STUDENT TEXT AND HOMEWORK HELPER

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1 UNIT 3 QUADRATIC FUNCTIONS AND EQUATIONS STUDENT TEXT AND HOMEWORK HELPER Randall I. Charles Allan E. Bellman Basia Hall William G. Handlin, Sr. Dan Kenned Stuart J. Murph Grant Wiggins Boston, Massachusetts Chandler, Arizona Glenview, Illinois Hoboken, New Jerse

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3 UNIT 3 QUADRATIC FUNCTIONS AND EQUATIONS TOPIC 8 Quadratic Functions and Equations TOPIC 8 Overview Quadratic Graphs and Their Properties Quadratic Functions Transformations of Quadratic Functions Verte Form of a Quadratic Function Solving Quadratic Equations Factoring to Solve Quadratic Equations Writing Quadratic Functions Completing the Square The Quadratic Formula and the Discriminant...37 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS (TEKS) FOCUS (6)(A) (6)(B) (6)(C) (7)(A) (7)(B) (7)(C) Determine the domain and range of quadratic functions and represent the domain and range using inequalities. Write equations of quadratic functions given the verte and another point on the graph, write the equation in verte form (f() = a( - h) + k), and rewrite the equation from verte form to standard form (f() = a + b + c). Write quadratic functions when given real solutions and graphs of their related equations. Graph quadratic functions on the coordinate plane and use the graph to identif ke attributes, if possible, including -intercept, -intercept, zeros, maimum value, minimum values, verte, and the equation of the ais of smmetr. Describe the relationship between the linear factors of quadratic epressions and the zeros of their associated quadratic functions. Determine the effects on the graph of the parent function f () = when f () is replaced b af (), f () + d, f ( - c), f (b) for specific values of a, b, c, and d. (8)(A) Solve quadratic equations having real solutions b factoring, taking square roots, completing the square, and appling the quadratic formula. (8)(B) Write, using technolog, quadratic functions that provide a reasonable fit to data to estimate solutions and make predictions for real-world problems. (1)(B) Evaluate functions, epressed in function notation, given one or more elements in their domains. i

4 Topic 8 Quadratic Functions and Equations TOPIC OVERVIEW VOCABULARY 8-1 Quadratic Graphs and Their Properties 8- Quadratic Functions 8-3 Transformations of Quadratic Functions 8- Verte Form of a Quadratic Function 8-5 Solving Quadratic Equations 8-6 Factoring to Solve Quadratic Equations 8-7 Writing Quadratic Functions 8-8 Completing the Square 8-9 The Quadratic Formula and the Discriminant English/Spanish Vocabular Audio Online: English Spanish ais of smmetr, p. 37 eje de simetría completing the square, p. 367 completar el cuadrado discriminant, p. 37 discriminante maimum, p. 37 máimo minimum, p. 37 mínimo parabola, p. 37 parábola quadratic equation, p. 350 ecuación cuadrática quadratic formula, p. 37 fórmula cuadrática quadratic function, p. 36 función cuadrática root of an equation, p. 350 raíz de una ecuación verte, p. 37 vértice zero of a function, p. 350 cero de una función DIGITAL APPS PRINT and ebook Access Your Homework... ONLINE HOMEWORK You can do all of our homework online with built-in eamples and Show Me How support! When ou log in to our account, ou ll see the homework our teacher has assigned ou. YOUR DIGITAL RESOURCES HOMEWORK TUTOR APP Do our homework anwhere! You can access the Practice and Application Eercises, as well as Virtual Nerd tutorials, with this Homework Tutor app, available on an mobile device. 3 Topic 8 Quadratic Functions and Equations STUDENT TEXT AND HOMEWORK HELPER Access the Practice and Application Eercises that ou are assigned for homework in the Student Tet and Homework Helper, which is also available as an electronic book.

5 The Long Shot 3-Act Math 3-Act Math Have ou ever been to a basketball game where the hold contests at halftime? A popular contest is one where the contestant needs to make a basket from half court to win the prize. Contestants often shoot the ball in different was. The might take a regular basketball shot, a hook shot, or an underhand toss. What s the best wa to shoot the basketball to make a basket? In this video ou ll see various shots and ou decide! Scan page to see a video for this 3-Act Math Task. If You Need Help... VOCABULARY ONLINE You ll find definitions of math terms in both English and Spanish. All of the terms have audio support. INTERACTIVE EXPLORATION You ll have access to a robust assortment of interactive eplorations, including interactive concept eplorations, dnamic activitites, and topiclevel eploration activities. LEARNING ANIMATIONS You can also access all of the stepped-out learning animations that ou studied in class. STUDENT COMPANION Refer to our notes and solutions in our Student Companion. Remember that our Student Companion is also available as an ACTIVebook accessible on an digital device. INTERACTIVE MATH TOOLS These interactive math tools give ou opportunities to eplore in greater depth ke concepts to help build understanding. VIRTUAL NERD Not sure how to do some of the practice eercises? Check out the Virtual Nerd videos for stepped-out, multi-level instructional support. 35

6 8-1 Quadratic Graphs and Their Properties TEKS FOCUS TEKS (6)(A) Determine the domain and range of quadratic functions and represent the domain and range using inequalities. TEKS (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including smbols, diagrams, graphs, and language as appropriate. Additional TEKS (1)(B), (7)(A) VOCABULARY Ais of smmetr the line that divides a parabola into two matching halves Falling object model The function h =-16t + c models the height of a falling object, where h is the object s height in feet, t is the time in seconds since the object began to fall, and c is the object s initial height. Maimum The verte of a parabola that opens downward is a maimum point. The -coordinate of the verte is the maimum value of the function. Minimum The verte of a parabola that opens upward is a minimum point. The -coordinate of the verte is the minimum value of the function. Parabola the graph of a quadratic function Quadratic function a function that can be written in the form = a + b + c, where a 0 Quadratic parent function the simplest quadratic function f() = or = Standard form of a quadratic function f() = a + b + c, where a 0 Verte the highest or lowest point on a parabola Implication a conclusion that follows from previousl stated ideas or reasoning without being eplicitl stated Representation a wa to displa or describe information. You can use a representation to present mathematical ideas and data. ESSENTIAL UNDERSTANDING A quadratic function is a tpe of nonlinear function that models certain situations where the rate of change is not constant. The graph of a quadratic function is a smmetric curve with a highest or lowest point corresponding to a maimum or minimum value. Ke Concept Standard Form of a Quadratic Function A quadratic function is a function that can be written in the form = a + b + c, where a 0. This form is called the standard form of a quadratic function. Eamples = 3 = + 9 = Lesson 8-1 Quadratic Graphs and Their Properties

7 Ke Concept Parabolas The simplest quadratic function f () = or = is the quadratic parent function. The graph of a quadratic function is a U-shaped curve called a parabola. The parabola with equation = is shown at the right. You can fold a parabola so that the two sides match eactl. This propert is called smmetr. The fold or line that divides the parabola into two matching halves is called the ais of smmetr. The highest or lowest point of a parabola is its verte, which is on the ais of smmetr. For graphs of functions of the form = a, the verte is at the origin. The ais of smmetr is the -ais, or = 0. O If a 7 0 in = a + b + c, the parabola opens upward. v The verte is the minimum point, or lowest point, of the parabola. If a 6 0 in = a + b + c, the parabola opens downward. v The verte is the maimum point, or highest point, of the parabola. Problem 1 Identifing Ke Attributes of a Quadratic Function Graph each function. Identif the verte and the ais of smmetr of the parabola. Tell whether the verte is a minimum or maimum point, and identif the minimum or maimum value of the function. What is the -intercept of the parabola? Can a parabola have both a minimum and a maimum point? No. A parabola either opens upward and has a minimum point or opens downward and has a maimum point. A = The function passes through the points ( -, - 1), ( -, ), (0, 3), (, ), and (, - 1). Since a =- 1 and , the parabola opens downward. O The verte is (0, 3) and the ais of smmetr is the line = 0. The verte is a maimum point, and the maimum value of the function is 3, the -coordinate of the verte. The -intercept is 3. B = The function passes through the points ( - 1, 3), (0, 0), (1, - 1), (, 0), and (3, 3). Since a = 1 and 1 7 0, the parabola opens upward. 1 O The verte is (1, - 1) and the ais of smmetr is the line = 1. The verte is a minimum point, and the minimum value of the function is - 1, the -coordinate of the verte. The -intercept is 0. 37

8 Problem TEKS Process Standard (1)(D) Graphing = a What are good values to choose for when making the table? Choose values of that make divisible b 3 so that the -values will be integers. Graph the function = 13. Make a table of values. What are the domain and range?! 13 (, ) 0 1 (0)! 0 3 (0, 0) 3 1 (3)! 3 3 (3, 3) 6 1 (6)! 1 3 (6, 1) 15 Reflect the points from the table over the ais of smmetr, = 0, to find more points on the graph. 9!1!6 O 6 1 The domain is all real numbers. The range is Ú 0. Problem 3 Comparing Widths of Parabolas Does the sign of the -term affect the parabola s width? No. The sign of the -term affects onl whether the parabola opens upward or downward. Use the graphs below. What is the order, from widest to narrowest, of the graphs of the quadratic functions f() =, f() = 1, and f() =? f () = - f () = 1 f () = Of the three graphs, f () = 1 is the widest and f () = - is the narrowest. So, the order from widest to narrowest is f () = 1, f () =, and f () = Lesson 8-1 Quadratic Graphs and Their Properties

