BIG IDEAS MATH. Ron Larson Laurie Boswell. A Common Core Curriculum

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1 BIG IDEAS MATH Ron Larson Laurie Boswell A Common Core Curriculum HS_Algebra_cover_pe.indd // :7 PM

2 Introducing A New Common Core High School Series b Ron Larson and Laurie Boswell Big Ideas Math is pleased to introduce a new high school program, Algebra, Geometr, and Algebra, developed for the Common Core State Standards and using the Standards for Mathematical Practice as its foundation. Big Ideas Math has been sstematicall developed using learning and instructional theor to ensure the qualit of instruction. Students gain a deeper understanding of math concepts b narrowing their focus to fewer topics at each grade level. Students master content through inductive reasoning opportunities, engaging activities that provide deeper understanding, concise stepped-out eamples, rich thought-provoking eercises, and a continual development of what has been previousl taught. The unique teaching editions provide teachers with complete instructional support from master teacher Laurie Boswell.

3 About the Authors Dr. Larson and Dr. Boswell began writing together in 99. Since that time the have authored over two dozen tetbooks. In their collaboration, Ron is primaril responsible for the pupil edition while Laurie is primaril responsible for the teaching edition of each tet. Ron Larson, Ph.D. is well known as the lead author of a comprehensive program for mathematics that spans middle school, high school, and college courses. He is a professor of mathematics at Penn State Erie, the Behrend College, where he has taught since receiving his Ph.D. in mathematics from the Universit of Colorado. Dr. Larson s numerous professional activities keep him in constant touch with the needs of students, teachers, and supervisors. Laurie Boswell, Ed.D. is the Head of School and a mathematics teacher at the Riverside School in Lndondale, Vermont. She received her Ed.D. from the Universit of Vermont. Dr. Boswell is a recipient of the Presidential Award for Ecellence in Mathematics Teaching and has taught math to students at all levels from elementar through college. Dr. Boswell was a Tand Technolog Scholar and served on the NCTM Board of Directors from 00 to 005. She currentl serves on the board of NSCM and is a popular national speaker. Copright Big Ideas Learning, LLC. All rights reserved.

4 For the Student Welcome to the Big Ideas Math High School Series.. From start to finish, this program was designed with ou, the learner, in mind. As ou work through this chapter, ou will be encouraged to think and to make conjectures while ou persevere through challenging problems and eercises. You will make errors and that is ok! Learning occurs and understanding develops when ou make errors and push through mental roadblocks to comprehend and solve new and challenging problems. In this program, ou will also be required to talk about and to eplain our thinking and analsis of problems and eercises. Being activel involved in learning will help ou develop mathematical reasoning and to use it in solving math and other everda challenges. We wish ou the best of luck as ou eplore this chapter and as ou work through the remainder of our math course. We are ecited to be a part of our preparation for the challenges ou will face in the remainder of our high school career and beond. Monke Business Images/Shutterstock.com Copright Big Ideas Learning, LLC. All rights reserved.

5 Table of Contents Algebra Chapter Solving Linear Equations Chapter Solving Linear Inequalities Chapter Graphing Linear Functions Chapter Writing Linear Functions Chapter 5 Solving Sstems of Linear Equations Chapter Eponential Functions and Sequences Chapter 7 Chapter 8 Chapter 9 Polnomial Equations and Factoring Graphing Quadratic Functions Solving Quadratic Equations Chapter 0 Radical Functions and Equations Chapter Data Analsis and Displas Geometr Chapter Basics of Geometr Chapter Reasoning and Proofs Chapter Parallel and Perpendicular Lines Chapter Transformations Chapter 5 Congruent Triangles Chapter Relationships within Triangles Chapter 7 Chapter 8 Chapter 9 Quadrilaterals and Other Polgons Similarit Chapter 0 Circles Right Triangles and Trigonometr Chapter Circumference, Area, and Volume Algebra Chapter Linear Functions Chapter Quadratic Functions Chapter Quadratic Equations and Comple Numbers Chapter Polnomial Functions Chapter 5 Rational Eponents and Radical Functions Chapter Eponential and Logarithmic Functions Chapter 7 Chapter 8 Chapter 9 Rational Functions Sequences and Series Chapter 0 Probabilit Trigonometric Ratios and Functions Chapter Data Analsis and Statistics Copright Big Ideas Learning, LLC. All rights reserved.

6 Chapter Quadratic Functions Copright Big Ideas Learning, LLC. All rights reserved.

7 Quadratic Functions. Transformations of Quadratic Functions. Characteristics of Quadratic Functions. Focus of a Parabola. Modeling with Quadratic Functions Meteorologist (p. 77) Electricit Generating Dish (p. 7) Gateshead Millennium Bridge (p. ) Soccer (p. ) Kangaroo (p. 5) Mathematical Practices: Mathematicall proficient students can appl the mathematics the know to solve problems arising in everda life, societ, and the workplace. Copright Big Ideas Learning, LLC. All rights reserved.

8 Maintaining Mathematical Proficienc Finding -intercepts (HSF-IF.C.7a) Eample Find the -intercept of the graph of the linear equation =. = Write the equation. 0 = Substitute 0 for. = Add to each side. = Divide each side b. The -intercept is. Find the -intercept of the graph of the linear equation.. = + 7. = + 8. = 0. = ( 5) 5. = ( + 0). + = The Distance Formula (8.G.B.8) The distance d between an two points (, ) and (, ) is given b the formula d = ( ) + ( ). Eample Find the distance between (, ) and (, ). Let (, ) = (, ) and (, ) = (, ). d = ( ) + ( ) Write the Distance Formula. = ( ) + ( ) Substitute. = ( ) + Simplif. = + Evaluate powers. = 0 Add..7 Use a calculator. Find the distance between the two points. 7. (, 5), (, 7) 8. (, 0), ( 8, ) 9. (, 0), (5, 9) 0. (7, ), ( 5, 0). (, 8), (, ). (0, 9), (, ). ABSTRACT REASONING Use the Distance Formula to write an epression for the distance between the two points (a, c) and (b, c). Is there an easier wa to find the distance when the -coordinates are equal? Eplain our reasoning. Copright Big Ideas Learning, LLC. All rights reserved. 5

9 Mathematical Practices Mathematicall profi cient students distinguish correct logic or reasoning from that which is fl awed. (MP) Using Correct Logic Core Concept Deductive Reasoning In deductive reasoning, ou start with two or more statements that ou know or assume to be true. From these, ou deduce or infer the truth of another statement. Here is an eample.. Premise: If this traffic does not clear, then I will be late for work.. Premise: The traffic has not cleared.. Conclusion: I will be late for work. This pattern for deductive reasoning is called a sllogism. Recognizing Flawed Reasoning The sllogisms below represent common tpes of fl awed reasoning. Eplain wh each conclusion is not valid. a. When it rains, the ground gets wet. The ground is wet. Therefore, it must have rained. c. Police, schools, and roads are necessar. Taes fund police, schools, and roads. Therefore, taes are necessar. b. When it rains, the ground gets wet. It is not raining. Therefore, the ground is not wet. d. All students use cell phones. M uncle uses a cell phone. Therefore, m uncle is a student. SOLUTION a. The ground ma be wet for another reason. b. The ground ma still be wet when the rain stops. c. The services could be funded another wa. d. People other than students use cell phones. Monitoring Progress Decide whether the sllogism represents correct or flawed reasoning. If flawed, eplain wh the conclusion is not valid.. All mammals are warm-blooded. All dogs are mammals. Therefore, all dogs are warm-blooded.. All mammals are warm-blooded. M pet is warm-blooded. Therefore, m pet is a mammal.. If I am sick, then I will miss school. I missed school. Therefore, I am sick.. If I am sick, then I will miss school. I did not miss school. Therefore, I am not sick. Chapter Quadratic Functions Copright Big Ideas Learning, LLC. All rights reserved.

10 . Transformations of Quadratic Functions COMMON CORE Learning Standards HSF-IF.C.7c HSF-BF.B. Essential Question How do the constants a, h, and k affect the graph of the quadratic function g() = a( h) + k? The parent function of the quadratic famil is f() =. A transformation of the graph of the parent function is represented b the function g() = a( h) + k, where a 0. Identifing Graphs of Quadratic Functions Work with a partner. Match the quadratic function with its graph. Eplain our reasoning. Then use a graphing calculator to verif that our answers are correct. a. g() = ( ) b. g() = ( ) + c. g() = ( + ) d. g() = 0.5( ) e. g() = ( ) f. g() = ( + ) + A. B. C. D. E. F. LOOKING FOR STRUCTURE To be proficient in math, ou need to look closel to discern a pattern or structure. Communicate Your Answer. How do the constants a, h, and k affect the graph of the quadratic function g() = a( h) + k?. Write the equation of the quadratic function whose graph is shown at the right. Eplain our reasoning. Then use a graphing calculator to verif that our equation is correct. Copright Big Ideas Learning, LLC. All rights reserved. Section. Transformations of Quadratic Functions 7

11 . Lesson What You Will Learn Core Vocabular quadratic function, p. 8 parabola, p. 8 verte of a parabola, p. 50 verte form, p. 50 Previous transformations Describe transformations of quadratic functions. Write transformations of quadratic functions. Describing Transformations of Quadratic Functions A quadratic function is a function that can be written in the form f() = a( h) + k, where a 0. The graph of a quadratic function is called a parabola. In Section., ou graphed quadratic functions using tables of values. You can also graph quadratic functions b appling transformations to the graph of the parent function f() =. Core Concept Horizontal Translations f() = f( h) = ( h) = ( h), h < 0 = Vertical Translations f() = f() + k = + k = + k, k > 0 = = ( h), h > 0 shifts left when h < 0 shifts right when h > 0 = + k, k < 0 shifts down when k < 0 shifts up when k > 0 Translations of a Quadratic Function Describe the transformation of f() = represented b g() = ( + ). Then graph each function. SOLUTION Notice that the function is of the form g() = ( h) + k. Rewrite the function to identif h and k. g() = ( ( )) + ( ) h Because h = and k =, the graph of g is a translation units left and unit down of the graph of f. k g f Monitoring Progress Help in English and Spanish at BigIdeasMath.com Describe the transformation of f() = represented b g. Then graph each function.. g() = ( ). g() = ( ). g() = ( + 5) + 8 Chapter Quadratic Functions Copright Big Ideas Learning, LLC. All rights reserved.

