height of the diving platform. Then preview the Lesson Performance Task. Houghton Mifflin Harcourt Publishing Compan A C D Find the ais of smmetr. - b _ a - _ The ais of smmetr is. Find the -intercept ( -5 ) B The -intercept is -5 ; the graph passes through (0, -5 ). Find the verte The verte is (, ). Find two more points on the same side of the ais of smmetr as the -intercept. a. Find when The first point is (, ). b. Find when ( ) ( ) The second point is (, ). 0 Module 7 59 Lesson Name Class Date 7. Connecting Intercepts and Zeros Essential Question: How can ou use the graph of a quadratic function to solve its related quadratic equation? F-IF.7a Graph linear and quadratic functions and show intercepts, maima, and minima. Also A-REI., A-APR.3, A-REI. Houghton Mifflin Harcourt Publishing Compan Eplore Graphing Quadratic Functions Find the ais of smmetr. a in Standard Form A parabola can be graphed using its verte and ais of smmetr. Use these characteristics, the -intercept, and smmetr to graph a quadratic function. Graph b completing the steps. - b_ _ The ais of smmetr is. Find the -intercept ( ) Find the verte. The verte is (, ). The -intercept is ; the graph passes through (0, ). Find two more points on the same side of the ais of smmetr as the -intercept. a. Find when. b. Find when The first point is (, ) Resource ( ) ( ) - 5 The second point is (, ). Module 7 59 Lesson HARDCOVER PAGES 95 0 Turn to these pages to find this lesson in the hardcover student edition. 59 Lesson 7.

2 Graph the ais of smmetr, the verte, the -intercept, and the two etra points on the same coordinate plane. Then reflect the graphed points over the ais of smmetr to create three more points, and sketch the graph. Reflect (-, 0) 0 (0, -5) ( ) (, ) (, -9). Discussion Wh is it important to find additional points before graphing a quadratic function? Additional points provide more information about the shape of the parabola, and the sketch of the quadratic function will be more accurate. Eplain Using Zeros to Solve Quadratic Equations Graphicall A zero of a function is an -value that makes the value of the function 0. The zeros of a function are the -intercepts of the graph of the function. A quadratic function ma have one, two, or no zeros. Quadratic equations can be solved b graphing the related function of the equation. To write the related function, rewrite the quadratic equation so that it equals zero on one side. Replace the zero with. Graph the related function. Find the -intercepts of the graph, which are the zeros of the function. The zeros of the function are the solutions to the original equation. Eample Solve b graphing the related function. a. Write the related function. Add 3 to both sides to get - 0. The related function is -. b. Make a table of values for the related function c. Graph the points represented b the table and connect the points. d. The zeros of the function are - and, so the solutions of the equation are - and. (-, 6) (, 6) (-, 0) (, 0) - 0 (0, -) Module 7 60 Lesson PROFESSIONAL DEVELOPMENT Math Background Quadratic equations are often used to model the motion of falling objects. The general formula for this motion is h -6t + v 0 t + h 0, where h represents the height of the object in feet, t is the number of seconds the object has been falling, v 0 is the initial vertical velocit of the object in feet per second, and h 0 is the initial height of the object in feet. The coefficient -6 is equal to half of the constant acceleration due to gravit, -3 ft/s. Students can compare the quadratic equations given for falling objects in this lesson to the general formula to determine the values of v 0 and h 0 in each case. PAGE 96 Houghton Mifflin Harcourt Publishing Compan EXPLORE Graphing Quadratic Functions in Standard Form QUESTIONING STRATEGIES How man points on the graph of a quadratic function are on the ais of smmetr? Eplain. One; the onl point on the graph of a quadratic function that is on the ais of smmetr is the verte of the function. Wh is it helpful to find the ais of smmetr when graphing a quadratic function? After ou find the ais of smmetr and a few points on one side of it, ou can use smmetr to quickl and easil find an equal number of points on the other side. EXPLAIN Using Zeros to Solve Quadratic Equations Graphicall AVOID COMMON ERRORS Students ma think that the zeros of a quadratic function can be found b substituting 0 for in the function. Make sure students understand that the zeros of a function are the values of when 0, not the values of when 0. Focus on Critical Thinking MP.3 For cases in which a quadratic function has two zeros, discuss with students how to use the zeros to determine the function s ais of smmetr. Students should realize that the ais of smmetr will be the -value of the point that is midwa between the two zeros. Connecting Intercepts and Zeros 60