9 Problem TEKS Process Standard (1)(D) Graphing = a + c Multiple Choice How is the graph of = + 3 different from the graph of =? What values should ou choose for? Use the same values of for graphing both functions so that ou can see the relationship between corresponding -coordinates. It is shifted 3 units up. It is shifted 3 units down It is shifted 3 units to the right. It is shifted 3 units to the left. 6 O The graph of = + 3 has the same shape as the graph of = but is shifted up 3 units. The correct answer is A. Problem 5 TEKS Process Standard (1)(B) Using the Falling Object Model A Nature An acorn drops from a tree branch 0 ft above the ground. The function h = 16t + 0 gives the height h of the acorn (in feet) after t seconds. What is the graph of this quadratic function? At about what time does the acorn hit the ground? Can ou choose negative values for t? No. t represents time, so it cannot be negative. t h 16t Height (ft), h Time (s), t Graph the function using the first three ordered pairs from the table. Do not plot (1.5, 16) because height cannot be negative. The acorn hits the ground when its height above the ground is 0 ft, or when h = 0. B estimating the t-intercept of the graph, ou can see that the acorn hits the ground after about 1.1 s. continued on net page 39

10 Problem 5 continued B What are a reasonable domain and range for the function h = 16t + 0 described in part (A)? Time t and height h cannot be negative. You can use the graph in part (A) to determine the domain and range of the function h = -16t + 0. Note that the maimum value of t, 1.1, is an approimate value. Domain: 0 t 1.1 Range: 0 h 0 ONLINE H O M E W O R K PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. For additional support when completing our homework, go to. Identif the verte and the ais of smmetr of each parabola. Tell whether the verte is a minimum or a maimum, and identif the minimum or maimum value of the function O O Graph each function. Then identif the domain and range of the function.. =- 5. f () = f () = 3 7. Use a Problem Solving Model (1)(B) A person walking across a bridge accidentall drops an orange into the river below from a height of 0 ft. The function h =-16t + 0 gives the orange s approimate height h above the water, in feet, after t seconds. Graph the function. In how man seconds will the orange hit the water? Use inequalities to describe a reasonable domain and range for the function. 8. Eplain Mathematical Ideas (1)(G) Describe and correct the error made in graphing the function = Use a Problem Solving Model (1)(B) A bird drops a stick to the ground from a height of 80 ft. The function h =-16t + 80 gives the stick s approimate height h above the ground, in feet, after t seconds. Graph the function. At about what time does the stick hit the ground? Use inequalities to describe a reasonable domain and range for the function. - O Identif the domain and range of each function. 10. f () = = = = Lesson 8-1 Quadratic Graphs and Their Properties

11 1. What information do the numbers a and c give ou about the graph of = a + c? Order each group of quadratic functions from widest to narrowest graph. 15. = 3, =, = 16. =- 1, = 5, = Appl Mathematics (1)(A) Suppose a person is riding in a hot-air balloon, 15 ft above the ground. He drops an apple. The height h, in feet, of the apple above the ground is given b the formula h =-16t + 15, where t is the time in seconds. To the nearest tenth of a second, at what time does the apple hit the ground? Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Graph each function. Identif the verte and the ais of smmetr of the parabola. Tell whether the verte is a minimum or a maimum point, and identif the minimum or maimum value of the function. Identif the -intercept of the parabola. 18. f () = = = f () = = f () = - 5 Match each function with its graph.. f () = f () = f () = A. B. C. 7. Using a graphing calculator, graph f () = +. a. If f () = + and g () = 3f (), write the equation for g (). Graph g () and compare the graph to the graph of f (). b. If f () = + and h() = f (3), write the equation for h(). Graph h() and compare the graph to the graph of f (). c. Compare how multipling a quadratic function b a number and multipling the -value of a quadratic function b a number change the graphs of the quadratic functions. Use a graphing calculator to graph each function. Identif the verte and ais of smmetr. 8. = f () = =

12 Three graphs are shown at the right. Identif the graph or graphs that fit each description. 31. a a a 0 has the greatest value a 0 has the least value. K L M STEM 35. Consider the graphs of = a and = (a). Assume a 0. a. For what values of a will both graphs lie in the same quadrant(s)? b. For what values of a will the graph of = a be wider than the graph of = (a)? 36. Complete each statement. Assume a 0. a. The graph of = a + c intersects the -ais in two places when?. b. The graph of = a + c does not intersect the -ais when?. 37. Appl Mathematics (1)(A) A blueprint is given for a rectangular wall with a square window in the center. If each side of the window is feet, the function = gives the area (in square feet) of the wall without the window. a. Graph the function. b. What is a reasonable domain for the function? Eplain. c. What is the range of the function? Eplain. d. Estimate the side length of the window if the area of the wall is 117 ft. Bedroom # South Facing Wall 9 ft 15 ft TEXAS End-of-Course PRACTICE 38. Which equation has a graph that is narrower than the graph of = + 5? A. = - 5 B. =-5 + C. = D. = Kristina is evaluating some formulas as part of a science eperiment. One of the formulas involves the epression - (-17). What is the value of this epression? F. -1 G. -7 H. 7 J Which epression is equivalent to 8( + 9)? A. + 7 B C D What is the solution of the equation ( + 3) + 7 =-11? F. -1 G. -1 H. 1 J. 1. A rectangular dog run has an area of What are possible dimensions of the dog run? Use factoring. Eplain how ou found the dimensions. 33 Lesson 8-1 Quadratic Graphs and Their Properties

13 8- Quadratic Functions TEKS FOCUS TEKS (7)(A) Graph quadratic functions on the coordinate plane and use the graph to identif ke attributes, if possible, including -intercept, -intercept, zeros, maimum value, minimum values, verte, and the equation of the ais of smmetr. TEKS (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including smbols, diagrams, graphs, and language as appropriate. Additional TEKS (1)(A), (1)(C), (8)(B) VOCABULARY Vertical motion model The function h =-16t + vt + c models the height h of an object, in feet, t seconds after it is projected into the air from initial height c, in feet, with an initial upward velocit v, in feet per second. Implication a conclusion that follows from previousl stated ideas or reasoning without being eplicitl stated Representation a wa to displa or describe information. You can use a representation to present mathematical ideas and data. ESSENTIAL UNDERSTANDING In the quadratic function = a + b + c, the value of b affects the position of the ais of smmetr. Ke Concept Ais of Smmetr of a Quadratic Function The graph of = a + b + c, where a 0, has the line = - a b as its ais of smmetr. The -coordinate of the verte is - a b. The eamples below show how the position of the ais of smmetr changes as the b-value changes. In each case, the equation of the ais of smmetr is = - b a. = + = + = O 3 1 O O 3 ais of smmetr: =- 1 =-1 =

14 Problem 1 TEKS Process Standard (1)(D) Graphing = a + b + c What is the graph of the function = 6 +? Step 1 Find the ais of smmetr and the coordinates of the verte. How are the verte and the ais of smmetr related? The verte is on the ais of smmetr. You can use the equation for the ais of smmetr to find the -coordinate of the verte. = - b a = - (-6) = 3 Find the equation of the ais of smmetr. (1) The ais of smmetr is = 3. So the -coordinate of the verte is 3. = = 3-6(3) + Substitute 3 for to find the -coordinate of the verte. =-5 Simplif. The verte is (3, -5). Step Find two other points on the graph. Find the -intercept. When = 0, =, so one point is (0, ). Find another point b choosing a value for on the same side of the verte as the -intercept. Let = 1. = = 1-6(1) + =-1 Substitute 1 for and simplif. When = 1, =-1, so another point is (1, -1). Step 3 Graph the verte and the points ou found in Step, (0, ) and (1, -1). Reflect the points from Step across the ais of smmetr to get two more points on the graph. Then connect the points with a parabola. (0, ) (6, ) O (1, 1) (5, 1) (3, 5) 33 Lesson 8- Quadratic Functions

15 Problem TEKS Process Standard (1)(C) Using the Vertical Motion Model Entertainment During halftime of a basketball game, a slingshot launches T-shirts at the crowd. A T-shirt is launched with an initial upward velocit of 7 ft/s. The T-shirt is caught 35 ft above the court. How long will it take the T-shirt to reach its maimum height? What is its maimum height? What is the range of the function that models the height of the T-shirt over time? 5 ft What are the values of v and c? The T-shirt is launched from a height of 5 ft, so c = 5. The T-shirt has an initial upward velocit of 7 ft/s, so v = 7. To solve this problem, use the vertical motion model h = 16t + vt + c, where h is the height of an object, in feet, t seconds after it is projected into the air from initial height c, in feet, with initial upward velocit v, in feet per second. The function h =-16t + 7t + 5 gives the T-shirt s height h, in feet, after t seconds. Since the coefficient of t is negative, the parabola opens downward, and the verte is the maimum point. Method 1 Use paper and pencil with a formula. t = -b a = - 7 (-16) = 9 h =-16( 9 ) + 7( 9 ) + 5 = 86 Find the t-coordinate of the verte. Find the h-coordinate of the verte. The T-shirt will reach its maimum height of 86 ft after 9 s, or.5 s. The range describes the height of the T-shirt during its flight. The T-shirt starts at 5 ft, peaks at 86 ft, and then is caught at 35 ft. The height of the T-shirt at an time is between 5 ft and 86 ft, inclusive, so the range is 5 h 86. Method Use technolog. Enter the function h =-16t + 7t + 5 as = on the Y = screen and graph the function. Use the CALC feature and select MAXIMUM. Set left and right bounds on the maimum point and calculate the point s coordinates. The coordinates of the maimum point are (.5, 86). Maimum X.9998 The T-shirt will reach its maimum height of 86 ft after.5 s. The range of the function is 5 h 86. Y