12 Core Concept Reflections in the -ais f() = f() = ( ) = = Reflections in the -ais f() = f( ) = ( ) = = flips over the -ais = Horizontal Stretches and Shrinks f() = f(a) = (a) = (a), a > = = is its own reflection in the -ais. Vertical Stretches and Shrinks f() = a f() = a = a, a > = = (a), 0 < a < = a, 0 < a < horizontal stretch (awa from -ais) when 0 < a < horizontal shrink (toward -ais) when a > vertical stretch (awa from -ais) when a > vertical shrink (toward -ais) when 0 < a < Transformations of Quadratic Functions MAKING USE OF STRUCTURE In Eample b, notice that g() = +. So, ou can also describe the graph of g as a vertical stretch b a factor of followed b a translation unit up of the graph of f. Describe the transformation of f() = represented b g. Then graph each function. a. g() = b. g() = () + SOLUTION a. Notice that the function is of the form g() = a, where a =. So, the graph of g is a reflection in the -ais and a vertical shrink b a factor of of the graph of f. f g b. Notice that the function is of the form g() = (a) + k, where a = and k =. So, the graph of g is a horizontal shrink b a factor of followed b a translation unit up of the graph of f. 5 g f Copright Big Ideas Learning, LLC. All rights reserved. Section. Transformations of Quadratic Functions 9

13 Monitoring Progress Help in English and Spanish at BigIdeasMath.com Describe the transformation of f() = represented b g. Then graph each function.. g() = ( ) 5. g() = ( ). g() = ( + ) + Writing Transformations of Quadratic Functions The lowest point on a parabola that opens up or the highest point on a parabola that opens down is the verte. The verte form of a quadratic function is f() = a( h) + k, where a 0 and the verte is (h, k). a indicates a reflection in the -ais and/or a vertical stretch or shrink. f() = a( h) + k h indicates a horizontal translation. k indicates a vertical translation. Writing a Transformed Quadratic Function Let the graph of g be a vertical stretch b a factor of and a reflection in the -ais, followed b a translation units down of the graph of f() =. Write a rule for g and identif the verte. SOLUTION Method Identif how the transformations affect the constants in verte form. reflection in -ais vertical stretch b a = translation units down} k = Write the transformed function. g() = a( h) + k Verte form of a quadratic function = ( 0) + ( ) Substitute for a, 0 for h, and for k. = Simplif. The transformed function is g() =. The verte is (0, ). Method Begin with the parent function and appl the transformations one at a time in the stated order. Check 0 f First write a function h that represents the reflection and vertical stretch of f. h() = f() Multipl the output b. = Substitute for f(). 5 5 Then write a function g that represents the translation of h. 0 g g() = h() Subtract from the output. = Substitute for h(). The transformed function is g() =. The verte is (0, ). 50 Chapter Quadratic Functions Copright Big Ideas Learning, LLC. All rights reserved.

14 Writing a Transformed Quadratic Function Let the graph of g be a translation units right and units up, followed b a reflection in the -ais of the graph of f() = 5. Write a rule for g. REMEMBER To multipl two binomials, use the FOIL Method. First Inner ( + )( + ) = Outer Last SOLUTION Step First write a function h that represents the translation of f. h() = f( ) + Subtract from the input. Add to the output. = ( ) 5( ) + Replace with in h(). = + Simplif. Step Then write a function g that represents the reflection of h. g() = h( ) Multipl the input b. = ( ) ( ) + Replace with in h(). = + + Simplif. 0 = Modeling with Mathematics The height h (in feet) of water spraing from a fire hose can be modeled b h() = , where is the horizontal distance (in feet) from the fire truck. The crew raises the ladder so that the water hits the ground 0 feet farther from the fire truck. Write a function that models the new path of the water. SOLUTION. Understand the Problem You are given a function that represents the path of water spraing from a fire hose. You are asked to write a function that represents the path of the water after the crew raises the ladder.. Make a Plan Analze the graph of the function to determine the translation of the ladder that causes water to travel 0 feet farther. Then write the function.. Solve the Problem Graph the transformed function. Because h(50) = 0, the water originall hits the ground 50 feet from the fire truck. The range of the function in this contet does not include negative values. However, b observing that h(0) =, ou can determine that a translation units (feet) up causes the water to travel 0 feet farther from the fire truck. g() = h() + Add to the output. = Substitute for h() and simplif. 0 X=50 Y= The new path of the water can be modeled b g() = Look Back To check that our solution is correct, verif that g(0) = 0. g(0) = 0.0(0) = = 0 Monitoring Progress Help in English and Spanish at BigIdeasMath.com 7. Let the graph of g be a vertical shrink b a factor of followed b a translation units up of the graph of f() =. Write a rule for g and identif the verte. 8. Let the graph of g be a translation units left followed b a horizontal shrink b a factor of of the graph of f() = +. Write a rule for g. 9. WHAT IF? In Eample 5, the water hits the ground 0 feet closer to the fire truck after lowering the ladder. Write a function that models the new path of the water. Copright Big Ideas Learning, LLC. All rights reserved. Section. Transformations of Quadratic Functions 5

15 . Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check. COMPLETE THE SENTENCE The graph of a quadratic function is called a(n).. VOCABULARY Identif the verte of the parabola given b f() = ( + ). Monitoring Progress and Modeling with Mathematics In Eercises, describe the transformation of f() = represented b g. Then graph each function. (See Eample.). g() =. g() = + 5. g() = ( + ). g() = ( ) 7. g() = ( ) 8. g() = ( + ) 9. g() = ( + ) 0. g() = ( 9) + 5. g() = ( 7) +. g() = ( + 0) ANALYZING RELATIONSHIPS In Eercises, match the function with the correct transformation of the graph of f. Eplain our reasoning.. = f( ). = f() + 5. = f( ) +. = f( + ) A. B. In Eercises 7, describe the transformation of f() = represented b g. Then graph each function. (See Eample.) 7. g() = 8. g() = ( ) 9. g() = 0. g() =. g() = (). g() = (). g() = 5. g() = ( ) ERROR ANALYSIS In Eercises 5 and, describe and correct the error in analzing the graph of f() = The graph is a reflection in the -ais and a vertical stretch b a factor of, followed b a translation units up of the graph of the parent quadratic function. The graph is a translation units down, followed b a vertical stretch b a factor of and a reflection in the -ais of the graph of the parent quadratic function. C. D. USING STRUCTURE In Eercises 7 0, describe the transformation of the graph of the parent quadratic function. Then identif the verte. 7. f() = ( + ) + 8. f() = ( + ) 5 9. f() = f() = ( ) 5 Chapter Quadratic Functions Copright Big Ideas Learning, LLC. All rights reserved.

16 In Eercises, write a rule for g described b the transformations of the graph of f. Then identif the verte. (See Eamples and.). f() = ; vertical stretch b a factor of and a reflection in the -ais, followed b a translation units up. f() = ; vertical shrink b a factor of and a reflection in the -ais, followed b a translation units right. f() = 8 ; horizontal stretch b a factor of and a translation units up, followed b a reflection in the -ais. f() = ( + ) + ; horizontal shrink b a factor of and a translation unit down, followed b a reflection in the -ais USING TOOLS In Eercises 5 0, match the function with its graph. Eplain our reasoning. 5. g() = ( ). g() = ( + ) 7. g() = ( ) + 8. g() = ( + ) + 9. g() = ( + ) 0. g() = ( ) + A. B. JUSTIFYING STEPS In Eercises and, justif each step in writing a function g based on the transformations of f() = +.. translation units down followed b a reflection in the -ais h() = f() = + g() = h() = ( + ) = +. reflection in the -ais followed b a translation units right h() = f( ) = ( ) + ( ) = () g() = h( ) = ( ) + ( ) = MODELING WITH MATHEMATICS The function h() = 0.0( ) + models the jump of a red kangaroo, where is the horizontal position (in feet) and h() is the height (in feet). When the kangaroo jumps from a higher location, it lands 5 feet farther awa. Write a function that models the second jump. (See Eample 5.) C. D. E. F.. MODELING WITH MATHEMATICS The function f(t) = t + 0 models the height (in feet) of an object t seconds after it is dropped from a height of 0 feet on Earth. The same object dropped from the same height on the moon is modeled b g(t) = 8 t + 0. Describe the transformation of the graph of f to obtain g. From what height must the object be dropped on the moon so it hits the ground at the same time as on Earth? Copright Big Ideas Learning, LLC. All rights reserved. Section. Transformations of Quadratic Functions 5