3 QUESTIONING STRATEGIES When a quadratic function has two zeros that are opposites, what must be true about the function? The ais of smmetr must be 0, and the verte must not be (0, 0). B a. Write the related function. Add to both sides to get The related function is b. Make a table of values for the related function (-5, 3) (-, 3) (, 0) (-, 0) 0 (-3, -) PAGE 97 c. Graph the points represented b the table and connect the points. d. The zeros of the function are and -, so the solutions of the equation are and -. Reflect. How would the graph of a quadratic equation look if the equation has one zero? If the quadratic equation has one zero, the graph will intersect the -ais at its verte. Your Turn Houghton Mifflin Harcourt Publishing Compan The related function is -. Graph: The zeros of the function are and -, so the solutions of the equation are - and. (-, 3) (, 3) (-, 0) (, 0) 0 (0, -) Module 7 6 Lesson COLLABORATIVE LEARNING Peer-to-Peer Activit Have students work in pairs. Have each student find the solution to the same quadratic equation b graphing. One student in each pair then solves the equation b finding the zeros of the function, and the other student uses points of intersection to find the solution. Students compare their answers; while the graphs ma be different, the solution should be the same regardless of the method used. 6 Lesson 7.

4 Eplain Using Points of Intersection to Solve Quadratic Equations Graphicall You can solve a quadratic equation b rewriting the equation in the form a + b c or a ( - h) k and then using the epressions on each side of the equal sign to define a function. Graph both functions and find the points of intersection. The solutions are the -coordinates of the points of intersection on the graph. As with using zeros, there ma be two, one, or no points of intersection. Eample Solve each equation b finding points of intersection of two related functions. ( - ) - 0 ( - ) Rewrite as a ( - h) k. Graph each side as related function. a. Let ƒ () ( - ). Let g (). b. Graph ƒ () and g () on the same graph. c. Determine the points at which the graphs of ƒ () and g () intersect. The graphs intersect at two locations: (3, ) and (5, ). This means ƒ () g () when 3 and 5. So the solutions of ( - ) - 0 are 3 and 5. 3 ( - 5) ( - 5) a. Let ƒ () 3 ( - 5). Let g (). b. Graph ƒ () and g () on the same graph. c. Determine the points at which the graphs of ƒ () and g () intersect. The graphs intersect at two locations: ( 3, ) and ( 7, ). This means ƒ () g () when 3 and 7. Therefore, the solutions of the equation ƒ () g () are 3 and 7. So the solutions of 3 ( - 5) - 0 are 3 and 7. g() (3, ) (5, ) 0 f() 6 (7, ) (3, ) 0 Module 7 6 Lesson -6 Houghton Mifflin Harcourt Publishing Compan PAGE 9 EXPLAIN Using Points of Intersection to Solve Quadratic Equations Graphicall Focus on Math Connections MP. Remind students that quadratic equations in the form a ( - h) + k will have the verte at (h, k). After identifing the verte, students can use substitution to find enough points to graph the quadratic function. Focus on Modeling MP. When using points of intersection to find solutions to quadratic equations, make sure that students understand that onl the -values of the points of intersection are solutions to the equation. The -values come from the function that was created to find the solutions; the are not part of the solution. QUESTIONING STRATEGIES How can ou use graphing to determine that a quadratic equation has no solutions? After rewriting the equation in the form a + b c, if the graphs of the functions f () a + b and g () c do not intersect, then the quadratic function has no solutions. DIFFERENTIATE INSTRUCTION Communicating Math Review the parameters of a parabola that students can use when graphing a function. Students should understand that being able to identif the verte of a parabola will make it easier to graph the function, but the ma have different ideas about how to find enough other points in order to make an accurate graph. Discuss how students can be sure that the have plotted enough points to sketch the function on a coordinate grid. Connecting Intercepts and Zeros 6