16 Will a quadratic function with all integer coefficients fit the data eactl? No. The table shows that for integer -values 1 to 5, the -values are not integers. This means that at least one of the coefficients a, b, and c is not an integer. Problem 3 Fitting a Quadratic Function to Data The table shows the run times, in seconds, a certain computer program requires to analze the given megabtes of data. A Find a quadratic function f () = a + b + c that provides a reasonable fit to the data. Use technolog to find a, b, and c. Let represents the data size in megabtes and let represent the run time in seconds. Enter the data into lists. Find the quadratic regression. L1 L L L(6)= Data (MB) Run Time (s) QuadReg = a +b+c a = b = c = A quadratic function that models the data is f () = Graphing the function with the data shows that the quadratic function provides a reasonable fit to the data, so the function can be used to estimate solutions to problems or to make predictions. B Estimate the run time to analze 3.5 MB of data. The variable represents megabtes of data. Evaluate f (3.5). f () = Write the quadratic function. f (3.5) = 0.51(3.5) +.59(3.5) Substitute 3.5 for. 8.7 Simplif. The run time to analze 3.5 MB of data is about 8.5 seconds. C Predict the run time for 10 MB of data. Evaluate f (10). f () = Write the quadratic function. f (10) = 0.51(10) (10) Substitute 10 for. = Simplif. The model predicts that the computer program will need to run for about 57.1 seconds to analze 10 MB of data. 336 Lesson 8- Quadratic Functions

17 ONLINE H O M E W O R K PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. Find the equation of the ais of smmetr and the coordinates of the verte of the graph of each function. For additional support when completing our homework, go to. 1. = + 3. = f () = = f () = = Match each function with its graph. 7. = = = = + 6 A. C. B. D. 11. Select Tools to Solve Problems (1)(C) A baseball is thrown into the air with an upward velocit of 30 ft/s. Its height h, in feet, after t seconds is given b the function h =-16t + 30t + 6. How long will it take the ball to reach its maimum height? What is the ball s maimum height? What is the range of the function? Select a tool, such as paper and pencil or technolog, to solve this problem. 1. Appl Mathematics (1)(A) Suppose ou have 100 ft of string to rope off a rectangular section for a bake sale at a school fair. The function A = gives the area of the section in square feet, where is the width in feet. What width gives ou the maimum area ou can rope off? What is the maimum area? What is the range of the function? Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Use the ais of smmetr, verte, and -intercept to graph each function. Label the ais of smmetr and the verte. 13. f () = = = Eplain Mathematical Ideas (1)(G) Describe and correct the error made in finding the ais of smmetr for the graph of = b 6 = = = 3 a ( 1) 337

18 17. a. What is the verte of the function = ? b. What is the verte of the function given in the table? Appl Mathematics (1)(A) The Riverside Geser in Yellowstone National Park erupts about ever 6.5 h. When the geser erupts, the water has an initial upward velocit of 69 ft/s. What is the maimum height of the geser? Round our answer to the nearest foot. 19. Appl Mathematics (1)(A) A cellphone compan sells about 500 phones each week when it charges $75 per phone. It sells about 0 more phones per week for each $1 decrease in price. The compan s revenue is the product of the number of phones sold and the price of each phone. What price should the compan charge to maimize its revenue? 0. Appl Mathematics (1)(A) Suppose a tennis plaer hits a ball over the net. The ball leaves the racket 0.5 m above the ground. The equation h =-.9t + 3.8t gives the ball s height h in meters after t seconds. a. When will the ball be at the highest point in its path? Round to the nearest tenth of a second. b. If ou double our answer from part (a), will ou find the amount of time the ball is in the air before it hits the court? Eplain. 1. The parabola at the right is of the form = + b + c. a. Use the graph to find the -intercept. b. Use the graph to find the equation of the ais of smmetr. c. Use the formula = - a b to find b. d. Write the equation of the parabola. e. Test one point using our equation from part (d). f. Would this method work if the value of a were not known? Eplain. 6 (, ). The table shows data a biologist gathered about the population of a bacteria colon. Use technolog to find a quadratic function that is a reasonable fit to the data. Use the function to estimate the bacteria population at 1 das. Round to the nearest million bacteria. O ( 1, ) Time (das) Population (millions) Lesson 8- Quadratic Functions

19 3. Select Tools to Solve Problems (1)(C) A laser tracks the flight of a hit baseball. The table shows the height of the ball in feet as a function of time in seconds. Use technolog to find a quadratic function that is a reasonable fit to the data. Use the function to predict the height of the ball at.5 seconds. Round to the nearest hundredth of a foot. Time (s) Height (ft) Select Tools to Solve Problems (1)(C) The table shows eamples of the epected return on an investment in a new compan. Use technolog to find a quadratic function that is a reasonable fit to the data. Use the function to predict the return on an investment of $000. Round to the nearest thousand dollars. Investment ($) Return ($) A meteorologist records the temperature data that is shown in the table. Does the data suggest that there is a quadratic model that is a reasonable fit? Justif our answer. Number of Das Temperature ( F) TEXAS End-of-Course PRACTICE 6. A half-pipe ramp at a skate park is approimatel parabolic in shape. It can be modeled b the quadratic function = At what point would a skater be at the lowest part of the ramp? A. (-3, 36) B. (36, -3) C. (3, 0) D. (0, 3) 7. What is the simplified form of the product (-8)(5)(-1)? F G. -80 H. 80 J Which of the following is equivalent to (-) 3? A. -6 B. -1 C. 1 D Tob needs to write an eample of the Commutative Propert of Multiplication for his homework. Which of the following epressions could he use? F. ab = ba G. a = a H. ab = ab J. a(bc) = (ab)c 30. Simplif the product (3r - 1)(r + r + ). Justif each step. 339

20 8-3 Transformations of Quadratic Functions VOCABULARY TEKS FOCUS Ě Compression A compression is a transformation that decreases TEKS (7)(C) Determine the effects on the graph of the parent function f() = when f() is replaced b af(), f() + d, f( - c), f(b) for specific values of a, b, c, and d. the distance between corresponding points of a graph and a line. Ě Stretch A stretch is a transformation that increases the distance between corresponding points of a graph and a line. TEKS (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including smbols, diagrams, graphs, and language as appropriate. Ě Implication a conclusion that follows from previousl stated ideas or reasoning without being eplicitl stated Ě Representation a wa to displa or describe information. You can use a representation to present mathematical ideas and data. Additional TEKS (1)(A), (1)(F), (7)(A) ESSENTIAL UNDERSTANDING The graph of an quadratic function is a transformation of the graph of the parent quadratic function, =. Ke Concept Reflection, Stretch, and Compression Vertical reflection, stretch, and compression The graphs of = a and = -a are reflections of each other across the -ais. When! a! 7 1, the graph of = is stretched verticall. When! a! 6 1, the graph of = is compressed verticall.!!! "!! Reflection, a and a! O Stretch, a " 1! 1! O Compression, 0 # a # 1 Horizontal reflection, stretch, and compression The graphs of = (b) and = ( -b) are reflections of each other across the -ais. When! b! 7 1, the graph of = is compressed horizontall. When! b! 6 1, the graph of = is stretched horizontall. = = ( ) - O Reflection, b and b 30 Lesson 8-3 = () - = (1 ) O Compression, b > 1 Transformations of Quadratic Functions - O Stretch, 0 < b < 1

21 Ke Concept Translation Vertical translation The graph of = + d is a vertical translation of the parent function =. When d 7 0, the graph of = shifts d units up. When d 6 0, the graph of = shifts 0 d 0 units down. When c 7 0, the graph of = shifts c units right. When c 6 0, the graph of = shifts 0 c 0 units left. = + = - O Horizontal translation The graph of = ( - c) is a horizontal translation of the parent function =. = - O = ( ) Problem 1 Graphing Quadratic Functions With Stretch and Compression How can ou use the new equation to identif the change to the graph of the parent function? Think about how the change in the equation affects the coordinates of a particular point, such as (, ). In part (A), the point (, ) becomes the point, 3. Graph each function. How is each graph a stretch or compression of f () =? A = 13 Graph the parent function, f () =. The -coordinate of each point on the graph of = 13 is one-third the -coordinate of the corresponding point on the graph of the parent function. The graph of = 13 is a vertical compression of the graph of f () =. 6 = = (, ) (, 3) O 6 ( ) B = 1 13 Graph the parent function, f () =. The -coordinate of each point on the graph of = 1 13 is 3 times the -coordinate of the corresponding point on the graph of the parent function. The graph of = 1 13 is a horizontal stretch of the graph of f () =. 6 = (6, ) (, ) = (13 ) O 6 31

22 Problem TEKS Process Standard (1)(D) Graphing Translations of f () = Graph each function. How is each graph a translation of f () =? A g() = How does g() differ from f()? For each value of, the value of g() is less than the value of f(). Verte (0, 0) Verte (0, -) f() = - O - g() = Ais of Smmetr = 0 Translate the graph of f () down units to get the graph of g() = -. B h() = ( 3) Ais of Smmetr = 0 Verte (3, 0) r f() = h() = ( 3) Verte (0, 0) O 6 Ais of Smmetr = 3 Translate the graph of f () to the right 3 units to get the graph of h() = ( - 3). 3 Lesson 8-3 Transformations of Quadratic Functions