17 5. MODELING WITH MATHEMATICS Fling fish use their pectoral fins like airplane wings to glide through the air. a. Write an equation of the form = a( h) + k with verte (, 5) that models the flight path, assuming the fish leaves the water at (0, 0). b. What are the domain and range of the function? What do the represent in this situation? c. Does the value of a change when the flight path has verte (0, )? Justif our answer. 7. COMPARING METHODS Let the graph of g be a translation units up and unit right followed b a vertical stretch b a factor of of the graph of f() =. a. Identif the values of a, h, and k and use verte form to write the transformed function. b. Use function notation to write the transformed function. Compare this function with our function in part (a). c. Suppose the vertical stretch was performed first, followed b the translations. Repeat parts (a) and (b). d. Which method do ou prefer when writing a transformed function? Eplain.. HOW DO YOU SEE IT? Describe the graph of g as a transformation of the graph of f() =. g 5 f 8. THOUGHT PROVOKING A jump on a pogo stick with a conventional spring can be modeled b f() = 0.5( ) + 8, where is the horizontal distance (in inches) and f() is the vertical distance (in inches). Write at least one transformation of the function and provide a possible reason for our transformation. 9. MATHEMATICAL CONNECTIONS The area of a circle as the radius r changes is shown in the graph. Describe two different transformations of the graph that model the area of the circle if the area is doubled. Area (square feet) Circle 9 A 8 7 A = πr r Radius (feet) Maintaining Mathematical Proficienc Reviewing what ou learned in previous grades and lessons A line of smmetr for the figure is shown in red. Find the coordinates of point A. (Skills Review Handbook) 50. (, ) = 5. (0, ) A 5. A = A = (, ) 5 Chapter Quadratic Functions Copright Big Ideas Learning, LLC. All rights reserved.

18 . Characteristics of Quadratic Functions COMMON CORE Learning Standards HSF-IF.B. HSF-IF.C.7c HSF-IF.C.9 HSA-APR.B. Essential Question What tpe of smmetr does the graph of f() = a( h) + k have and how can ou describe this smmetr? Parabolas and Smmetr Work with a partner. a. Complete the table. Then use the values in the table to sketch the graph of the function f() = on graph paper f() 5 5 f() b. Use the results in part (a) to identif the verte of the parabola. c. Find a vertical line on our graph paper so that when ou fold the paper, the left portion of the graph coincides with the right portion of the graph. What is the equation of this line? How does it relate to the verte? d. Show that the verte form f() = ( ) is equivalent to the function given in part (a) ATTENDING TO PRECISION To be proficient in math, ou need to use clear definitions in our reasoning and discussions with others. Parabolas and Smmetr Work with a partner. Repeat Eploration for the function given b f() = + + = ( ) +. Communicate Your Answer. What tpe of smmetr does the graph of the parabola f() = a( h) + k have and how can ou describe this smmetr?. Describe the smmetr of each graph. Then use a graphing calculator to verif our answer. a. f() = ( ) + b. f() = ( + ) c. f() = ( ) + d. f() = ( + ) e. f() = + f. f() = ( 5) + Copright Big Ideas Learning, LLC. All rights reserved. Section. Characteristics of Quadratic Functions 55

19 . Lesson What You Will Learn Core Vocabular ais of smmetr, p. 5 standard form, p. 5 maimum value, p. 58 minimum value, p. 58 intercept form, p. 59 Previous -intercept Eplore properties of parabolas. Find maimum and minimum values of quadratic functions. Graph quadratic functions using -intercepts. Solve real-life problems. Eploring Properties of Parabolas An ais of smmetr is a line that divides a parabola into mirror images and passes through the verte. Because the verte of f() = a( h) + k is (h, k), the ais of smmetr is the vertical line = h. Previousl, ou used transformations to graph quadratic functions in verte form. You can also use the ais of smmetr and the verte to graph quadratic functions written in verte form. (h, k) = h Using Smmetr to Graph Quadratic Functions Graph f() = ( + ) +. Label the verte and ais of smmetr. SOLUTION Step Identif the constants a =, h =, and k =. Step Plot the verte (h, k) = (, ) and draw the ais of smmetr =. Step Evaluate the function for two values of. (, ) 5 = : f( ) = ( + ) + = = : f( ) = ( + ) + = 5 Plot the points (, ), (, ), and their reflections in the ais of smmetr. Step Draw a parabola through the plotted points. = Quadratic functions can also be written in standard form, f() = a + b + c, where a 0. You can derive standard form b epanding verte form. f() = a( h) + k Verte form f() = a( h + h ) + k Epand ( h). f() = a ah + ah + k Distributive Propert f() = a + ( ah) + (ah + k) Group like terms. f() = a + b + c Let b = ah and let c = ah + k. This allows ou to make the following observations. a = a: So, a has the same meaning in verte form and standard form. b = ah: Solve for h to obtain h = b a. So, the ais of smmetr is = b a. c = ah + k: In verte form f() = a( h) + k, notice that f(0) = ah + k. So, c is the -intercept. 5 Chapter Quadratic Functions Copright Big Ideas Learning, LLC. All rights reserved.

20 Core Concept Properties of the Graph of f() = a + b + c = a + b + c, a > 0 = a + b + c, a < 0 (0, c) = b a b = a (0, c) The parabola opens up when a > 0 and opens down when a < 0. The graph is narrower than the graph of f() = when a > and wider when a <. The ais of smmetr is = b a ( and the verte is b a (, f a) b ). The -intercept is c. So, the point (0, c) is on the parabola. Graphing a Quadratic Function in Standard Form COMMON ERROR Be sure to include the negative sign when writing the epression for the -coordinate of the verte. Graph f() = +. Label the verte and ais of smmetr. SOLUTION Step Identif the coefficients a =, b =, and c =. Because a > 0, the parabola opens up. Step Find the verte. First calculate the -coordinate. = b a = () = Then find the -coordinate of the verte. f() = () () + = So, the verte is (, ). Plot this point. Step Draw the ais of smmetr =. Step Identif the -intercept c, which is. Plot the point (0, ) and its reflection in the ais of smmetr, (, ). Step 5 Evaluate the function for another value of, such as =. f() = () () + = 0 Plot the point (, 0) and its reflection in the ais of smmetr, (, 0). Step Draw a parabola through the plotted points. 9 5 (, )) = Monitoring Progress Help in English and Spanish at BigIdeasMath.com Graph the function. Label the verte and ais of smmetr.. f() = ( + ). g() = ( ) + 5. h() = +. p() = 8 + Copright Big Ideas Learning, LLC. All rights reserved. Section. Characteristics of Quadratic Functions 57

21 STUDY TIP When a function f is written in verte form, ou can use h = b a and k = f ( a) b to state the properties shown. Finding Maimum and Minimum Values Because the verte is the highest or lowest point on a parabola, its -coordinate is the maimum value or minimum value of the function. The verte lies on the ais of smmetr, so the function is increasing on one side of the ais of smmetr and decreasing on the other side. Core Concept Maimum and Minimum Values For the quadratic function f() = a + b + c, the -coordinate of the verte is the minimum value of the function when a > 0 and the maimum value when a < 0. a > 0 a < 0 decreasing increasing minimum = b a Minimum value: f ( a) b Domain: All real numbers Range: f ( a) b increasing = b a maimum decreasing Maimum value: f ( a) b Domain: All real numbers Range: f ( a) b Decreasing to the left of = b a Increasing to the right of = b a Increasing to the left of = b a Decreasing to the right of = b a Check 0 0 Minimum X= Y=- 0 0 Finding a Minimum or a Maimum Value Find the minimum value or maimum value of f() =. Describe the domain and range of the function, and where the function is increasing and decreasing. SOLUTION Identif the coefficients a =, b =, and c =. Because a > 0, the parabola opens up and the function has a minimum value. To find the minimum value, calculate the coordinates of the verte. = b a = = f() = ( () () = ) The minimum value is. So, the domain is all real numbers and the range is. The function is decreasing to the left of = and increasing to the right of =. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 5. Find the minimum value or maimum value of (a) f() = + and (b) h() = Describe the domain and range of each function, and where each function is increasing and decreasing. 58 Chapter Quadratic Functions Copright Big Ideas Learning, LLC. All rights reserved.