5 EXPLAIN 3 Modeling a Real-World Problem INTEGRATE TECHNOLOGY Before students are familiar with using the quadratic formula, graphing calculators offer the most accessible wa to find the solutions to quadratic equations that do not have simple whole-number solutions. Focus on Modeling MP. The functions used to determine the height of a thrown object can be difficult to understand. Discuss different was to rewrite the functions used to determine height. Students ma wish to rewrite a function like h (t) -6t + 00 as h (t) 00 6 t to make it more evident that the object loses height for each second of time. PAGE 99 Houghton Mifflin Harcourt Publishing Compan Image Credits: USBFCO/ Shutterstock Reflect. In Part B above, wh is the -coordinates the answer to the equation and not the -coordinates? The -coordinates are the solution because the -values are the unknown amount being solved for in the original equation. The -values are used to create a related function to find the values, but since the are not part of the original equation, the values are not part of the solution. Your Turn 5. Solve 3 ( - ) -3 0 b finding the points of intersection of the two related functions. 3( - ) Eplain 3 Modeling a Real-World Problem Man real-world problems can be modeled b quadratic functions. Eample 3 3( - ) 3 Let f () 3 ( - ) and let g () 3. The graphs intersect at two locations: (, 3) and (3, 3). This means f () g () when and 3. So the solutions of 3 (- ) are and 3. Create a quadratic function for each problem and then solve it b using a graphing calculator. Nature A squirrel is in a tree holding a chestnut at a height of 6 feet above the ground. It drops the chestnut, which lands on top of a bush that is 36 feet below the squirrel. The function h (t) -6t + 6 gives the height in feet of the chestnut as it falls, where t represents time. When will the chestnut reach the top of the bush? Analze Information Identif the important information. The chestnut is 6 feet above the ground, and the top of the bush is 36 feet below the chestnut. The chestnut s height as a function of time can be represented b h (t) -6 t + 6, where (h) t is the height of the chestnut in feet as it is falling. (, 3) (3, 3) Formulate a Plan Create a related quadratic equation to find the height of the chestnut in relation to time. Use h (t) -6 t + 6 and insert the known value for h. Module 7 63 Lesson LANGUAGE SUPPORT Connect Vocabular Students ma epect that problems involving points of intersection will alwas have solutions. Relate the word intersection to its uses outside the math classroom. Just as two streets in a town ma or ma not intersect, the graphs of two functions ma or ma not intersect. When dealing with the intersection of a quadratic function and a linear function, remind students that there ma be 0,, or points of intersection. 63 Lesson 7.

6 Solve Write the equation that needs to be solved. Since the top of the bush is 36 feet below the squirrel, it is 0 feet above the ground. -6t Separate the function into f (t) and g (t). f (t) -6 t + 6 and g (t) 0. To graph each function on a graphing calculator, rewrite them in terms of and and 0 Graph both functions. Use the intersect feature to find the amount of time it takes for the chestnut to hit the top of the bush. The chestnut will reach the top of the bush in Justif and Evaluate -6 (.5 ) When t is replaced b.5.5 seconds. in the original equation, -6 t is true. QUESTIONING STRATEGIES When finding the time it takes for an object to fall to the ground, wh is onl the positive zero of the function used as an answer? Since the value for time will alwas be a positive number, the negative zero of the function can be ignored as a solution. Focus on Technolog MP.5 Students who have the minimum and maimum - and -values set incorrectl on their graphing calculators ma not be able to see the point of intersection when the graph two functions. Discuss how to change the dimensions of the graph b accessing the WINDOW menu. Reflect 6. In Eample 3 above, the graphs also intersect to the left of the -ais. Wh is that point irrelevant to the problem? That point is irrelevant to the problem since negative time has no meaning in this problem. Your Turn 7. Nature An egg falls from a nest in a tree 5 feet off the ground and lands on a potted plant that is 0 feet below the nest. The function h (t) -6t + 5 gives the height in feet of the egg as it drops, where t represents time. When will the egg land on the plant? and 5. The egg will hit the plant after about. seconds. Houghton Mifflin Harcourt Publishing Compan PAGE 00 Module 7 6 Lesson Connecting Intercepts and Zeros 6