23 Problem 3 TEKS Process Standard (1)(F) What does the value of a tell ou? The value of a tells ou whether the graph of the parent function, =, is stretched or compressed verticall. If a is negative, the graph is a reflection across the -ais. Describing a Combination of Transformations Multiple Choice What steps transform the graph of = to = 3( + ) +? Reflect across the -ais, stretch b the factor 3, translate units to the right and units up. Stretch b the factor 3, translate units to the right and units up. Reflect across the -ais, translate units to the left and units up. Stretch b the factor 3, reflect across the -ais, translate units to the left and units up. 0 a and a is negative, so the graph is stretched verticall and reflects across the -ais. Since + = - (-), ou know that c 6 0, so the graph shifts c units to the left. d 7 0, so the graph shifts d units up. The correct choice is D. Problem TEKS Process Standard (1)(A) Appling Transformations of Quadratic Functions A compan makes satellite dishes with cross sections that can be modeled b quadratic functions. The parent function f () = models a cross section of a satellite dish that is 1 ft deep and has a radius of 1 ft. On the graph, the radius of the satellite dish is the distance of 1 ft from the -ais and the depth is the distance of 1 ft from the -ais. f() = O 1 The compan needs to make a new satellite dish that has a cross section 1 ft deep with a radius of ft. What is a quadratic function that models the cross section of the new dish? Draw a graph of the parent function and the new function. The new dish has the same verte as the parent function, (0, 0), but is stretched times as wide. So the equation is in the form = (b), because the graph has been stretched horizontall. A horizontal stretch means that 0 b A horizontal reflection will not change the graph because the ais of smmetr is the -ais, so ou can choose b 7 0. Since the new graph is twice as wide and b 6 1, b = 1. A quadratic function modeling the cross section is = ( 1 ), or = 1. f() = 1 1 O 1 33

24 ONLINE H O M E W O R K PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. In Eercises 1 8, f () =. Write an equation of the function g() and describe the effects on the graph of the parent function f () =. For additional support when completing our homework, go to. 1. g() = f () + d, where d =-1. g() = f ( - c), where c =-3 3. g() = af (), where a =-1. g() = af (), where a = g() = f (b), where b = 3 6. g() = f (b), where b =-1 7. g() = f () + d, where d = 8. g() = f ( - c), where c = Use Representations to Communicate Mathematical Ideas (1)(E) Sketch the graph of the function. Identif the verte and ais of smmetr. 9. f () = 10. f () = f () = ( + 3) - 1. f () = 3( + 1) f () =-(-1.5) f () = 1 ( - 3) Sketch the graphs of each pair of functions on the same coordinate grid. Identif the verte of each graph. If the second graph is a transformation of the first, identif the transformation as a stretch, compression, reflection, or translation. 15. f () =, g() = 16. f () =- 1 3 ; g() = f () =, g() = ( - 3) f () = ( + ) ; g() = ( + ) 19. f () = (3) + 1; g() = (-3) f () = 1 ( + ) ; g () =- 1 ( + ) 1. Analze Mathematical Relationships (1)(F) When a ball is tossed into the air, its height is a quadratic function of time. One student models the height of the ball as h(t) =-16t. Another student models it as h(t) =-16t Could both models be correct? Eplain. Match the function to the graph.. = =-( - 3) + 3. = ( + 3) - 3 A. B. C. - - O - - O O Lesson 8-3 Transformations of Quadratic Functions

25 5. Eplain Mathematical Ideas (1)(G) Let a be a nonzero real number such that a 1. The graphs of f () = and g () = (a) have the same verte of (0, 0). Do the graphs share an other points in common? Eplain our reasoning. 6. Evaluate Reasonableness (1)(B) An engineer is designing a section of a roller coaster. The section runs 0 meters along the ground, and its shape is described b a quadratic function. In each function below, represents the distance in meters along the ground, and is the height of the track in meters. Which of the functions describes the most reasonable track? Eplain our choice. =-( - 10) + 80 =- 1 ( - 10) + 8 = 5( - 10) + 8 TEXAS End-of-Course PRACTICE 7. The graph shows two parabolas, one of which is described b f () =. Which function describes the other parabola? A. =-( - ) + B. = () - C. = 0.( + ) D. =-0.( - ) 8. Which transformation maps the graph of f () = to the graph of g () = ( + )? F. a reflection across the line =- G. a reflection across the line =- H. a translation shifting f () units to the left J. a translation shifting f () units to the right 9. Which describes the transformation of the graph of f () = + to the graph of g () = (-3) +? A. a horizontal compression without reflection B. a horizontal stretch without reflection C. a vertical compression without reflection D. a vertical stretch and a translation 30. Let n be a nonzero real number. Of the functions = + n, = ( - n), and = (n), which has the same ais of smmetr as the parent function =? Eplain. - 6 O 35

26 8- Verte Form of a Quadratic Function TEKS FOCUS TEKS (6)(B) Write equations of quadratic functions given the verte and another point on the graph, write the equation in verte form (f() = a( - h) + k), and rewrite the equation from verte form to standard form (f() = a + b + c). TEKS (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including smbols, diagrams, graphs, and language as appropriate. Additional TEKS (6)(A), (7)(A), (1)(B) VOCABULARY Verte form The verte form of a quadratic function is f() = a( - h) + k, where a 0 and (h, k) is the verte of the function. Implication a conclusion that follows from previousl stated ideas or reasoning without being eplicitl stated ESSENTIAL UNDERSTANDING A quadratic function written in verte form clearl indicates the verte and ais of smmetr of the parabola, and the maimum or minimum value of the function. Ke Concept Verte Form of a Quadratic Function The quadratic function f () = a( - h) + k is written in verte form. The line = h is the ais of smmetr of the parabola. The verte of the parabola is (h, k). The maimum or minimum value of the function is k. f() = ( ) + 1 a = 1 h = k = O - Verte (, 1) Ais of Smmetr = Problem 1 Interpreting Verte Form How do ou use verte form? Compare = 6( - 5) - 1 to verte form = a( - h) + k to find values for a, h, and k. For = 6( 5) 1, what are the verte, the ais of smmetr, the maimum or minimum value, and the domain and range? Step 1 Compare: = 6( - 5) - 1 Step The verte is (h, k) = (5, -1). = a( - h) + k Step 3 The ais of smmetr is = h, or = 5. Step Step 5 Since a 7 0, the parabola opens upward. k = - 1 is the minimum value. There is no restriction on the value of, so the domain is all real numbers. Since the minimum value of the function is -1, the range is Ú Lesson 8- Verte Form of a Quadratic Function

27 Problem TEKS Process Standard (1)(D) Using Verte Form What is the graph of f () = 3( 1) +? How can ou use the values of a, h, and k to check our graph? Check that the graph is a stretched reflection of =, shifted 1 unit right and units up. Step 1 Identif the constants a = -3, h = 1, and k =. Because a 6 0, the parabola opens downward. Step Plot the verte (h, k) = (1, ) and draw the ais of smmetr = 1. f() = 3( 1) + - Step 3 Plot two points. f() = -3( - 1) + = 1. Plot (, 1) and the reflected point (0, 1). Step Sketch the curve. Problem 3 Writing a Quadratic Function in Verte Form and in Standard Form A Nature The picture shows the jump of a cricket. What quadratic function models the path of the cricket s jump? Write the function in verte form. What is the verte? Use the verte to identif h and k. Choose another point, (6, 5), from the path. Substitute for h, k,, and f() in the verte form. Solve for a. The verte is (3, 6). h = 3, k = 6 f() = a( h) + k 5 = a(6 3) = 9a = 9a a = Substitute for a, h, and k in the verte form. f() = 1 9 ( 3) + 6 models the path of the cricket s jump. continued on net page 37

28 Problem 3 continued B Write the function in standard form. f () =- 1 9 ( - 3) + 6 f () =- 1 9 ( ) + 6 f () = ( ) + 6 f () = Write the equation in verte form. Square the binomial. Use the Distributive Propert. Simplif. The function in standard form is f () = ONLINE H O M E W O R K PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. For additional support when completing our homework, go to. For each quadratic function in verte form, identif the verte of the parabola, the ais of smmetr, whether the parabola opens upward or downward, and the domain and range of the function. 1. f () = ( - 3) f () =- 1 3 ( + 6) f () = 3 ( - 5 ) - 9. f () =-1( - 1) f () = f () = ( + 11) Sketch the graph of each of these functions. Label the verte. 7. f () = ( - ) f () =-( + ) f () = 1 ( - 3.5) Use Representations to Communicate Mathematical Ideas (1)(E) Compare the graphs of the functions f () = ( - 1) + 5 and g () =-( - 1) + 5. In our answer, use the concepts of verte, range, minimum, and maimum. Rewrite these functions from verte form to standard form. Then use either form to calculate f (1) and f ( 1). 11. f () = ( + 7) - 1. f () =- 1 ( + 3 ) f () = 3( - 3) Eplain Mathematical Ideas (1)(G) Does the graph of a quadratic function depend on whether it is written in verte form or standard form? Eplain. 38 Lesson 8- Verte Form of a Quadratic Function