22 REMEMBER An -intercept of a graph is the -coordinate of a point where the graph intersects the -ais. It occurs where f() = 0. COMMON ERROR Remember that the -intercepts of the graph of f() = a( p)( q) are p and q, not p and q. Graphing Quadratic Functions Using -Intercepts When the graph of a quadratic function has at least one -intercept, the function can be written in intercept form, f() = a( p)( q), where a 0. Core Concept Properties of the Graph of f() = a( p)( q) Because f(p) = 0 and f(q) = 0, p and = q are the -intercepts of the graph of the function. The ais of smmetr is halfwa between (p, 0) and (q, 0). So, the ais of smmetr is = p + q. The parabola opens up when a > 0 and opens down when a < 0. Graphing a Quadratic Function in Intercept Form Graph f() = ( + )( ). Label the -intercepts, verte, and ais of smmetr. SOLUTION Step Identif the -intercepts. The -intercepts are p = and q =, so the parabola passes through the points (, 0) and (, 0). Step Find the coordinates of the verte. = p + q = + = f( ) = ( + )( ) = 8 So, the ais of smmetr is = and the verte is (, 8). Step Draw a parabola through the verte and the points where the -intercepts occur. (p, 0) p + q (q, 0) (, 8) (, 0) 5 = a( p)( q) 9 5 = (, 0) Check You can check our answer b generating a table of values for f on a graphing calculator. -intercept -intercept X X=- Y The values show smmetr about =. So, the verte is (, 8). Monitoring Progress Help in English and Spanish at BigIdeasMath.com Graph the function. Label the -intercepts, verte, and ais of smmetr.. f() = ( + )( + 5) 7. g() = ( )( ) Copright Big Ideas Learning, LLC. All rights reserved. Section. Characteristics of Quadratic Functions 59

23 Solving Real-Life Problems Modeling with Mathematics (50, 5) The parabola shows the path of our first golf shot, where is the horizontal distance (in ards) and is the corresponding height (in ards). The path of our second shot can be modeled b the function f() = 0.0( 80). Which shot travels farther before hitting the ground? Which travels higher? SOLUTION (0, 0) (00, 0). Understand the Problem You are given a graph and a function that represent the paths of two golf shots. You are asked to determine which shot travels farther before hitting the ground and which shot travels higher.. Make a Plan Determine how far each shot travels b interpreting the -intercepts. Determine how high each shot travels b finding the maimum value of each function. Then compare the values.. Solve the Problem First shot: The graph shows that the -intercepts are 0 and 00. So, the ball travels 00 ards before hitting the ground. 5 d 00 d Because the ais of smmetr is halfwa between (0, 0) and (00, 0), the ais of smmetr is = = 50. So, the verte is (50, 5) and the maimum height is 5 ards. Second shot: B rewriting the function in intercept form as f() = 0.0( 0)( 80), ou can see that p = 0 and q = 80. So, the ball travels 80 ards before hitting the ground. To find the maimum height, find the coordinates of the verte. = p + q = = 0 f(0) = 0.0(0)(0 80) = 0 = 5 The maimum height of the second shot is ards. Because 00 ards > 80 ards, the first shot travels farther. Because ards > 5 ards, the second shot travels higher. 0 0 f 90. Look Back To check that the second shot travels higher, graph the function representing the path of the second shot and the line = 5, which represents the maimum height of the first shot. The graph rises above = 5, so the second shot travels higher. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 8. WHAT IF? The graph of our third shot is a parabola through the origin that reaches a maimum height of 8 ards when = 5. Compare the distance it travels before it hits the ground with the first two shots. 0 Chapter Quadratic Functions Copright Big Ideas Learning, LLC. All rights reserved.

24 . Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check. WRITING Eplain how to determine whether a quadratic function will have a minimum value or a maimum value.. WHICH ONE DOESN T BELONG? The graph of which function does not belong with the other three? Eplain. f() = + f() = + f() = ( )( + ) f() = ( + ) 7 Monitoring Progress and Modeling with Mathematics In Eercises, graph the function. Label the verte and ais of smmetr. (See Eample.). f() = ( ). h() = ( + ) 5. g() = ( + ) + 5. = ( 7) 7. = ( ) + 8. g() = ( + ) 9. f() = ( ) 5 0. h() = ( + ) + REASONING In Eercises 9 and 0, use the ais of smmetr to plot the reflection of each point and complete the parabola (, ) (, ) 5 (0, ) = = (, ) 7 5 (, ) (, ). = ( + ) +. = ( ) +. f() = 0.( ). g() = ANALYZING RELATIONSHIPS In Eercises 5 8, use the ais of smmetr to match the equation with its graph. 5. = ( ) +. = ( + ) 7. = ( + ) + 8. = ( ) A. C. 5 5 = = 5 B. D. = 5 5 = In Eercises 0, graph the function. Label the verte and ais of smmetr. (See Eample.). = + +. = +. = f() = + 5. g() =. f() = 5 7. g() = f() = = + 0. = 5. WRITING Two quadratic functions have graphs with vertices (, ) and (, ). Eplain wh ou can not use the aes of smmetr to distinguish between the two functions.. WRITING A quadratic function is increasing to the left of = and decreasing to the right of =. Will the verte be the highest or lowest point on the graph of the parabola? Eplain. Copright Big Ideas Learning, LLC. All rights reserved. Section. Characteristics of Quadratic Functions

25 ERROR ANALYSIS In Eercises and, describe and correct the error in analzing the graph of = The -coordinate of the verte is = b a = () =. In Eercises 9 8, find the minimum or maimum value of the function. Describe the domain and range of the function, and where the function is increasing and decreasing. (See Eample.) 9. = 0. = =. g() = + 5. The -intercept of the graph is the value of c, which is 7.. f() = g() = h() =. h() = MODELING WITH MATHEMATICS In Eercises 5 and, is the horizontal distance (in feet) and is the vertical distance (in feet). Find and interpret the coordinates of the verte. 5. The path of a basketball thrown at an angle of 5 can be modeled b = The path of a shot put released at an angle of 5 can be modeled b = = + 8. f() = PROBLEM SOLVING The path of a diver is modeled b the function f() = , where f() is the height of the diver (in meters) above the water and is the horizontal distance (in meters) from the end of the diving board. a. What is the height of the diving board? b. What is the maimum height of the diver? c. Describe where the diver is ascending and where the diver is descending ANALYZING EQUATIONS The graph of which function has the same ais of smmetr as the graph of = + +? A = + + B = + C = + D = USING STRUCTURE Which function represents the parabola with the widest graph? Eplain our reasoning. A = ( + ) B = 5 C = 0.5( ) + D = PROBLEM SOLVING The engine torque (in foot-pounds) of one model of car is given b = , where is the speed of the engine (in thousands of revolutions per minute). a. Find the engine speed that maimizes torque. What is the maimum torque? b. Eplain what happens to the engine torque as the speed of the engine increases. MATHEMATICAL CONNECTIONS In Eercises 5 and 5, write an equation for the area of the figure. Then determine the maimum possible area of the figure w b w b Chapter Quadratic Functions Copright Big Ideas Learning, LLC. All rights reserved.

26 In Eercises 5 0, graph the function. Label the -intercept(s), verte, and ais of smmetr. (See Eample.) 5. = ( + )( ) 5. = ( + )( ) 55. = ( + )( + ) 5. f() = ( 5)( ) 57. g() = ( + ) 58. = ( + 7) 59. f() = ( ) 0. = ( 7) USING TOOLS In Eercises, identif the -intercepts of the function and describe where the graph is increasing and decreasing. Use a graphing calculator to verif our answer.. f() = ( )( + ). = ( + )( ). g() = ( )( ). h() = 5( + 5)( + ) 5. MODELING WITH MATHEMATICS A soccer plaer kicks a ball down field. The height of the ball increases until it reaches a maimum height of 8 ards, 0 ards awa from the plaer. A second kick is modeled b = ( ). Which kick travels farther before hitting the ground? Which kick travels higher? (See Eample 5.). MODELING WITH MATHEMATICS Although a football field appears to be flat, some are actuall shaped like a parabola so that rain runs off to both sides. The cross section of a field can be modeled b = 0.000( 0), where and are measured in feet. What is the width of the field? What is the maimum height of the surface of the field? 8. OPEN-ENDED Write two different quadratic functions in intercept form whose graphs have the ais of smmetr =. 9. PROBLEM SOLVING An online music store sells about 000 songs each da when it charges $ per song. For each $.05 increase in price, about 80 fewer songs per da are sold. Use the verbal model and quadratic function to determine how much the store should charge per song to maimize dail revenue. Revenue (dollars) = Price (dollars/song) Sales (songs) R() = ( ) (000 80) 70. PROBLEM SOLVING An electronics store sells 70 digital cameras per month at a price of $0 each. For each $0 decrease in price, about 5 more cameras per month are sold. Use the verbal model and quadratic function to determine how much the store should charge per camera to maimize monthl revenue. Revenue (dollars) = Price (dollars/camera) R() = (0 0) Sales (cameras) (70 + 5) 7. DRAWING CONCLUSIONS Compare the graphs of the three quadratic functions. What do ou notice? Rewrite the functions f and g in standard form to justif our answer. f() = ( + )( + ) g() = ( + ) h() = USING STRUCTURE Write the quadratic function f() = + in intercept form. Graph the function. Label the -intercepts, -intercept, verte, and ais of smmetr. surface of football field Not drawn to scale 7. PROBLEM SOLVING A woodland jumping mouse hops along a parabolic path given b = 0. +., where is the mouse s horizontal position (in feet) and is the corresponding height (in feet). Can the mouse jump over a fence that is feet high? Justif our answer. 7. REASONING The points (, ) and (, ) lie on the graph of a quadratic function. Determine whether ou can use these points to find the ais of smmetr. If not, eplain. If so, write the equation of the ais of smmetr. Not drawn to scale Copright Big Ideas Learning, LLC. All rights reserved. Section. Characteristics of Quadratic Functions