7 EXPLAIN Interpreting a Quadratic Model AVOID COMMON ERRORS When viewing graphs of quadratic functions modeling height, students ma believe that the shape of the graph represents the path an object takes while in the air. Remind students that while the -ais represents the height of the object, the -ais does not show distance, but rather time. Eplain Interpreting a Quadratic Model The solutions of a quadratic equation can be used to find other information about the situation modeled b the related function. Eample Use the given quadratic function model to answer questions about the situation it models. Nature A dolphin jumps out of the water. The quadratic function h (t) -6 t + 0t models the dolphin s height above the water in feet after t seconds. How long is the dolphin out of the water? Use the level of the water as a height of 0 feet. h(0) 0, so the dolphin leaves the water at t 0. When the dolphin reenters the water again, its height is 0 feet. Solve 0-6 t + 0t to find the time when the dolphin reenters the water. Graph the function on a graphing calculator, and find the other zero that occurs at > 0. The zeros appear to be 0 and.5. Check.5. QUESTIONING STRATEGIES For quadratic functions that model the height of a thrown object, how can ou use the zeros of the function to determine when the object is at its maimum height? Eplain. Since the ais of smmetr for a quadratic function lies midwa between the two zeros of the function, the maimum height is reached at a time halfwa between the two zeros. For a quadratic function modeling the height of a thrown object, when one of the zeros of the function is 0, what must be true about the object? The object must have been thrown from ground level, so that the height of the object equals 0 at time 0. PAGE 0 Houghton Mifflin Harcourt Publishing Compan -6 (.5) + 0 (.5) 0 so.5 is a solution. The dolphin is out of the water for.5 seconds. Sports A baseball coach uses a pitching machine to simulate pop flies during practice. The quadratic function h(t) 6 t + 0t + 5 models the height in feet of the baseball after t seconds. The ball leaves the pitching machine and is caught at a height of 5 feet. How long is the baseball in the air? To find when the ball is caught at a height of 5 feet, ou need to solve 5 6 t + 0t + 5. Graph Y and Y 5, and use the intersection feature to find -values when 5. From the graph, it appears that the ball is 5 feet above the ground when 0 or 5. Therefore, the ball is in the air for seconds. Module 7 65 Lesson 65 Lesson 7.

8 Your Turn. Nature The quadratic function models the height, in feet, of a fling fish above the water after seconds. How long is the fling fish out of the water? The graph of shows a zero at about The fish is out of the water for 0.35 second. Elaborate 9. How is graphing quadratic functions in standard form similar to using zeros to solve quadratic equations graphicall? If there are two solutions for a quadratic function, the reflection of the point representing one solution across the ais of smmetr will be the other point, which represents the other solution. Both methods use the value of the function at 0 to find the second point. 0. How can graphing calculators be used to solve real-world problems represented b quadratic equations? Graphing calculators can be used to solve real-world quadratic equations b writing the equation and then finding either an intersection or the zeros of the equation s graph.. Essential Question Check-In How can ou use the graph of a quadratic function to solve a related quadratic equation b wa of intersection? You can graph the two sides of the equation as two functions and find their points of intersection. The -coordinate or coordinates of the intersection(s) will be the solution to the quadratic equation. Houghton Mifflin Harcourt Publishing Compan ELABORATE Focus on Communication MP.3 Discuss with students what kinds of realworld problems can be solved b identifing the zeros of a quadratic function. Students should understand that the zeros of a quadratic function will not be the solution for ever real-world problem. SUMMARIZE THE LESSON How do ou solve equations graphicall using zeros and points of intersection? To solve quadratic equations graphicall using zeros, rewrite the equation so one side is equal to zero, then graph the other side of the equation and identif the zeros. To solve quadratic equations graphicall using points of intersection, rewrite the equation in the form a + b c, graph both sides of the equation, then find the -values of the points of intersection. Module 7 66 Lesson Connecting Intercepts and Zeros 66

9 EVALUATE PAGE 0 Evaluate: Homework and Practice Solve each equation b graphing the related function and finding its zeros Online Homework Hints and Help Etra Practice ASSIGNMENT GUIDE Concepts and Skills Eplore Graphing Quadratic Functions in Standard Form Eample Using Zeros to Solve Quadratic Equations Graphicall Eample Using Points of Intersection to Solve Quadratic Equations Graphicall Eample 3 Modeling a Real-World Problem Eample Interpreting a Quadratic Model Practice Eercise Eercises 6, 5 Eercises 7,, 3 Eercises 3 6, Eercises (, 9) (-, 9) (, 0) - 0 (-, 0) -0 (0, -3) -0 6 (, ) (-, ) (, 0) - 0 (-, 0) (0, ) -6 The zeros of 3-3 are and -, so - or. 3 (, ) (, ) 6 (, 0) 0 (-, 0) (0, ) The zeros - are and -, so - or. (-6, ) (-5, 0) (-, 0) 0 (-3, -) (, -) Houghton Mifflin Harcourt Publishing Compan The zeros of - are and -, so - or (-3, 0) (, 0) 0 (-, -3) (0, -3) (-, ) The zeros of are -5 and -, so -5 or -. (-, 3) (, 3) (-, 0) (, 0) (0, -) The zeros of are -3 and -, so -3 or. The zeros of - are - and, - or. Module 7 67 Lesson IN_MNLESE3930_U3M07L.indd 67 Eercise Depth of Knowledge (D.O.K.) Mathematical Practices 0/0/ 3:3 PM Recall of Information MP. Modeling 3 0 Recall of Information MP. Modeling Recall of Information MP.5 Using Tools Recall of Information MP. Modeling 3 5 Skills/Concepts MP.3 Logic 67 Lesson 7.