29 For each graph, write a quadratic function in verte form and standard form (1, 6) (, 5) O ( 3, ) (0, 0.5) - - O - For each description, write a quadratic function in verte form and standard form. 17. The parabola with verte (3, - ) and that passes through (, -.5) 18. The parabola with verte (0, 5) and that passes through the point (-3, -13) 19. The parabola with verte (-10, 0) and that passes through (1, 60.5) TEXAS End-of-Course PRACTICE 0. Which equation describes the parabola shown in the graph? - - O - - A. f () =-( - 3) - 1 C. f () =-( + 3) B. f () = ( - 3) - 1 D. f () =-( + 3) Which function is equivalent to f () =-( - ) +? F. f () =- + 8 H. f () =- + 0 G. f () = J. f () = Find the range of the function f () =-3( +.5) A. f () 5.9 C. f () -.5 B. f () Ú 5.9 D. f () Ú Identif the verte and one other point on the graph of the function f () =-( + 1) -. Then use the ais of smmetr to identif a second point on the graph. 39

30 8-5 Solving Quadratic Equations TEKS FOCUS TEKS (8)(A) Solve quadratic equations having real solutions b factoring, taking square roots, completing the square, and appling the quadratic formula. TEKS (1)(C) Select tools, including real objects, manipulatives, paper and pencil, and technolog as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems. Additional TEKS (1)(D), (7)(A) VOCABULARY Quadratic equation A quadratic equation is one that can be written in the standard form a + b + c = 0, where a 0. Root of an equation a solution of an equation of the form f() = 0 Standard form of a quadratic equation The standard form of a quadratic equation is a + b + c = 0, where a 0. Zero of a function A zero of a function f() is an value of for which f() = 0. Number sense the understanding of what numbers mean and how the are related ESSENTIAL UNDERSTANDING Quadratic equations can be solved b a variet of methods, including graphing and finding square roots. Ke Concept Standard Form of a Quadratic Equation A quadratic equation is an equation that can be written in the form a + b + c = 0, where a 0. This form is called the standard form of a quadratic equation. Ke Concept Solve Quadratic Equations b Graphing One wa to solve a quadratic equation a + b + c = 0 is to graph the related quadratic function = a + b + c. The solutions of the equation are the -intercepts of the related function. To solve 0, graph. 3 1 O 3 The solutions of 0 are the -intercepts and. A quadratic equation can have two, one, or no real-number solutions. In a future course ou will learn about solutions of quadratic equations that are not real numbers. In this course, solutions refers to real-number solutions. The solutions of a quadratic equation and the -intercepts of the graph of the related function are often called roots of the equation or zeros of the function. 350 Lesson 8-5 Solving Quadratic Equations

31 Problem 1 TEKS Process Standard (1)(D) Solving b Graphing What are the solutions of each equation? Use a graph of the related function. A 1 = 0 B Graph = - 1. = 0 C Graph =. 3! O!! There are two solutions, {1. O! Graph = + 1. What feature of the graph shows the solutions of the equation? The -intercepts show the solutions of the equation. + 1 = 0 O!! There is one solution, 0. There is no real-number solution. Problem TEKS Process Standard (1)(C) Using the Graph of a Quadratic Function A What tool would ou use to solve the quadratic equation = 0? Eplain our choice. The solutions of the equation are the zeros of the function (f )() = A graphing calculator is a good tool to use to identif the zeros of a quadratic function, which correspond to the -intercepts of the graph of the function. If ou graph the function with paper and pencil, ou ma not be able to determine the intercepts ver accuratel. When ou use technolog, ou can get a good approimation. B Use a graphing calculator to solve = 0. Step 1 Enter = on the Y = screen. Use the CALC feature. Select ZERO. Step Left bound? X! " Y! Move the cursor to the left of the first -intercept. Press enter to set the left bound. Step 3 Right bound? X! Step Y! ".515 Move the cursor slightl to the right of the intercept. Press enter to set the right bound. Zero X! Y!0 Press enter to displa the first zero, which is about Repeat Steps to identif the second zero from the other -intercept. The second zero is about 5.5. So the solutions of the equation are about 0.55 and

32 Problem 3 Solving Using Square Roots What are the solutions of 3 75 = 0? How do ou know ou can solve using square roots? The equation has an -term and a constant term, but no -term. So, ou can write the equation in the form = k and then find the square roots of each side. Write the original equation. Isolate on one side of the equation. Find the square roots of each side and simplif = 0 3 = 75 = 5 = t 5 = t5 Problem Choosing a Reasonable Solution Aquarium An aquarium is designing a new ehibit to showcase tropical fish. The ehibit will include a tank that is a rectangular prism with a length O that is twice the width w. The volume of the tank is 0 ft 3. What is the width of the tank to the nearest tenth of a foot? < w 3 ft How can ou write the length of the tank? The length / is twice the width w, so write the length as w. V = /wh Use the formula for volume of a rectangular prism. 0 = (w)(w)(3) Substitute 0 for V, w for /, and 3 for h. 0 = 6w Simplif. 70 = w Divide each side b 6. { 70 = w Find the square roots of each side. { w Use a calculator. A tank cannot have a negative width, so onl the positive square root makes sense. The tank will have a width of about 8. ft. 35 Lesson 8-5 Solving Quadratic Equations

33 HO ME RK O NLINE WO PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. Use a graphing calculator to graph the function and approimate the -intercepts. Then write the zeros of the function. Round to the nearest hundredth. For additional support when completing our homework, go to. 1. f () = f () = f () = Select Tools to Solve Problems (1)(C) Eplain wh a graphing calculator is a good tool to use to find the -intercepts of the function f () = Use a graphing calculator to graph the function and approimate the -intercepts to the nearest hundredth. What are the zeros of the function? 5. Appl Mathematics (1)(A) You have enough paint to cover an area of 50 ft. What is the side length of the largest square that ou could paint? Round our answer to the nearest tenth of a foot. 6. Appl Mathematics (1)(A) You have enough shrubs to cover an area of 100 ft. What is the radius of the largest circular region ou can plant with these shrubs? Round our answer to the nearest tenth of a foot. Tell how man solutions each equation has. 7. h = c - 18 = 9 9. s - 35 = Appl Mathematics (1)(A) A circular aboveground pool has a height of 5 in. and a volume of 1100 ft3. What is the radius of the pool to the nearest tenth of a foot? Use the equation V = pr h, where V is the volume, r is the radius, and h is the height. 11. Eplain Mathematical Ideas (1)(G) For what values of n will the equation = n have two solutions? Eactl one solution? No solution? 5 in. Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Solve each equation b graphing the related function. If the equation has no real-number solution, write no solution = = = = = = 0 Model each problem with a quadratic equation. Then solve. If necessar, round to the nearest tenth. 18. Find the length of a side of a square with an area of 169 m. 19. Find the length of a side of a square with an area of 75 ft. 0. Find the radius of a circle with an area of 90 cm. 353

34 1. The trapezoid has an area of 1960 cm. Use the formula A = 1 h(b 1 + b ), where A represents the area of the trapezoid, h represents its height, and b 1 and b represent its bases, to find the value of. Solve each equation b finding square roots. If the equation has no real-number solution, write no solution.. n = a = 3. k = 0 5. r + 9 = 9 6. w - 36 =-6 7. g = 5 8. Appl Mathematics (1)(A) You are making a square quilt with the design shown at the right. Find the side length of the inner square that would make its area equal to 50% of the total area of the quilt. Round to the nearest tenth of a foot. Solve each equation b finding square roots. If the equation has no real-number solution, write no solution. If a solution is irrational, round to the nearest tenth z - 7 = p - 16 = m = 0 6 ft STEM 3. Find the value of c such that the equation - c = 0 has 1 and -1 as solutions. 33. Appl Mathematics (1)(A) The equation d = 1 at gives the distance d that an object starting at rest travels given acceleration a and time t. Suppose a ball rolls down the ramp shown at the right with acceleration a = ft>s. Find the time it will take the ball to roll from the top of the ramp to the bottom. Round to the nearest tenth of a second. 3. Analze Mathematical Relationships (1)(F) Describe and correct the error made in solving the equation. 35. Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Write and solve an equation in the form a + c = 0, where a 0, that satisfies the given condition. a. The equation has no solution. b. The equation has eactl one solution. c. The equation has two solutions. Find the value of h for each triangle. If necessar, round to the nearest tenth ft = 0 = 100 = ± 10 A = 0 ft h h A = 10 cm h h 35 Lesson 8-5 Solving Quadratic Equations

35 38. a. Solve the equation ( - 7) = 0. b. Find the verte of the graph of the related function = ( - 7). c. Choose a value for h and repeat parts (a) and (b) using ( - h) = 0 and = ( - h). d. Where would ou epect to find the verte of the graph of = ( + )? Eplain. 39. Select Tools to Solve Problems (1)(C) You can use a spreadsheet like the one at the right to solve a quadratic equation. a. What spreadsheet formula would ou use to find the value in cell B? b. Use a spreadsheet to find the solutions of the quadratic equation 6 - = 0. Eplain how ou used the spreadsheet to find the solutions. c. Suppose a quadratic equation has solutions that are not integers. How could ou use a spreadsheet to approimate the solutions? A B 6^ 0 TEXAS End-of-Course PRACTICE 0. A package is shaped like a rectangular prism. The length and the width are equal. The volume of the package is 3 ft 3. The height is ft. What is its length? A. - ft B. ft C. 8 ft D. 16 ft 1. What is the -intercept of the line with equation = 3 -? F. - G. -3 H. 3 J.. What is the domain of the relation {(3, -1), (, ), ( -, 5), (1, 0)}? A. {-1, 0,, 5} B. {0,, 5} C. {-, 1, 3, } D. {1, 3, } 3. What is the solution of the inequalit ? F. - G. Ú- H. J. Ú. The surface area of a cube is 96 ft. a. What is the length of each edge? Show our work. b. Suppose ou double the length of each edge. What happens to the surface area of the cube? Show our work. 355