27 7. HOW DO YOU SEE IT? Consider the graph of the function f() = a( p)( q). 77. MAKING AN ARGUMENT The point (, 5) lies on the graph of a quadratic function with ais of smmetr =. Your friend sas the verte could be the point (0, 5). Is our friend correct? Eplain. 78. CRITICAL THINKING Find the -intercept in terms of a, p, and q for the quadratic function f() = a( p)( q). a. What does f ( p + q ) represent in the graph? b. If a < 0, how does our answer in part (a) change? Eplain. 79. MODELING WITH MATHEMATICS A kernel of popcorn contains water that epands when the kernel is heated, causing it to pop. The equations below represent the popping volume (in cubic centimeters per gram) of popcorn with moisture content (as a percent of the popcorn s weight). Hot-air popping: = 0.7( 5.5)(.) Hot-oil popping: = 0.5( 5.5)(.8) 75. MODELING WITH MATHEMATICS The Gateshead Millennium Bridge spans the River Tne. The arch of the bridge can be modeled b a parabola. The arch reaches a maimum height of 50 meters at a point roughl meters across the river. Graph the curve of the arch. What are the domain and range? What do the represent in this situation? 7. THOUGHT PROVOKING You have 00 feet of fencing to enclose a garden. Draw three possible designs for the garden. Of these, which has the greatest area? Of all possible rectangular gardens, which has the greatest area? Eplain our reasoning. a. For hot-air popping, what moisture content maimizes popping volume? What is the maimum volume? b. For hot-oil popping, what moisture content maimizes popping volume? What is the maimum volume? c. Use a graphing calculator to graph both functions in the same coordinate plane. What are the domain and range of each function in this situation? Eplain. 80. ABSTRACT REASONING A function is written in intercept form with a > 0. If the -intercepts of the graph do not change, what happens to the verte of the graph as a increases? as a approaches 0? Maintaining Mathematical Proficienc Solve the equation. Check for etraneous solutions. (Skills Review Handbook) 8. = 0 8. = = = + Solve the proportion. (Skills Review Handbook) Reviewing what ou learned in previous grades and lessons 85. = 8. = = = 0 Chapter Quadratic Functions Copright Big Ideas Learning, LLC. All rights reserved.

28 .. What Did You Learn? Core Vocabular quadratic function, p. 8 standard form, p. 5 parabola, p. 8 minimum value, p. 58 verte of a parabola, p. 50 maimum value, p. 58 verte form, p. 50 intercept form, p. 59 ais of smmetr, p. 5 Core Concepts Section. Horizontal Translations, p. 8 Reflections in the -ais, p. 9 Vertical Translations, p. 8 Horizontal Stretches and Shrinks, p. 9 Reflections in the -ais, p. 9 Vertical Stretches and Shrinks, p. 9 Section. Properties of the Graph of f() = a + b + c, Properties of the Graph of f() = a( p)( q), p. 57 p. 59 Minimum and Maimum Values, p. 58 Mathematical Practices. Wh does the height ou found in Eercise on page 5 make sense in the contet of the situation?. How can ou effectivel communicate our preference in methods to others in Eercise 7 on page 5?. How can ou use technolog to deepen our understanding of the concepts in Eercise 79 on page? Stud Skills Reworking Your Notes It is impossible to write down in our notes all the detailed information ou are taught in class. A good wa to reinforce the concepts and put them in our long-term memor is to rework our notes. Leave etra space when ou take notes. Go back after class and fill in: important definitions and rules additional eamples questions ou have about the material Copright Big Ideas Learning, LLC. All rights reserved. 5

29 .. Quiz Describe the transformation of f() = represented b g. (Section.). 5 f g. g f. g f Write a rule for g and identif the verte. (Section.). Let g be a translation units up followed b a reflection in the -ais and a vertical stretch b a factor of of the graph of f() =. 5. Let g be a translation unit left and units down, followed b a vertical shrink b a factor of of the graph of f() = ( + ).. Let g be a horizontal shrink b a factor of, followed b a translation unit up and units right of the graph of f() = ( + ). Graph the function. Label the verte and ais of smmetr. (Section.) 7. f() = ( ) 5 8. h() = + 9. f() = 7 8 Find the -intercepts of the graph of the function. Then describe where the function is increasing and decreasing. (Section.) 0. g() = ( + )( + ). g() = ( 5)( + ). f() = 0.( ). A grasshopper can jump incredible distances, up to 0 times its length. The height (in inches) of the jump above the ground of a one-inch-long grasshopper is given b h() = 0 +, where is the horizontal distance (in inches) of the jump. When the grasshopper jumps off a rock, it lands on the ground inches farther. Write a function that models the new path of the jump. (Section.) (0, 0) (0, 0) Not drawn to scale. A passenger on a stranded life boat shoots a distress flare into the air. The height (in feet) of the flare above the water is given b f(t) = t(t 8), where t is time (in seconds) since the flare was shot. The passenger shoots a second flare, whose path is modeled in the graph. Which flare travels higher? Which remains in the air longer? Justif our answer. (Section.) (.5, 9) (0, 0) (7, 0) Chapter Quadratic Functions Copright Big Ideas Learning, LLC. All rights reserved.

30 . Focus of a Parabola COMMON CORE Learning Standards HSF-IF.B. HSF-IF.C.7c HSG-GPE.A. Essential Question What is the focus of a parabola? Analzing Satellite Dishes Work with a partner. Vertical ras enter a satellite dish whose cross section is a parabola. When the ras hit the parabola, the reflect at the same angle at which the entered. (See Ra in the figure.) a. Draw the reflected ras so that the intersect the -ais. b. What do the reflected ras have in common? c. The optimal location for the receiver of the satellite dish is at a point called the focus of the parabola. Determine the location of the focus. Eplain wh this makes sense in this situation. Ra Ra Ra CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, ou need to make conjectures and build logical progressions of statements to eplore the truth of our conjectures. incoming angle outgoing angle = Analzing Spotlights Work with a partner. Beams of light are coming from the bulb in a spotlight, located at the focus of the parabola. When the beams hit the parabola, the reflect at the same angle at which the hit. (See Beam in the figure.) Draw the reflected beams. What do the have in common? Would ou consider this to be the optimal result? Eplain. outgoing angle = Beam bulb incoming angle Beam Beam Communicate Your Answer. What is the focus of a parabola?. Describe some of the properties of the focus of a parabola. Copright Big Ideas Learning, LLC. All rights reserved. Section. Focus of a Parabola 7

31 . Lesson What You Will Learn Core Vocabular focus, p. 8 directri, p. 8 Previous perpendicular distance formula congruent Eplore the focus and the directri of a parabola. Write equations of parabolas. Solve real-life problems. Eploring the Focus and Directri Previousl, ou learned that the graph of a quadratic function is a parabola that opens up or down. A parabola can also be defined as the set of all points (, ) in a plane that are equidistant from a fied point called the focus and a fied line called the directri. The focus is in the interior of the parabola and lies on the ais of smmetr. The verte lies halfwa between the focus and the directri. ais of smmetr The directri is perpendicular to the ais of smmetr. STUDY TIP The distance from a point to a line is defined as the length of the perpendicular segment from the point to the line. Using the Distance Formula to Write an Equation Use the Distance Formula to write an equation of the parabola with focus F(0, ) and directri =. SOLUTION Notice the line segments drawn from point F to point P and from point P to point D. B the definition of a parabola, these line segments must be congruent. F(0, ) = D(, ) P(, ) PD = PF Definition of a parabola ( ) + ( ) = ( ) + ( ) Distance Formula ( ) + ( ( )) = ( 0) + ( ) Substitute for,,, and. ( + ) = + ( ) Simplif. ( + ) = + ( ) Square each side. + + = = Epand. Combine like terms. = 8 Divide each side b 8. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. Use the Distance Formula to write an equation of the parabola with focus F(0, ) and directri =. = D(, ) F(0, ) P(, ) 8 Chapter Quadratic Functions Copright Big Ideas Learning, LLC. All rights reserved.

32 MAKING USE OF STRUCTURE Notice that = p is of the form = a. So, changing the value of p verticall stretches or shrinks the parabola. You can derive the equation of a parabola that opens up or down with verte (0, 0), focus (0, p), and directri = p using the procedure in Eample. F(0, p) = p ( ) + ( ( p)) = ( 0) + ( p) ( + p) = + ( p) + p + p = + p + p p = = p The focus and directri each lie p units from the verte. Parabolas can also open left or right, in which case the equation has the form = p when the verte is (0, 0). Core Concept Standard Equations of a Parabola with Verte at the Origin Vertical ais of smmetr ( = 0) Equation: = p Focus: (0, p) Directri: = p D(, p) P(, ) Horizontal ais of smmetr ( = 0) focus: (0, p) directri: = p verte: (0, 0) focus: directri: (0, p) = p p > 0 p < 0 verte: (0, 0) STUDY TIP Notice that parabolas opening left or right do not represent functions. Equation: = p Focus: (p, 0) Directri: = p directri: = p focus: (0, p) verte: (0, 0) focus: (0, p) verte: (0, 0) p > 0 p < 0 directri: = p Graphing an Equation of a Parabola Identif the focus, directri, and ais of smmetr of =. Graph the equation. SOLUTION Step Rewrite the equation in standard form. = Write the original equation. (, 0) = = Divide each side b. Step Identif the focus, directri, and ais of smmetr. The equation has the form = p, where p =. The focus is (p, 0), or (, 0). The directri is = p, or =. Because is squared, the ais of smmetr is the -ais. Step Use a table of values to graph the equation. Notice that it is easier to substitute -values and solve for. Opposite -values result in the same -value. 0 ± ± ± ± Copright Big Ideas Learning, LLC. All rights reserved. Section. Focus of a Parabola 9

