D: all real; R: y g (x) = 3 _ 2 x 2 5. g (x) = 5 x g (x) = - 4 x 2 7. g (x) = -4 x 2. Houghton Mifflin Harcourt Publishing Company.

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1 AVOID COMMON ERRORS Watch for students who do not graph points on both sides of the verte of the parabola. Remind these students that a parabola is U-shaped and smmetric, and the can use that smmetr to locate points on both sides of the verte. Graph each quadratic function. State the domain and range.. g () = 3. g () = D: all real; R: D: all real; R:. g () = 3 _ 5. g () = Houghton Mifflin Harcourt Publishing Compan D: all real; R:. g () = - 7. g () = D: all real; R: D: all real; R: D: all real; R: Module 19 Lesson 1 IN_MNLESE393_U3ML1.indd 19 Eercise Depth of Knowledge (D.O.K.) Mathematical Practices //1 1:3 AM 3 3 Strategic Thinking MP.3 Logic 5 3 Strategic Thinking MP. Reasoning 19 Lesson.1

2 . g () = - 3_ D: all real; R: 9. g () = -5 D: all real; R: Determine the equation of the parabola graphed (-1, 3) (, -) MULTIPLE REPRESENTATIONS Show students the graph of the function g () = -1.5, and instruct them to make a table showing several - and -values for both the given graph and the parent function ƒ () =. Have them compare the values in the table and describe how the values of g () are related to the values of ƒ () for an value of. Then have them write the equation for g (). The should find that for a given value of, the value of g () is -1.5 times the value of the parent function, so the can conclude that g () = Use the point (-1, 3). g () = a 3 = a (-1) 3 = a g () = (3, -) Use the point (3, -). g () = a - = a (3) - = 9a - _ 9 = a g () = - _ 9 Use the point (, -). g () = a - = a () - = 1a - = a g () = Use the point (, 5). g () = a 5 = a () 5 = a 5_ = a g () = 5_ (, 5) Houghton Mifflin Harcourt Publishing Compan Module Lesson 1 IN_MNLESE393_U3ML1.indd //1 1:3 AM Understanding Quadratic Functions

3 INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Give students the graphs of two parabolas, f () = -a and g () = a, that are smmetric about the -ais. Have them consider the sum of their a-values. Students should see that the values of a are opposites, so the sum of these values is. Have them verif that for an value of the sum of the function values is. PAGE 13 A cannonball fired horizontall appears to travel in a straight line, but drops to earth due to gravit, just like an other object in freefall. The height of the cannonball in freefall is parabolic. The graph shows the change in height of the cannonball (in meters) as a function of distance traveled (in kilometers). Refer to this graph for questions 1 and 15. Height (m) - - h (., -5) Distance (km) d Houghton Mifflin Harcourt Publishing Compan Image Credits: (t) Brandon Alms/Alam; (b) Olegusk/Shutterstock 1. Describe what the verte, -intercept, and 15. Find the function h (d) that describes these coordinates. The verte is the position of the cannon that Use the point (., -5). fired the cannonball. h (d) = ad The -intercept represents the height of the -5 = a (.) cannonball relative to the cannon at d =. -5 =.1a The endpoint is the end of the cannonball s = a trajector. h (d) = d A slingshot stores energ in the stretched elastic band when it is pulled back. The amount of stored energ versus the pull length is approimatel parabolic. Questions 1 and 17 refer to this graph of the stored energ in millijoules versus pull length in centimeters. Energ (mj) E (, ) Pull Length (cm) 1. Describe what the verte, -intercept, and The verte is the point at which the slingshot is relaed and stores no energ. The -intercept is the energ, mj, when the pull length is cm at the beginning. The endpoint is at the maimum etent the slingshot is pulled back and the maimum stored energ. d 17. Determine the function, E (d), that describes this plot. Use the point (, ). E (d) = ad = a () = a = a E (d) = d Module 1 Lesson 1 1 Lesson.1

4 Newer clean energ sources like solar and wind suffer from unstead availabilit of energ. This makes it impractical to eliminate more traditional nuclear and fossil fuel plants without finding a wa to store etra energ when it is not available. One solution being investigated is storing energ in mechanical flwheels. Mechanical flwheels are heav disks that store energ b spinning rapidl. The graph shows how much energ is in a flwheel, as a function of revolution speed. Height (feet) h Energ (kwh) 1 (1, ) E (1, 1) 1 Rotation Speed (rps) 1. Describe what the verte, -intercept, and 19. Determine the function, E (r), that describes this plot. The verte represents the flwheel at rest Use the point (1, 1). with no rotation and no stored energ. E (r) = ar The -intercept is the energ, kwh, when 1 = a (1) the rotation speed is rps at the beginning. 1 = 1,a The endpoint represents the maimum.1 = a rotation speed and energ storage. E (r) =.1 r Phineas is building a homemade skate ramp and wants to model the shape as a parabola. He sketches out a cross section shown in the graph. l 5 1 Length (feet). Describe what the verte -intercept, and 1. Determine the function, h (l), that describes this plot. Use the point (1, ). The verte is the bottom of the ramp. h (l) = al The -intercept represents the height of = a (1) the ramp relative to the length at l =.. = a The endpoint is the highest point on the curved portion. E (l) =. l Module Lesson 1 r Houghton Mifflin Harcourt Publishing Compan Image Credits: Tusumaru/Shutterstock PAGE 1 QUESTIONING STRATEGIES For a function in the form = a, where a, what is the relationship between the value of a and the graph of the function? The value of a determines the direction and shape of the parabola. If a >, the parabola opens upward; if a <, the parabola opens downward. If a > 1, the parabola is narrower than the graph of = ; if a < 1, the parabola is wider than the graph of =. Understanding Quadratic Functions

5 JOURNAL In their journals, have students eplain how the graphs of = 5 and = _ 5 compare to the graph of =. PAGE 15 H.O.T. Focus on Higher Order Thinking. Multipart Classification f() g() - Mark the following statements about ƒ () = and g () = a as true or. a. a > 1 b. a < c. a > d. a < e. a < 1 true true f. The graphs of ƒ () and g () share a verte. true g. The graphs share an ais of smmetr. true h. The graphs share a minimum. i. The graphs share a maimum. 3. Check for Reasonableness The graph of g () = a is a parabola that passes through the point (-, ). Kle sas the value of a must be - _. Eplain wh this value of a is not reasonable. When a is negative, the values cannot be positive. Since the value is positive, a must be positive. Houghton Mifflin Harcourt Publishing Compan. Communicate Mathematical Ideas Eplain how ou know, without graphing, what the graph of g () = 1 1 looks like. Compared to the graph of the parent function, f () =, the graph of g () would be wider (verticall compressed). It will open upwards because a is positive. 5. Critical Thinking A quadratic function has a minimum value when the function s graph opens upward, and it has a maimum value when the function s graph opens downward. In each case, the minimum or maimum value is the -coordinate of the verte of the function s graph. What can ou sa about a when the function ƒ () = a has a minimum value? A maimum value? What is the minimum or maimum value in each case? When f () has a minimum value, it means a >. When it has a maimum value, a <. In either case, the minimum or the maimum value will be. Module 3 Lesson 1 3 Lesson.1