36 TEKS FOCUS 8-6 Factoring to Solve Quadratic Equations VOCABULARY TEKS (8)(A) Solve quadratic equations having real solutions b factoring, taking square roots, completing the square, and appling the quadratic formula. TEKS (1)(E) Create and use representations to organize, record, and communicate mathematical ideas. Additional TEKS (1)(A) Zero-Product Propert For all real numbers a and b, if ab = 0, then a = 0 or b = 0. Representation a wa to displa or describe information. You can use a representation to present mathematical ideas and data. ESSENTIAL UNDERSTANDING You can solve some quadratic equations a + b + c = 0, including equations where b 0, b using the Zero-Product Propert. Propert Zero-Product Propert For an real numbers a and b, if ab = 0, then a = 0 or b = 0. Eample If ( + 3)( + ) = 0, then + 3 = 0 or + = 0. Problem 1 TEKS Process Standard (1)(E) Using the Zero-Product Propert What are the solutions of the equation (t + 1)(t ) = 0? (t + 1)(t - ) = 0 t + 1 = 0 or t - = 0 Use the Zero-Product Propert. t =-1 or t = Solve for t. How else can ou write the solutions? You can write the solutions using set notation: 5-1, 6. t =- 1 or t = The solutions are - 1 and. 356 Lesson 8-6 Factoring to Solve Quadratic Equations

37 Problem Solving b Factoring Multiple Choice What are the solutions of the equation = 0? How can ou factor ? Find two integers with a product of 15 and a sum of 8. -5, -3-3, 5-5, 3 3, = 0 ( + 3)( + 5) = 0 Factor = 0 or + 5 = 0 Use the Zero-Product Propert. =-3 or =-5 Solve for. The solutions are -3 and -5. The correct answer is A. Problem 3 Writing in Standard Form First What are the solutions of 1 = 18? Wh do ou need to subtract 18 from each side before ou factor? To use the Zero-Product Propert, one side of the equation must be zero. - 1 = = 0 Subtract 18 from each side. ( + 3)( - 6) = 0 Factor = 0 or - 6 = 0 Use the Zero-Product Propert. =-3 or = 6 Solve for. =- 3 or = 6 The solutions are - 3 and 6. Problem TEKS Process Standard (1)(A) Using Factoring to Solve a Real-World Problem Photograph You are constructing a frame for the rectangular photo shown. You want the frame to be the same width all the wa around and the total area of the frame and photo to be 315 in.. What should the outer dimensions of the frame be? 17 in. 11 in. The size of the photo is 11 in. b 17 in. The total area is 315 in.. The outer dimensions of the frame Write the frame s outer dimensions in terms of its width. Use these dimensions to write an equation for the area of the frame and photo. continued on net page 357

38 Problem continued ( + 11)( + 17) = 315 Width * Length = Area = 315 Find the product ( + 11)( + 17). Wh can ou ignore the factor of? B the Zero-Product Propert, one of the factors,, + 16, or -, must equal 0. Since 0, either + 16 or - equals = 0 Subtract 315 from each side. ( + 1-3) = 0 Factor out. ( + 16)( - ) = 0 Factor = 0 or - = 0 Use the Zero-Product Propert. =-16 or = Solve for. The onl reasonable solution is. So the outer dimensions of the frame are () + 11 in. b () + 17 in., or 15 in. b 1 in. ONLINE H O M E W O R K PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. For additional support when completing our homework, go to. STEM Use Representations to Communicate Mathematical Ideas (1)(E) Use the Zero-Product Propert to solve each equation. 1. ( - 9)( - 8) = 0. (k + 5)(k + 7) = 0 3. n(n + ) = 0. Eplain Mathematical Ideas (1)(G) Describe and correct the error made in solving the equation. 5. A bo shaped like a rectangular prism has a volume of 80 in. 3. Its dimensions are in. b (n + ) in. b (n + 5) in. Find n. 6. Eplain Mathematical Ideas (1)(G) How man solutions does an equation of the form - k = 0 have? Eplain. 7. Appl Mathematics (1)(A) You are knitting a blanket. You want the area of the blanket to be ft. You want the length of the blanket to be ft longer than its width. What should the dimensions of the blanket be? 8. Appl Mathematics (1)(A) You are building a rectangular deck. The area of the deck should be 50 ft. You want the length of the deck to be 5 ft longer than twice its width. What should the dimensions of the deck be? 9. Appl Mathematics (1)(A) You have a rectangular koi pond that measures 6 ft b 8 ft. You have enough concrete to cover 7 ft for a walkwa, as shown in the diagram. What should the width of the walkwa be? 10. Find the zeros of the function f () = b factoring. How can ou verif the zeros of the function are correct b looking at the graph? + 3 = 0 ( + 3) = 0 = 0 or + 3 = 0 = 0 or = ft 6 ft 358 Lesson 8-6 Factoring to Solve Quadratic Equations

39 Write each equation in standard form. Then solve n + 16n + 15 = n q + 3q = 3q - q + 18 Solve b factoring = 0 1. z - 1z - 36 = p - p = c = 5c 17. w - 11w = b = Use a Problem-Solving Model (1)(B) You throw a softball into the air with an initial upward velocit of 38 ft/s and an initial height of 5 ft. a. Use the vertical motion model to write an equation that gives the ball s height h, in feet, at time t, in seconds. b. The ball s height is 0 ft when it is on the ground. Solve the equation ou wrote in part (a) for h = 0 to find when the ball lands. Solve each cubic equation b factoring out the GCF first = = 0 Solve. Factor b grouping = = = 0 TEXAS End-of-Course PRACTICE 5. Phil, Tob, and Sam bowled four games last weekend. Their scores are shown in the Venn diagram at the right. What is the highest score that onl Tob and Sam have in common? 6. What is the negative solution of the equation = 0? 7. What is the -intercept of the line with equation 3 - = 9? 8. A rectangular card has a length 1 in. longer than twice the width and an area of 15 in.. What is the width of the card, in inches? Bowling Scores Phil Tob Sam 359

40 8-7 Writing Quadratic Functions TEKS FOCUS TEKS (6)(C) Write quadratic functions when given real solutions and graphs of their related equations. TEKS (1)(B) Use a problem-solving model that incorporates analzing given information, formulating a plan or strateg, determining a solution, justifing the solution, and evaluating the problem-solving process and the reasonableness of the solution. Additional TEKS (1)(D), (7)(B) VOCABULARY Linear factor A linear factor of an epression is a factor in the form a + b, a 0. Formulate create with careful effort and purpose. You can formulate a plan or strateg to solve a problem. Reasonableness the qualit of being within the realm of common sense or sound reasoning. The reasonableness of a solution is whether or not the solution makes sense. Strateg a plan or method for solving a problem. ESSENTIAL UNDERSTANDING You can use the solutions of a quadratic equation to write an equation for a related quadratic function. Ke Concept Relating Linear Factors, Solutions, and Zeros Each linear factor of a quadratic epression corresponds to a solution of a related quadratic equation and a zero of a related quadratic function. quadratic epression quadratic equation quadratic function = 0 f () = ( + )( - 1) You use the linear factors of the quadratic epression to find the solutions of the quadratic equation = 0 ( + )( - 1) = 0 + = 0 or - 1 = 0 =- or = 1 The solutions of the equation are the zeros of the related quadratic function, and the -intercepts of its graph. f () = f (-) = (-) + 3(-) - = = 0 f (1) = (1) + 3(1) - = = 0 (, 0) - O - (1, 0) 360 Lesson 8-7 Writing Quadratic Functions

41 Ke Concept Writing Quadratic Functions If a real number r is a solution of a quadratic equation in standard form, then - r is a factor of a related quadratic function. 8 Since each solution r corresponds to a zero of the related quadratic function, ou can use the zeros to find the factors of the function. For eample, suppose a quadratic function = f () has zeros 6 and 8. Then - 6 and - 8 are factors of f (). So the factored form of an quadratic function with zeros 6 and 8 can be written as shown below. f () = a( - 6)( - 8) 6 O (6, 0) (8, 0) Problem 1 Relating Linear Factors and Zeros In addition to -intercepts and the verte, what other point helps ou sketch the graph? When = 0, =-15. The -intercept is (0, -15). A Describe the relationship between the linear factors of the quadratic epression 15 and the zeros of the related quadratic function f () = 15. The zeros of the quadratic function f () = are the solutions of the quadratic equation 0 = You can solve 0 = b factoring the epression and using the Zero-Product Propert, so each linear factor of leads to one zero of the function. 0 = = ( - 5)( + 3) - 5 = 0 or + 3 = 0 = 5 or =-3 The zeros of the function f () = are 5 and -3. B Use the solutions of 0 = 15 that ou found in part (A) to help ou graph = 15. The solutions of the equation, 5 and -3, are the zeros of the function = and the -intercepts of its graph. Plot the -intercepts and the verte. Then sketch the parabola. 10 ( 3, 0) (5, 0) - O 6 (1, 16) The -coordinate of the verte is b = 1, a and the -coordinate is 1 (1) 15 =