33 Writing Equations of Parabolas Writing an Equation of a Parabola directri Write an equation of the parabola shown. SOLUTION verte Because the verte is at the origin and the ais of smmetr is vertical, the equation has the form = p. The directri is = p =, so p =. Substitute for p to write an equation of the parabola. = ( ) = So, an equation of the parabola is =. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Identif the focus, directri, and ais of smmetr of the parabola. Then graph the equation.. = 0.5. =. = Write an equation of the parabola with verte at (0, 0) and the given directri or focus. 5. directri: =. focus: (, 0) 7. focus: ( 0, ) STUDY TIP The standard form for a vertical ais of smmetr looks like verte form. To remember the standard form for a horizontal ais of smmetr, switch and, and h and k. The verte of a parabola is not alwas at the origin. As in previous transformations, adding a value to the input or output of a function translates its graph. Core Concept Standard Equations of a Parabola with Verte at (h, k) Vertical ais of smmetr ( = h) Equation: = p ( = h h) + k (h, k + p) Focus: (h, k + p) Directri: = k p = k p Horizontal ais of smmetr ( = k) Equation: = p ( k) + h Focus: (h + p, k) Directri: = h p = k (h, k) = k p (h, k) = h (h, k + p) p > 0 p < 0 = h p (h, k) (h + p, k) = k (h + p, k) (h, k) = h p p > 0 p < 0 70 Chapter Quadratic Functions Copright Big Ideas Learning, LLC. All rights reserved.

34 8 verte focus Write an equation of the parabola shown. SOLUTION Writing an Equation of a Translated Parabola Because the verte is not at the origin and the ais of smmetr is horizontal, the equation has the form = p ( k) + h. The verte (h, k) is (, ) and the focus (h + p, k) is (0, ), so h =, k =, and p =. Substitute these values to write an equation of the parabola. = () ( ) + = ( ) + So, an equation of the parabola is = ( ) +. Solving Real-Life Problems Parabolic refl ectors have cross sections that are parabolas. Incoming sound, light, or other energ that arrives at a parabolic reflector parallel to the ais of smmetr is directed to the focus Focus Diagram Focus Diagram (Diagram ). Similarl, energ that is emitted from the focus of a parabolic reflector and then strikes the reflector is directed parallel to the ais of smmetr (Diagram ). Solving a Real-Life Problem engine An electricit generating dish uses a parabolic reflector to concentrate sunlight onto a high-frequenc engine located at the focus of the reflector. The sunlight heats helium to 50 C to power the engine. Write an equation that represents the cross section of the dish shown with its verte at (0, 0). What is the depth of the dish? 8.5 m.5 m SOLUTION Because the verte is at the origin, and the ais of smmetr is vertical, the equation has the form = p. The engine is at the focus, which is.5 meters above the verte. So, p =.5. Substitute.5 for p to write the equation. = (.5) = 8 The depth of the dish is the -value at the dish s outside edge. The dish etends 8.5 =.5 meters to either side of the verte (0, 0), so find when =.5. = 8 (.5) The depth of the dish is about meter. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 8. Write an equation of a parabola with verte (, ) and focus (, ). 9. A parabolic microwave antenna is feet in diameter. Write an equation that represents the cross section of the antenna with its verte at (0, 0) and its focus 0 feet to the right of the verte. What is the depth of the antenna? Copright Big Ideas Learning, LLC. All rights reserved. Section. Focus of a Parabola 7

35 . Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check. COMPLETE THE SENTENCE A parabola is the set of all points in a plane equidistant from a fied point called the and a fied line called the.. WRITING Eplain how to find the coordinates of the focus of a parabola with verte ( 0, 0 ) and directri = 5. Monitoring Progress and Modeling with Mathematics In Eercises 0, use the Distance Formula to write an equation of the parabola. (See Eample.).. F(0, ) = D(, ) P(, ) 5. focus: (0, ). directri: = 7 directri: = focus: (0, 7) 7. verte: (0, 0) 8. verte: (0, 0) directri: = focus: (0, 5) 9. verte: (0, 0) 0. verte: (0, 0) focus: (0, 0) directri: = 9 = F(0, ). ANALYZING RELATIONSHIPS Which of the given characteristics describe parabolas that open down? Eplain our reasoning. D(, ) P(, ) In Eercises 0, identif the focus, directri, and ais of smmetr of the parabola. Graph the equation. (See Eample.). = 8. = 5. = 0. = 7. = 8. = = = 0 ERROR ANALYSIS In Eercises and, describe and correct the error in graphing the parabola.. + = 0 8 (0,.5) =.5 A focus: (0, ) B focus: (0, ) directri: = directri: = C focus: (0, ) D focus: (0, ) directri: = directri: = = 0 = 0.5. REASONING Which of the following are possible coordinates of the point P in the graph below? Eplain. F(0, 9) V(0, 0) P(, ) (0.5, 0) A (, ) B (, ) C (, ) 9 ) 8 D (, ) E (, ) F (,. ANALYZING EQUATIONS The cross section (with units in inches) of a parabolic satellite dish can be modeled b the equation = 8. How far is the receiver from the verte of the cross section? Eplain. 7 Chapter Quadratic Functions Copright Big Ideas Learning, LLC. All rights reserved.

36 . ANALYZING EQUATIONS The cross section (with units in inches) of a parabolic spotlight can be modeled b the equation = 0. How far is the bulb from the verte of the cross section? Eplain. In Eercises 7 0, write an equation of the parabola shown. (See Eample.) focus verte 8 5 verte focus 5 In Eercises 5 8, write an equation of the parabola shown. (See Eample.) 5. = 8 verte directri. = = = verte directri directri verte In Eercises 9, write an equation of the parabola with the given characteristics. 9. focus: (, 0) 0. focus: (, 0 ) directri: = directri: =. directri: = 0. directri: = 8 verte: (0, 0) verte: (0, 0). focus: ( 0, 5 ). focus: ( 0, 5 ) directri: = 5 directri: = 5 5. focus: ( 0, 7 ). focus: ( 5, 0 ) verte: (0,0) verte: (0, 0) directri verte focus verte 0 verte focus In Eercises, identif the verte, focus, directri, and ais of smmetr of the parabola. Describe the transformations of the graph of the standard equation with verte (0, 0).. = 8 ( ) +. = ( + ) +. = ( ) +. = ( + ) 5 5. = ( + ) +. = ( + 5) 7. MODELING WITH MATHEMATICS Scientists studing dolphin echolocation simulate the projection of a bottlenose dolphin s clicking sounds using computer models. The models originate the sounds at the focus of a parabolic reflector. The parabola in the graph shows the cross section of the reflector with focal length of. inches and aperture width of 8 inches. Write an equation to represent the cross section of the reflector. What is the depth of the reflector? (See Eample 5.) F focal length aperture Copright Big Ideas Learning, LLC. All rights reserved. Section. Focus of a Parabola 7

37 8. MODELING WITH MATHEMATICS Solar energ can be concentrated using long troughs that have a parabolic cross section as shown in the figure. Write an equation to represent the cross section of the trough. What are the domain and range in this situation? What do the represent? 5.8 m 5. CRITICAL THINKING The distance from point P to the directri is units. Write an equation of the parabola. P(, ) V(0, 0).7 m 5. THOUGHT PROVOKING Two parabolas have the same focus (a, b) and focal length of units. Write an equation of each parabola. Identif the directri of each parabola. 9. ABSTRACT REASONING As p increases, how does the width of the graph of the equation = p change? Eplain our reasoning. 50. HOW DO YOU SEE IT? The graph shows the path of a volleball served from an initial height of feet as it travels over a net. A 5. REPEATED REASONING Use the Distance Formula to derive the equation of a parabola that opens to the right with verte (0, 0), focus (p, 0), and directri = p. = p F(p, 0) D( p, ) P(, ) B a. Label the verte, focus, and a point on the directri. C b. An underhand serve follows the same parabolic path but is hit from a height of feet. How does this affect the focus? the directri? 5. PROBLEM SOLVING The latus rectum of a parabola is the line segment that is parallel to the directri, passes through the focus, and has endpoints that lie on the parabola. Find the length of the latus rectum of the parabola shown. A latus rectum F(0, ) V(0, 0) B = Maintaining Mathematical Proficienc Reviewing what ou learned in previous grades and lessons Write an equation of the line that passes through the points. (Section.) 55. (, ), (, ) 5. (, ), (0, ) 57. (, ), ( 5, 5) 58. (, ), (0, ) Use a graphing calculator to find an equation for the line of best fit. (Section.) Chapter Quadratic Functions Copright Big Ideas Learning, LLC. All rights reserved.