10 Solve each equation b finding points of intersection of two functions. 7. ( - 3) - 0. ( + ) - 0 (.59, ) (., ) The graphs of f () ( - 3) and g () intersect at (.59, ) and (., ). So.59 or.. (, ) (0, ) The graphs of f () ( + ) and g () intersect at (, ) and (0, ). So 0 or. Focus on Communication MP.3 Students ma feel that certain equations must be solved using certain methods. Discuss the fact that most problems can be solved using more than one of the methods outlined in this lesson, and that the will learn additional methods as the continue in algebra ( - 3) ( + ) - 0 (, ) (5, ) 0 The graphs of f () ( - 3) and g () intersect at (5, ) and (, ). So 5 or The graphs of f () ( + ) and g () - do not intersect. So there is no solution.. ( + ) - 0. ( + ) - 0 (-3., ) (-.59, ) (-, ) (0, ) Houghton Mifflin Harcourt Publishing Compan The graphs of f () ( + ) and g () intersect at (-, ) and (0, ). So - or 0. The graphs of f () ( + ) and g () intersect at (-3., ) and (-0.59, ). So -3. or Module 7 6 Lesson IN_MNLESE3930_U3M07L.indd 6 // : PM Connecting Intercepts and Zeros 6

11 AVOID COMMON ERRORS When solving quadratic equations graphicall, students ma mistake the zeros of a quadratic function for the places where the function intersects the -ais. Remind students that the zeros of a function are the -values at the points where the function intersects the -ais. Create a quadratic equation for each problem and then solve the equation with a related function using a graphing calculator. 3. Nature A bird is in a tree 30 feet off the ground and drops a twig that lands on a rosebush 5 feet below. The function h (t) -6 t + 30, where t represents the time in seconds, gives the height h, in feet, of the twig above the ground as it falls. When will the twig land on the bush? The graphs of and 5 intersect at about (.5, 5). The twig will hit the rosebush after about.5 seconds.. Nature A monke is in a tree 50 feet off the ground and drops a banana, which lands on a shrub 30 feet below. The function h (t) -6 t + 50, where t represents the time in seconds, gives the height h, in feet, of the banana above the ground as it falls. When will the banana land on the shrub? The graphs of and 0 intersect at about (., 0). The banana will hit the shrub after about. seconds. 5. Sports A trampolinist steps off from 5 feet above ground to a trampoline 3 feet below. The function h (t) -6 t + 5, where t represents the time in seconds, gives the height h, in feet, of the trampolinist above the ground as he falls. When will the trampolinist land on the trampoline? PAGE 03 Houghton Mifflin Harcourt Publishing Compan Dmitr Laudin/ Shutterstock The graphs of and intersect at about (0.9, ). The trampolinist will land on the trampoline after about 0.9 second. 6. Phsics A ball is dropped from 0 feet above the ground. The function h (t) -6 t + 0, where t represents the time in seconds, gives the height h, in feet, of the ball above the ground. When will the ball be feet above the ground? The graphs of and intersect at about (0.6, ). The ball will be feet above the ground in about 0.6 second. Use the given quadratic function model to answer questions about the situation it models. 7. Nature A shark jumps out of the water. The quadratic function ƒ () -6 + models the shark s height, in feet, above the water after seconds. How long is the shark out of the water? The graph of f () -6 + shows a zero of about.5. The shark is out of the water for about.5 seconds. Module 7 69 Lesson 69 Lesson 7.