42 Problem TEKS Process Standard (1)(B) Using Solutions to Write Quadratic Functions The solutions of a quadratic equation are and 8. What is the standard form of a related quadratic function? Analze Given Information You know the solutions of the equation and need to use this to write the function in standard form. Formulate a Plan Use each solution r to write one linear factor - r of the function. Then multipl the two linear factors to find the standard form of the quadratic function. Determine a Solution The solutions are - and 8. Each solution r corresponds to a linear factor ( - r ). Is this the onl quadratic function with zeros of and 8? No; ever function of the form f() = a( + )( - 8), where a 0 has zeros of - and 8. f () = ( - (- ))( - 8) f () = ( + )( - 8) f () = f () = Write the function in factored form. Simplif. Multipl the factors. Simplif to write the function in standard form. Justif the Solution Check our work b graphing f () = and verifing that the -intercepts of the graph match the given solutions of the equation. The -intercepts of the parabola are - and 8, which are the given solutions of the equation. Evaluate the Problem Solving Process The plan to use the solutions to write a function in factored form and then rewriting the function in standard form worked. Checking the solution b graphing confirms that the function is correct. You can also check b substituting - and 8 in the standard form of the equation to see if both values satisf the equation. ( -) - 6(-) - 16 = 0 (8) - 6(8) - 16 = = = 0 Both values make the equation true. scale: 1 scale: 5 0 = 0 0 = 0 36 Lesson 8-7 Writing Quadratic Functions

43 Problem 3 TEKS Process Standard (1)(B) Using a Graph to Write a Quadratic Function What is a quadratic function in standard form for the parabola shown? Wh do ou multipl the product of the factors b a constant a? There are infinitel man quadratic functions with roots - 1 and, each determined b a unique constant factor. Analze Given Information The graph gives the -intercepts, which are the zeros of the function, and shows that f (0) = -. Formulate a Plan Use the zeros to write the factored form of the function. O!! Determine a Solution The zeros are -1 and. f () = a( - ( -1))( - ) Write the function in factored form. f () = a( - - ) Mulitpl and simplif. Use the fact that f (0) = - to solve for a. f (0) = a(0-0 - ) Substitute 0 for. - = - a Substitute - for f (0) and simplif. 1=a So f () = Solve for a Justif the Solution Check the solution b graphing the function in standard form on a graphing calculator. Evaluate the Reasonableness of the Solution The graph of the function in standard form is the same as the graph above. The solution is reasonable. Problem Writing a Quadratic Function Using Transformations The blue parabola is a translation of the graph of the parent quadratic function f () =. What is the function written in standard form for the translated parabola? 8 6 What is the verte form of a quadratic function? If the verte of the function = is shifted to (h, k), then the equation of the transformed parabola is = ( - h) + k. f() = O A translation shifts the parent function f () = to the left 8 units and up units. = ( + 8) + Write the equation in verte form. = Simplif. A quadratic function in standard form for the blue parabola is h () =

44 ONLINE H O M E W O R K PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. For additional support when completing our homework, go to. 1. Connect Mathematical Ideas (1)(F) What are the linear factors of the quadratic epression ? What are the zeros of the related quadratic function? Eplain the relationship between the factors of the epression and the zeros of the function.. Create Representations to Communicate Mathematical Ideas (1)(E) What are the solutions of = 0? Use the solutions to help ou graph the related quadratic function. Tell whether each of the following is a quadratic epression, a quadratic equation, or a quadratic function. State the linear factors of an epressions, the solutions of an equations, and the zeros of an functions. 3. f () = = = = Use the -intercepts of the parabola and the given point to write a quadratic function in factored form O (0, 8) O - ( 5, ) The solutions of a quadratic equation are given. Write a related quadratic function in factored form , , 5 Write a quadratic function in standard form for the parabola shown O O Lesson 8-7 Writing Quadratic Functions

45 15. Use a Problem-Solving Model (1)(B) The solutions of a quadratic equation are -7 and 7. Write a related quadratic function in standard form. Use a problemsolving model b analzing the given information, formulating a plan or strateg, determining a solution, justifing the solution, and evaluating the problem-solving process. 16. Evaluate Reasonableness (1)(B) Write a quadratic function in standard form for the parabola at the right. Use 6 a problem-solving model b analzing the given information, formulating a plan or strateg, determining a solution, justifing the solution, and evaluating the reasonableness of the solution. 17. Select Tools to Solve Problems (1)(C) Describe two different methods ou could use to sketch a graph of a quadratic function with zeros -8 and 3, and -intercept -. Which tool(s) would ou need for each method? 18. Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Eplain the relationship among the terms solution, zero, and linear factor as the relate to quadratic equations, functions, and epressions. 19. Appl Mathematics (1)(A) The height h, in feet, of a coin being tossed at a football game is given b the function h =-16t + 1t +, where t is the time in seconds. Sketch the graph of this function. When will the coin hit the ground? O 6 365

46 Use translations of the parent function f () = to write a quadratic function in standard form for each parabola below O O 6 8 TEXAS End-of-Course PRACTICE. What are the solutions of the equation = 0? A. -10 and -3 B. 8 and 5 C. 8 and -5 D. 10 and - 3. Which function has a graph with intercepts 1 and -5? F. f () = G. f () = H. f () = J. f () = Which function can be graphed b shifting the graph of = right 3 units and down units? A. f () = ( + 3) + B. f () = ( - ) + 3 C. f () = ( + ) - 3 D. f () = ( - 3) - 5. Sketch the graph of the function f () = Use the graph to find the zeros of the function. 6. Eplain how to use the solutions of a quadratic equation to write an equation of a related quadratic function in standard form. Is the function unique? Eplain. 366 Lesson 8-7 Writing Quadratic Functions

47 8-8 Completing the Square TEKS FOCUS TEKS (8)(A) Solve quadratic equations having real solutions b factoring, taking square roots, completing the square, and appling the quadratic formula. TEKS (1)(E) Create and use representations to organize, record, and communicate mathematical ideas. Additional TEKS (1)(A) VOCABULARY Completing the square a method of solving quadratic equations. Completing the square is a wa to rewrite a quadratic equation into the form m = n. Representation a wa to displa or describe information. You can use a representation to present mathematical ideas and data. ESSENTIAL UNDERSTANDING You can solve an quadratic equation b first writing it in the form m = n. Ke Concept Completing the Square You can model a quadratic epression using algebra tiles. The algebra tiles below represent the epression + 8. Here is the same epression rearranged to form part of a square. Notice that the -tiles have been split evenl into two groups of four. You can complete the square b adding, or 16, 1-tiles. The completed square is , or ( + ). In general, ou can change the epression + b into a perfect-square trinomial b adding 1 b to + b. This process is called completing the square. The process is the same whether b is positive or negative. 367

48 Problem 1 Can c be negative? No. c is the square of a real number, which is never negative. Finding c to Complete the Square What is the value of c such that 16 + c is a perfect-square trinomial? The value of b is -16. The term to add to - 16 is ( - 16 ), or 6. So c = 6. Problem Solving + b + c = 0 What are the solutions of the equation = 0? = 0-1 =-16 Subtract 16 from each side = Add ( - 1 ), or 9, to each side. ( - 7) = 33 Write as a square. Wh do ou write 7? t5.7 as two equations? Recall that the smbol { means plus or minus. That means - 7 equals 5.7 or = { 133 Find square roots of each side. - 7 { 5.7 Use a calculator to approimate or Write as two equations or Add 7 to each side. 1.7 or 1.6 Simplif. Problem 3 TEKS Process Standard (1)(E) Finding the Verte b Completing the Square Find the verte of = b completing the square. = Wh do ou need to add the same number to each side? You need to add the same number to each side to keep the equation balanced. - 8 = = = ( + 3) = ( + 3) - 1 The verte of = is (-3, -1). Subtract 8 from each side. Add ( 6 ), or 9, to each side. Simplif the left side and write the right side as a square. Subtract 1 from each side so that the equation is in verte form. 368 Lesson 8-8 Completing the Square

49 Problem TEKS Process Standard (1)(A) Completing the Square When a 1 Gardening You are planning a flower garden consisting of three square plots surrounded b a 1-ft border. The total area of the garden and the border is 100 ft. What is the side length of each square plot? 1 ft 1 ft 1 ft 1 ft Red tulips Yellow tulips Area of garden and border Epressions for the dimensions of the garden and border The side length of each square plot Write and solve an equation that relates the dimensions and area of the garden and border. Step 1 Write an equation that ou can use to solve the problem. ( 3 + )( + ) = 100 Length * Width = Area = 100 Find the product (3 + )( + ) = 96 Subtract from each side = 3 Divide each side b 3. Wh do ou need to find 1 ( 8 3 )? To make a perfect-square trinomial on the left side of + 8 = 3, find 3 1 ( 8 3). Then square the result and add to each side of the equation. Step Step 3 Complete the square = ( + 3 ) = 30 9 Solve the equation. + 3 = { Add ( 3 ), or 16 9, to each side. Write left side as a square and right side as a fraction. Find square roots of each side. + 3 {5.81 Use a calculator to approimate or Write as two equations..8 or -7.1 Solve for. The negative answer does not make sense in this problem. So, the side length of each square plot is about.8 ft. 369