38 . Modeling with Quadratic Functions COMMON CORE Learning Standards HSA-CED.A. HSF-IF.B. HSF-BF.A.a HSS-ID.B.a 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 00 and 0. 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) at time t (in seconds) of a wrench that has been dropped from a building under construction. Time, t 0 Height, h MODELING WITH 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. When does the wrench hit the ground? Eplain Communicate Your Answer. 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. Copright Big Ideas Learning, LLC. All rights reserved. Section. Modeling with Quadratic Functions 75

39 . 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 Given three points Use verte form: = a( h) + k Use intercept form: = a( p)( q) Write and solve a sstem of three equations in three variables. Writing an Equation Using a Verte and a Point Height (feet) Human Cannonball 0 (50, 5) (0,5) Horizontal distance (feet) The graph shows the parabolic path of a performer that 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 90 feet from the cannon. What is the height of the net to the nearest foot? SOLUTION From the graph, ou can see that the verte (h, k) is (50, 5) and the parabola passes through the point (0, 5). Use the verte and the point to solve for a in verte form. = a( h) + k Verte form 5 = a(0 50) + 5 Substitute for h, k,, and. 0 = 500a Simplif = a Divide each side b 500. Because a = 0.008, h = 50, and k = 5, the path can be modeled b the equation = 0.008( 50) + 5, where Find the height when = 90. = 0.008(90 50) + 5 Substitute 90 for. = 0.008(00) + 5 Simplif. =. Simplif. So, the height of the net is about feet. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. WHAT IF? The verte of the parabola is (50, 7.5). What is the height of the net to the nearest foot?. Write an equation of the parabola that passes through the point (, ) and has verte (, 9). 7 Chapter Quadratic Functions Copright Big Ideas Learning, LLC. All rights reserved.

40 Writing an Equation Using a Point and -Intercepts Temperature ( C) Temperature Forecast (0, 9.) REMEMBER (, 0) (, 0) 9 5 Hours after midnight The average rate of change of a function f from to is the slope of the segment connecting f( ) and f( ): f( ) f( ). A meteorologist creates the 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 on 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 (0, 9.). Use the -intercepts and the point to solve for a in intercept form. = a( p)( q) Intercept form 9. = a(0 )(0 ) Substitute for p, q,, and. 9. = 9a Simplif. 0. = a Divide each side b 9. Because a = 0., p =, and q =, the temperature over time can be modeled b f() = 0.( )( ), where 0. The coldest temperature is the minimum value. So, find f() when = + =. f() = 0.( )( ) Substitute for. = 0 Simplif. So, the coldest temperature is 0 C at hours after midnight, or p.m. b. The parabola opens up and the ais of smmetr is =. So, the function is decreasing on the interval 0 < < and increasing on the interval < <. Average rate of change Average rate of change over 0 < < : over < < : f() f(0) = =. f() f() = 0 ( 0) = 0 5 (0, 9.) 0 5 (, 0) (, 0) Because. >, 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. WHAT IF? The -intercept is.8. 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. Copright Big Ideas Learning, LLC. All rights reserved. Section. Modeling with Quadratic Functions 77

41 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 0 f() first differences: 5 5 second differences: Writing a Quadratic Equation Using Three Points Time, t Height, h 0, ,05 0 0,00 5,5 0,00 5,05 0,00 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 0.8 seconds, passengers begin to eperience a weightless environment. Write and evaluate a function to approimate the height at which this occurs. SOLUTION Step 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(0) h(5) h(0) h(5) h(0) h(5) h(0),900 9,05 0,00,5,00,05, 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 (0,,900): 00a + 0b + c =,900 Equation Use (0, 0,00): 00a + 0b + c = 0,00 Equation Use (0,,00): 900a + 0b + c =,00 Equation Use the elimination method to solve the sstem. Subtract Equation from Equation. Subtract Equation from Equation. 00a + 0b = 700 New Equation 800a + 0b = 500 New Equation 00a = 00 Subtract times new Equation from new Equation. a = Solve for a. b = 700 Substitute into new Equation to find b. c =,000 Substitute into Equation to find c. The data can be modeled b the function h(t) = t + 700t +,000. Step Evaluate the function when t = 0.8. h(0.8) = (0.8) + 700(0.8) +,000 = 0,800.9 Passengers begin to eperience a weightless environment at about 0,800 feet. 78 Chapter Quadratic Functions Copright Big Ideas Learning, LLC. All rights reserved.

42 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, The coefficient of determination R shows how well an equation fits a set of data. The closer R is to, 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 Enter the data in a graphing calculator using two lists and create a scatter plot. The data show a quadratic relationship Step 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. miles per gallon when driving about 8.9 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 = QuadReg =a +b+c a= b=.887 c=-7.05 R = Maimum 0 X= Y= Monitoring Progress Help in English and Spanish at BigIdeasMath.com 5. Write an equation of the parabola that passes through the points (, ), (0, ), and (, 7).. 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, Profit, 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, Maimum Weight, Copright Big Ideas Learning, LLC. All rights reserved. Section. Modeling with Quadratic Functions 79

43 . Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check. 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 0? What is the distance from f(0) to f()? What is the slope of the line segment? f() f(0) What is? 0 5 Monitoring Progress and Modeling with Mathematics In Eercises 8, write an equation of the parabola in verte form. (See Eample.). 0 8 (, ) (, ). 8 (8, ) 8 0 (, ) 5. passes through (, 8) and has verte (, ). passes through ( 7, 5) and has verte ( 5, 9) 7. passes through (0, ) and has verte (, ) 8. passes through (, 5) and has verte (, ) In Eercises 9, write an equation of the parabola in intercept form. (See Eample.) 9. (, ) (, 0) 8 (, 0) 0. (, 0) 5 (, 0) (, ). -intercepts of and ; passes through (, ). -intercepts of 9 and ; passes through (0, 8). -intercepts of and ; passes through ( 8, 7) 5. WRITING Eplain when to use intercept form and when to use verte form when writing an equation of a parabola.. ANALYZING EQUATIONS Which of the following equations represent the parabola? (, 0) 5 A = ( )( + ) B = ( + 0.5).5 C = ( 0.5).5 D = ( + )( ) (, 0) (0.5,.5) In Eercises 7 0, write an equation of the parabola in verte form or intercept form Height (feet) Flare Signal 00 (, 50) (, 8) Time (seconds) Height (feet) New Ride (0, 80) (, ) 0 5 Time (seconds). -intercepts of 7 and ; passes through (, 0.05) 80 Chapter Quadratic Functions Copright Big Ideas Learning, LLC. All rights reserved.

44 9. Human Jump 0. Height (feet) (,.5) (0, 0) (, 0) Distance (feet) Height (feet) Frog Jump (, ) (, 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, 0 Baseball Height,. ERROR ANALYSIS Describe and correct the error in writing an equation of the parabola. (, 0) (, ) (, 0) = a( p)( q) = a( )( + ) a = 5 = ( )( + ) 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 0 8 (, ) (0, 0) (7, 0) Width (meters). 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 0 inches. (See Eample.) Circumference, C 0 Safe Working Load, S COMPARING METHODS You use a sstem with three variables to find the equation of a parabola that passes through the points ( 8, 0), (, 0), and (, 0). Your friend uses intercept form to find the equation. Whose method is easier? Justif our answer.. MODELING WITH MATHEMATICS The table shows the distances a motorcclist is from home after hours. Time (hours), 0 Distance (miles), 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 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 Height, h 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. 8. 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 s statement correct? Eplain our reasoning. Copright Big Ideas Learning, LLC. All rights reserved. Section. Modeling with Quadratic Functions 8

45 In Eercises 9, 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.. THOUGHT PROVOKING Describe a real-life situation that can be modeled b a quadratic equation. Justif our answer Price Decrease (dollars), Revenue ($000s), Time (hours), 0 Height (feet), 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), Height (feet), Time (hours), 5 Population (hundreds), 8 Time (das), 0 Height (feet), PROBLEM SOLVING The graph shows the number of students absent from school due to the flu each da,.. HOW DO YOU SEE IT? Use the graph to determine whether the rate of change over each interval is positive, negative, or zero Flu Epidemic Number of students (0, ) (, 9) Das a. 0 b. 5 c. d 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 th figure. a. Interpret the meaning of the verte in this situation. b. Write an equation for the parabola to predict the number of students that are absent on da 0. c. Compare the average rates of change in the students with the flu from 0 to das and to das. Figure Figure Figure Figure Figure Number of Tiles 5 9 Maintaining Mathematical Proficienc Factor the trinomial. (Skills Review Handbook) Reviewing what ou learned in previous grades and lessons Chapter Quadratic Functions Copright Big Ideas Learning, LLC. All rights reserved.