12 . Sports A baseball coach uses a pitching machine to simulate pop flies during practice. The quadratic function ƒ () models the height in feet of the baseball after seconds. How long is the baseball in the air if the ball is not caught? The ball is pitched at t 0, and f(0) 0, so the ball is pitched from a height of 0 feet. The graph of f () shows a positive zero of about.5. The ball is in the air for.5 seconds. CRITICAL THINKING Challenge students to solve a quadratic equation graphicall using two different methods: b finding the zeros of the function and b finding points of intersection. The should find that the graphs look similar, but that one is translated verticall relative to the other. Students should understand that when solving one problem using two different methods, the graphs ma look different, but the final solutions will be the same. 9. The quadratic function ƒ () -6 + models the height, in feet, of a fling fish above the water after seconds. How long is the fling fish out of the water? The graph of f () -6 + shows a zero of about 0.7. The fling fish is in the air for about 0.7 seconds. 0. A football coach uses a passing machine to simulate 50-ard passes during practice. The quadratic function ƒ () models the height in feet of the football after seconds. How long is the football in the air if the ball is not caught? The ball is passed at t 0, and f(0) 5, so the ball is pitched from a height of 5 feet. The graph of f () shows a positive zero of about 3.. The football is in the air for about 3. seconds. Houghton Mifflin Harcourt Publishing Compan Radharc Images/Alam Module 7 70 Lesson Connecting Intercepts and Zeros 70

13 JOURNAL Have students write a paragraph outlining the steps the take when graphing a quadratic function b hand. Students should be sure to describe how to find enough points so that the function can be accuratel graphed on a coordinate grid.. In each polnomial function in standard form, identif a, b, and c. a b. + c. d. 5 e a. b. c. d. e. a 3, b, c a 0, b, c a, b 0, c 0 a 0, b 0, c 5 a 3, b, c. Identif the ais of smmetr, -intercept, and verte of the quadratic function and then graph the function on a graphing calculator to confirm. Ais of smmetr: - b a - -Intercept: Verte: ( -, -6 ) because ( _ - ) _ - _ PAGE 0 Houghton Mifflin Harcourt Publishing Compan H.O.T. Focus on Higher Order Thinking 3. Countereamples Pamela sas that if the graph of a function opens upward, then the related quadratic equation has two solutions. Provide a countereample to refute Pamela s claim. Sample answer: The graph of f () opens upward, but the related equation, 0, has onl one solution.. Eplain the Error Rodne was given the function h (t) -6 t + 50 representing the height above the ground (in feet) of a water balloon t seconds after being dropped from a roof 50 feet above the ground. He was asked to find how long it took the balloon to fall 0 feet. Rodne used the equation -6 t to solve the problem. What was his error? Falling 0 feet means that the balloon would have been feet above the ground, so Rodne should have solved the equation -6 t instead. 5. Critical Thinking If Jamie is given the graph of a quadratic function with onl the -intercepts and a random point labeled, can she determine an equation for the function? Eplain. Yes; Jamie can find the zeros of the function using the -intercepts, and write the equation in the factored form f () k( - a)( - b), where a and b are the zeros. She can then substitute the coordinates of the given point to find the value of k. Module 7 7 Lesson 7 Lesson 7.

14 Lesson Performance Task QUESTIONING STRATEGIES Stella is competing in a diving competition. Her height in feet above the water is modeled b the function ƒ () , where is the time in seconds after she jumps from the diving board. Graph the function and solve the related equation What do the solutions mean in the contet of the problem? Are there solutions that do not make sense? Eplain. The graph crosses the -ais at about (, 0) so the -intercept is about.9. So, one solution of the equation is about. This solution means that Stella reaches the surface of the water almost seconds after she starts the dive. If the graph is etended on the left, it will cross the -ais once more. Since time cannot be negative, this solution does not make sense. Height (ft) 36 0 Height of Dive Time (s) How can ou find the height at which the diver starts her dive? Find the value of the function when t 0. How can ou find the maimum height that the diver reaches? Use the graph to find the -value of the verte of the parabola, or find the -value halfwa between the two zeros and calculate the corresponding -value. Focus on Reasoning MP. Have students eamine how the zeros, intercepts, function, and graph modeling the dive would be different if the diver had started the dive at the level of the water, that is, at height 0. Houghton Mifflin Harcourt Publishing Compan Module 7 7 Lesson EXTENSION ACTIVITY Give students the functions g (t) - 6t + t + and h (t) - 6t + 6 and eplain that the represent the height in feet of two different divers, where t is time in seconds from the start of each dive. Have students compare the initial height, maimum height, and time to reach the water for the two divers. Students should find that g (t) represents a dive starting at feet, reaching a maimum of 5 feet, and hitting the water after.5 seconds, and h (t) represents a dive starting at 6 feet, which is also its maimum height, and hitting the water after second. Scoring Rubric points: Student correctl solves the problem and eplains his/her reasoning. point: Student shows good understanding of the problem but does not full solve or eplain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Connecting Intercepts and Zeros 7

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