50 ONLINE H O M E W O R K PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. Find the value of c such that each epression is a perfect-square trinomial. For additional support when completing our homework, go to c. p - 30p + c 3. k - 5k + c Solve each equation b completing the square. If necessar, round to the nearest hundredth.. g + 7g = 1 5. m + 16m = w - 1w + 13 = 0 7. Appl Mathematics (1)(A) The painting shown at the right has an area of 0 in.. What is the value of to the nearest tenth? 8. Create Representations to Communicate Mathematical Ideas (1)(E) A park is installing a rectangular reflecting pool surrounded b a concrete ( + 5) in. walkwa of uniform width. The reflecting pool will measure ft b 6 ft. There is enough concrete to cover 60 ft for the walkwa. What is the maimum width of the walkwa to the nearest tenth? 9. Eplain Mathematical Ideas (1)(G) A classmate was completing the square to solve + 10 = 8. For her first step she wrote = What was her error? 10. Eplain Mathematical Ideas (1)(G) Eplain wh completing the square is a better strateg for solving = 0 than graphing or factoring. 11. Write a quadratic equation and solve it b completing the square. Show our work. 1. Appl Mathematics (1)(A) A zoo is fencing in a rectangular area for the capbaras. It plans to enclose this area using fencing on three sides, as shown at the right. The zoo has budgeted enough mone for 75 ft of fencing material and would like to make an enclosure with an area of 600 ft. a. Let w represent the width of the enclosure. Write an epression in terms of w for the length of the enclosure. b. Write and solve an equation to find the width w. Round to the nearest tenth of a foot. c. What should the length of the enclosure be? Create Representations to Communicate Mathematical Ideas (1)(E) Find the verte of each parabola b completing the square. 13. = = = Solve each equation b completing the square. If necessar, round to the nearest hundredth. 16. w + 1w - = = n - 3n - 15 = 10 wall w w in. 600 ft² 370 Lesson 8-8 Completing the Square

51 Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Use each graph to estimate the values of for which f () = 5. Then write and solve an equation to find the values of such that f () = 5. Round to the nearest hundredth. 19. f () = f () = O 6 3 O Solve each equation. If necessar, round to the nearest hundredth. If there is no real-number solution, write no solution. 1. q + 3q + 1 = 0. s + 5s = s + 9s + 0 = 0. Suppose the prism at the right has the same surface area as a cube with edges 8 in. long. a. Write an epression for the surface area of the prism shown. b. Write an equation that relates the surface area of the prism to the surface area of the 8-in. cube. c. Solve the equation ou wrote in part (b). What are the dimensions of the prism? 7 TEXAS End-of-Course PRACTICE 5. The rectangular poster has an area 0 ft. What is the value of to the nearest tenth of a foot? 6. The width of a notebook is.15 * 10 - m. In decimal form, how man meters wide is the notebook? 7. What is the solution of the equation 19 + = 35? 8. A ribbon with straight edges has an area of in.. Its width is and its length is What is the width of the ribbon in inches? 9. What is the -intercept of the graph of + 3 = 9? 30. The sum of two numbers is 0. The difference between three times the larger number and twice the smaller number is 0. What is the larger number? ( + 1) ft ( + ) ft 371

52 8-9 The Quadratic Formula and the Discriminant TEKS FOCUS TEKS (8)(A) Solve quadratic equations having real solutions b factoring, taking square roots, completing the square, and appling the quadratic formula. TEKS (1)(C) Select tools, including real objects, manipulatives, paper and pencil, and technolog as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems. Additional TEKS (1)(A), (1)(G) VOCABULARY Discriminant The discriminant of a quadratic equation of the form a + b + c = 0 is b - ac. The value of the discriminant determines the number of solutions of the equation. Quadratic formula If a + b + c = 0 and a 0, then = - b{ 1b - ac. a Number sense the understanding of what numbers mean and how the are related. ESSENTIAL UNDERSTANDING You can find the solution(s) of an quadratic equation using the quadratic formula. Ke Concept Quadratic Formula Algebra If a + b + c = 0, and a 0, then = -b { b - ac a Eample Suppose = 0. Then a =, b = 3, and c = -5. Therefore = - (3) { (3) - ()(-5) () Ke Concept Methods for Solving a Quadratic Equation There are man methods for solving a quadratic equation. Method Graphing Square roots Factoring Completing the square Quadratic formula When to Use Use if ou have a graphing calculator hand. Use if the equation has no -term. Use if ou can factor the equation easil. Use if the coefficient of is 1, but ou cannot easil factor the equation. Use if the equation cannot be factored easil or at all. 37 Lesson 8-9 The Quadratic Formula and the Discriminant

53 Ke Concept Deriving the Quadratic Formula Here s Wh It Works If ou complete the square for the general equation a + b + c = 0, with a 0, ou can derive the quadratic formula. Step 1 Write a + b + c = 0 so the coefficient of is 1. a + b + c = 0 + a b + a c = 0 Divide each side b a, a 0. Step Complete the square. + a b =- a c + b a + b ( a ) =- c a + ( b a ) ( + a) b =- c a + b a ( + a) b =- ac a + b a Step 3 Solve the equation for. Subtract c a from each side. Add ( b a ) to each side. Write the left side as a square. Multipl - a c b a a to get like denominators. ( + a) b = b - ac a Simplif the right side. 5 ( + a) b = { 5 b - ac a Take square roots of each side. + b a = { b - ac a =- b a { b - ac a = -b { b - ac a Simplif the right side. Subtract b a from each side. Simplif. Ke Concept Using the Discriminant of a + b + c = 0 Discriminant b - ac 7 0 b - ac = 0 b - ac 6 0 Eample Graph of = a + b + c = 0 The discriminant is (-6) - (1)(7) = 8, which is positive. O = 0 The discriminant is (-6) - (1)(9) = 0. O = 0 The discriminant is (-6) - (1)(11) =-8, which is negative. O Number of Solutions There are two real-number solutions. There is one realnumber solution. There are no realnumber solutions. 373

54 Problem 1 Wh do ou need to write the equation in standard form? You can onl use the quadratic formula with equations in the form a + b + c = 0. Using the Quadratic Formula What are the solutions of 8 =? Use the quadratic formula = 0 = -b { b - ac a = - (-) { (-) - (1)(-8) (1) Write the equation in standard form. Use the quadratic formula. Substitute 1 for a, - for b, and -8 for c. = { 136 Simplif. = + 6 or = - 6 Write as two equations. = or =- Simplif. Problem TEKS Process Standard (1)(A) Finding Approimate Solutions Sports In the shot put, an athlete throws a heav metal ball through the air. The arc of the ball can be modeled b the equation = , where is the horizontal distance, in meters, from the athlete and is the height, in meters, of the ball. How far from the athlete will the ball land? Wh do ou substitute 0 for? When the ball hits the ground, its height will be 0. 0 = Substitute 0 for in the given equation. = -b { b - ac a = { (-0.0)() (-0.0) = = { or = Use the quadratic formula. Substitute -0.0 for a, 0.8 for b, and for c. Simplif Write as two equations or 3.16 Simplif. Onl the positive answer makes sense in this situation. The ball will land about 3.16 m from the athlete. 37 Lesson 8-9 The Quadratic Formula and the Discriminant

55 Problem 3 TEKS Process Standard (1)(C) Choosing an Appropriate Method Which method(s) would ou choose to solve each equation? Eplain our reasoning. Can ou use the quadratic formula to solve part (A)? Yes. You can use the quadratic formula with a = 3, b = 0, and c =-9. However, it is faster to use square roots. Using number sense and mental math to identif relationships among the coefficients of a quadratic equation can help ou choose a solution method. A 3 9 = 0 B 30 = 0 C = 0 D = 0 Square roots; there is no -term. Factoring; the equation is easil factorable. Quadratic formula, graphing; the equation cannot be factored. Quadratic formula, completing the square, or graphing; the coefficient of the -term is 1, but the equation cannot be factored. E = 0 Quadratic formula, graphing; the equation cannot be factored easil since the numbers are large. Problem Using the Discriminant How man real-number solutions does 3 = 5 have? Can ou solve this problem another wa? Yes. You could actuall solve the equation to find an solutions. However, ou onl need to know the number of solutions, so use the discriminant. Write the equation in standard form. Evaluate the discriminant b substituting for a, - 3 for b, and 5 for c. Draw a conclusion = 0 b ac = ( 3) ()(5) = 31 Because the discriminant is negative, the equation has no real-number solutions. 375

56 ONLINE H O M E W O R K PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. Use the quadratic formula to solve each equation. For additional support when completing our homework, go to = = = = = = Appl Mathematics (1)(A) A football plaer punts a ball. The path of the ball can be modeled b the equation = , where is the horizontal distance, in feet, the ball travels and is the height, in feet, of the ball. How far from the football plaer will the ball land? Round to the nearest tenth of a foot. Use the quadratic formula to solve each equation. Round our answer to the nearest hundredth = = =-5 Select Techniques to Solve Problems (1)(C) Use techniques such as number sense and mental math to help ou choose one or more methods of solving each equation. Eplain our reasoning = = = 0 1. Appl Mathematics (1)(A) Your school wants to take out an ad in the paper congratulating the basketball team on a successful season, as shown at the right. The area of the photo will be half the area of the entire ad. What is the value of? 15. Eplain Mathematical Ideas (1)(G) You operate a dog-walking service. You have 50 customers per week when ou charge $1 per walk. For each $1 decrease in our fee for walking a dog, ou get 5 more customers per week. Can ou ever earn $750 in a week? Eplain. 16. Eplain Mathematical Ideas (1)(G) How can ou use the discriminant to write a quadratic equation that has two solutions? 17. Eplain Mathematical Ideas (1)(G) Describe and correct the error at the right that a student made in finding the discriminant of = Find the discriminant and the solution of each equation in parts (a) (c). If necessar, round to the nearest hundredth. a = 0 b = 0 c = 0 d. Eplain Mathematical Ideas (1)(G) When the discriminant is a perfect square, are the solutions rational or irrational? Eplain. 7 in. Photo 5 in. a =, b = 5, c = 6 b ac = 5 ()(-6) = 5 8 = Lesson 8-9 The Quadratic Formula and the Discriminant