46 .. What Did You Learn? Core Vocabular focus, p. 8 directri, p. 8 Core Concepts Section. Standard Equations of a Parabola with Verte at the Origin, p. 9 Standard Equations of a Parabola with Verte at (h, k), p. 70 Section. Writing Quadratic Equations, p. 7 Writing Quadratic Equations to Model Data, p. 78 Mathematical Practices. Eplain the solution pathwa ou used to solve Eercise 7 on page 7.. Eplain how ou used definitions to derive the equation in Eercise 5 on page 7.. Eplain the shortcut ou found to write the equation in Eercise 5 on page 8.. Describe how ou were able to construct a viable argument in Eercise 8 on page 8. Performance Task Accident Reconstruction Was the driver of a car speeding when the brakes were applied? What do skid marks at the scene of an accident reveal about the moments before the collision? To eplore the answers to these questions and more, go to BigIdeasMath.com. Copright Big Ideas Learning, LLC. All rights reserved. 8

47 Chapter Review. Transformations of Quadratic Functions (pp. 7 5) Let the graph of g be a translation unit left and units up of the function f() = +. Write a rule for g. g() = f( ( )) + Subtract from the input. Add to the output. = ( + ) + + Replace with + in g(). = + + Simplif. The transformed function is g() = + +. Describe the transformation of f() = represented b g. Then graph each function.. g() = ( + ). g() = ( 7) +. g() = ( + ) Write a rule for g.. Let g be a horizontal shrink b a factor of, followed b a translation 5 units left and units down of the graph of f() =. 5. Let g be a translation units left and units up, followed b a reflection in the -ais of the graph of f() =.. Characteristics of Quadratic Functions (pp. 55 ) Graph f() = 8 +. Label the verte and ais of smmetr. Step Identif the coefficients: a =, b = 8, c =. Because a > 0, the parabola opens up. Step Find the verte. First calculate the -coordinate. = b a = 8 () = Then find the -coordinate of the verte. f() = () 8() + = 7 So, the verte is (, 7). Plot this point. Step Draw the ais of smmetr =. 5 7 = (, 7) 5 Step Identif the -intercept c, which is. Plot the point (0, ) and its reflection in the ais of smmetr, (, ). Evaluate the function for another value of, such as =. f() = () 8() + = 5 Plot the point (, 5) and its reflection in the ais of smmetr, (, 5). Step 5 Draw a parabola through the plotted points. Graph the function. Label the verte and ais of smmetr. Find the minimum or maimum value of f. Describe where the function is increasing and decreasing.. f() = ( ) 7. g() = h() = ( )( + 7) 8 Chapter Quadratic Functions Copright Big Ideas Learning, LLC. All rights reserved.

48 . Focus of a Parabola (pp. 7 7) a. Identif the focus, directri, and ais of smmetr of 8 =. Graph the equation. Step Rewrite the equation in standard form. 8 = Write the original equation. = 8 Divide each side b 8. Step Identif the focus, directri, and ais of smmetr. The equation has the form = p, where p =. The focus is (p, 0), or (, 0). The directri is = p, or =. Because is squared, the ais of smmetr is the -ais. Step Use a table of values to graph the equation. Notice that it is easier to substitute -values and solve for. 0 ± ± ± = 8 (, 0) 8 b. Write an equation of the parabola shown. Because the verte is not at the origin and the ais of smmetr is vertical, the equation has the form = p ( h) + k. The verte (h, k) is (, ) and the focus (h + p, k) is (, ), so h =, k =, and p =. Substitute the values to write an equation of the parabola. = () ( ) + = ( ) + An equation of the parabola is = ( ) focus verte 5 9. You can make a solar hot dog cooker b shaping foil-lined cardboard into a parabolic trough and passing a wire through the focus of each end piece. For the trough shown, how far from the bottom should the wire be placed? in. 0. Graph the equation =. Identif the focus, directri, and ais of smmetr. in. Write an equation of the parabola with the given characteristics. acteristics.. verte: (0, 0). focus: (, ) directri: = verte: (, ) Copright Big Ideas Learning, LLC. All rights reserved. Chapter Chapter Review 85

49 . Modeling with Quadratic Functions (pp. 75 8) The graph shows the parabolic path of a stunt motorcclist jumping off a ramp, where is the height (in feet) and is the horizontal distance traveled (in feet). Write an equation of the parabola. The motorcclist lands on another ramp 0 feet from the first ramp. What is the height of the second ramp? Height (feet) (0, 0) (80, 0) Horizontal distance (feet) Step First write an equation of the parabola. From the graph, ou can see that the verte (h, k) is (80, 0) and the parabola passes through the point (0, 0). Use the verte and the point to solve for a in verte form. = a( h) + k Verte form 0 = a(0 80) + 0 Substitute for h, k,, and. 0 = 00a Simplif. = a Divide each side b Because a =, h = 80, and k = 0, the path can be modeled b 0 = 0 ( 80) + 0, where 0 0. Step Then find the height of the second ramp. = 0 (0 80) + 0 Substitute 0 for. = 0 Simplif. So, the height of the second ramp is 0 feet. Write an equation for the parabola with the given characteristics.. passes through (, ) and has verte (0, ). passes through (, ) and has -intercepts of and 5 5. passes through (, 7), (, 0), and (, 7). Compare the average rates of change from the point (, ) to the verte (, ) and from the verte to the point (, 7) on the graph of a parabola. 7. The table shows the heights of a dropped object after seconds. Verif that the data shows a quadratic relationship. Write a function that models the data. How long is the object in the air? Time (seconds), Height (feet), Chapter Quadratic Functions Copright Big Ideas Learning, LLC. All rights reserved.

50 Chapter Test. A parabola has an ais of smmetr = and passes through the point (, ). Find another point that lies on the graph of the parabola. Eplain our reasoning.. Let the graph of g be a translation units left and unit down, followed b a reflection in the -ais of the graph of f() = ( + ). Write a rule for g.. Identif the focus, directri, and ais of smmetr of =. Graph the equation.. Eplain wh a quadratic function models the data. Then use a linear sstem to find the model. 8 0 f() 0 00 Write the equation of the parabola. Justif our answer. 5. verte 0 focus verte 5 directri 8. A surfboard shop sells 0 surfboards per month when it charges $500 per surfboard. Each time the shop decreases the price b $0, it sells one additional surfboard per month. How much should the shop charge per surfboard to maimize the amount of mone earned? What is the maimum amount the shop can earn per month? Eplain. 9. Graph f() = 8 +. Label the verte and ais of smmetr. Describe where the function is increasing and decreasing. 0. Sunfire is a machine with a parabolic cross section used to collect solar energ. The Sun s ras are reflected from the mirrors toward two boilers located at the focus of the parabola. The boilers produce steam that powers an alternator to produce electricit. boilers 0 ft depth a. Write an equation that represents the cross section of the dish shown with its verte at (0, 0). 7 ft b. What is the depth of Sunfire? Justif our answer.. In 0, the price of gold reached an all-time high. The table shows the prices (in dollars per tro ounce) of gold each ear since 00 (t = 0 represents 00). Find a quadratic function that best models the data. Use the model to predict the price of gold in the ear 0 to the nearest cent. Year, t 0 5 Price, p $0. $95.9 $87.9 $97.5 $.5 $57.5 Copright Big Ideas Learning, LLC. All rights reserved. Chapter Chapter Test 87

51 Standards Assessment. You and our friend are throwing a football. The parabola shows the path of our friend s throw, where is the horizontal distance (in feet) and is the corresponding height (in feet). The path of our throw can be modeled b h() = Choose the correct inequalit smbol to indicate whose throw travels higher. Eplain our reasoning. (HSF-IF.C.9) height of our throw height of our friend s throw. The function g() = + is a combination of transformations of f() =. Which combinations describe the transformation from the graph of f to the graph of g? (HSF-IF.B.) A translation units right and vertical shrink b a factor of, followed b a translation units up B translation units right and units up, followed b a vertical shrink b a factor of C vertical shrink b a factor of, followed b a translation units up and units right D translation units right and 8 units up, followed b a vertical shrink b a factor of. Your school decides to have a dance in the school cafeteria to raise mone. There is no fee to use the cafeteria, but the DJ charges a fee of $750. The table shows the profits (in dollars) when students attend the dance. (HSF-LE.A.) Students, Profit, a. What is the cost per student? b. Your school epects 00 students to attend and finds another DJ who onl charges $50. How much should our school charge per student to still make the same profit? c. Your school decides to charge the amount in part (a) and use the less epensive DJ. How much more mone will the school raise?. Order the following parabolas from greatest width to least width. (HSF-IF.B.) a. focus: (0, ); directri: = b. = + c. = 8 d. = ( ) + 88 Chapter Quadratic Functions Copright Big Ideas Learning, LLC. All rights reserved.

52 5. Your friend claims that for g() = b, where b is a real number, there is a transformation in the graph that is impossible to notice. Is our friend correct? Eplain our reasoning. (HSF-BF.B.). Let the graph of g represent a vertical stretch and a reflection in the -ais, followed b a translation left and down of the graph of f() =. Use the tiles to write a rule for g. (HSF-BF.B.) 0 g() + = 7. Two balls are thrown in the air. The path of the first ball is represented in the graph. The second ball is released.5 feet higher than the first ball and after seconds reaches its maimum height 5 feet lower than the first ball. (HSF-IF.B.) Height (feet) Ball Toss 0 (, 5.5) (0, 5) Time (seconds) a. Write an equation for the path of the second ball. b. Do the balls hit the ground at the same time? If so, how long are the balls in the air? If not, which ball hits the ground first? Eplain our reasoning. 8. Let the graph of g be a translation units right of the graph of f. The points (, ), (, ), and (, ) lie on the graph of f. Which points lie on the graph of g? (HSF-BF.B.) A (, ) B (, ) C (, ) D (, 9) E (9, ) F (9, ) 9. Gm A charges $0 per month plus an initiation fee of $00. Gm B charges $0 per month, but due to a special promotion, is not currentl charging an initiation fee. (HSF-LE.A.) a. Write an equation for each gm modeling the total cost for a membership lasting months. b. When is it more economical for a person to choose Gm A over Gm B? c. Gm A lowers its initiation fee to $5. Describe the transformation this change represents and how it affects our decision in part (b). Copright Big Ideas Learning, LLC. All rights reserved. Chapter Standards Assessment 89