Algebra 2 Unit 2 Practice

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1 Algebra Unit Practice LESSON Consider a rectangle that has a perimeter of 80 cm. a. Write a function A(l) that represents the area of the rectangle with length l.. A rectangle has a perimeter of 160 cm. What is the maimum area? A. 160 cm B. 800 cm C cm D cm b. Graph the function A(l). Use an appropriate scale and label the aes. 3. Make sense of problems. Madison purchased 40 ft of fencing to build a corral for her horses. If each horse requires 600 ft of space, what is the maimum number of horses Madison can put in the corral she builds with the fencing? Eplain. c. Is an area of 6 cm possible? How do ou know? What is the length and width of the rectangle? 4. Chance has 60 ft of fencing to build a dog pen. He plans to build the pen using one side of a 0-ft-long building. He will use all of the fencing for the other three sides of the pen. Use the area function for this rectangle to determine the area of the pen. d. What are the dimensions of a rectangle with a perimeter of 80 cm and an area of 300 cm?. How is the maimum value of a quadratic function represented on the graph of the function? e. What are the reasonable domain and range of A(l )? Epress our answers as inequalities, in interval notation, and in set notation. f. What is the greatest area that the rectangle can have? Eplain. Give the dimensions of the rectangle with the greatest area and describe its shape. LESSON 7-6. Factor b coping and completing the graphic organizer. Then check College Board. All rights reserved. 1 SpringBoard Algebra, Unit Practice

2 7. Factor each quadratic epression. a. 3 b c d For each set of solutions, write a quadratic equation in standard form. a. 3, b. 4, 1 c., 3 d. 7, 3 e. 1 1 f e. 1, 1 f. 3, 1 g h g. 3 4, 3 h. 3, 1 3 i j Which of the following is the factored form of ? A. (6 3)( 1 6) B. ( 3)(6 1 6) C. (4 6)(3 1 3) D. (3 6)(4 1 3) 9. Reason abstractl. Given that b is positive and c is negative in the quadratic epression a 1 b 1 c, what can ou conclude about the constant terms in the factored form? 13. Julio has 00 ft of fencing to put around a field that has an area of 00 sq ft. Which equation can be used to find the length of the field? A. l 0l B. l 00l C. l 0l 00 0 D. l 0l Make use of structure. What propert do ou use to solve a quadratic equation b factoring? Eplain.. Make use of structure. Can the quadratic epression 1 1 be factored using integers? Eplain. LESSON Solve each equation b factoring. a b. 3 0 c d Make sense of problems. Akiko wants to fence in her 1 ft b ft vegetable garden. There will be a path ft wide between the garden and the fence. The area to be enclosed b the fence will be 360 sq ft. a. Model with mathematics. Draw a diagram of the situation. b. Write a quadratic equation that can be used to determine the value of. e f. 1 3 c. Solve the equation b factoring. g h d. Interpret the solutions. i j College Board. All rights reserved. SpringBoard Algebra, Unit Practice

3 LESSON For what values of is the product ( 1 )( 1) positive? Eplain. 17. Use the number line provided to solve each inequalit. a. 1 $ 0 0. Model with mathematics. Simon wants to enclose a rectangular corral net to the barn. The side of the barn will form one side of the corral. The other three sides will be fencing. Simon purchased 10 ft of fencing and wishes to enclose an area of at least 00 sq ft. a. Write an inequalit in terms of l that represents the possible area of the pen. b. Write the inequalit in standard form with integer coefficients b. 1 8, 0 c. Factor the inequalit. d. Determine the possible lengths and widths of the corral Which of the following is the solution set to the quadratic inequalit, 1 6? A., or. 3 B. 3,, C.,, 3 D., 3 or. 19. Solve each inequalit. a. ( 3)( 1 ), 0 b. ( 4)( 1) $ 0 c. 4 8 $ 0 d. 9 # 0 LESSON Write each number in terms of i. a. 36 b. 11 c. d. 4 e. 7 f. 98 g. 48 h. 900 e f , 0 g h $ 0. Make use of structure. Which of the following numbers can be written as i? A. B. C. i D. i 3 01 College Board. All rights reserved. SpringBoard Algebra, Unit Practice

4 3. Graph each comple number on the comple plane. a i b. 3 i c. 4i d. i imaginar ais. The sum of two numbers is 1, and their product is 40. a. Let represent one of the numbers, and write an epression for the other number in terms of. Use the epression to write an equation that models the situation given above. b. Use the Quadratic Formula to solve the equation. Write the solutions in terms of i. real ais LESSON 8-6. Find each sum or difference. a. (3 1 7i) 1 (9 i) b. ( 3i) (6 7i) c. ( 1 8i) 1 (7 3i) 4. Name the comple number represented b each labeled point on the comple plane. 8 6 C 4 D imaginar ais E B A real ais d. (1 1 6i) (9 4i) 1 3 e. 13i 1i f. (3 i) ( 1 6) i g. (3 14 i) (3 4) i h. i 1 (3 i) 7. Multipl. Write each product in the form a 1 bi. a. ( 1 i)(3 i) b. (8 3i)( 1 i) c. ( 1 i)( i) d. (7 1 3i)(3 1 i) a. point A b. point B e. (3 4i)(6 i) f. ( 1 i)(3 4i) c. point C d. point D g. (1 i)(3 1 i) h. ( 1 4i)( 3i) e. point E 4 01 College Board. All rights reserved. SpringBoard Algebra, Unit Practice

5 8. Divide. Write each quotient in the form a 1 bi. 3 i 1 3i a. b. 1i 4 i 33. Which are the solutions of the quadratic function f () 4 1 9? 1 A. i 3, 1 i 3 4 B. i 9, 4 i 9 c. 3 1i 3i d. 8 7i 1 i C. i 3, 3 i 3 D. i, 3 i 34. What are the solutions of each quadratic function? e. 11 4i 1i f. 3i i a b g. 1 3i h. 1i 3i c d Make use of structure. Give an eample of a comple number ou could subtract from i that would result in a real number. Show that the difference of the comple numbers is equal to a real number. 3. Attend to precision. What are the solutions of the equation 9 1 3? Show our work. LESSON Attend to precision. Solve the equation 3( ) 7 0, and eplain each of our steps. 30. Which of the following is the comple conjugate of 3 1 7i? A i B. 3 7i C. 3 7i D i LESSON Use comple conjugates to factor each epression. a b Solve for. a b. 9 0 c d e. 4( 3) 81 0 f. ( 1 9) c d Solve each equation b factoring. a. 1 0 b c d g. 3( 4) 0 h. 7( 1 ) Which is NOT a perfect square trinomial? A. 1 1 B C D College Board. All rights reserved. SpringBoard Algebra, Unit Practice

6 39. Use the method of completing the square to make a perfect square trinomial. Then factor the perfect square trinomial. a. 1 8 b Which quadratic equation would be best solved b using the Quadratic Formula to find the solutions? A B C D Solve for b completing the square. a. 8 1 b Solve each quadratic equation b using an of the methods ou have learned. For each equation, tell which method ou used and wh ou chose that method. a b. 1 0 c d LESSON Write the Quadratic Formula. c d. ( ) Solve each equation using the Quadratic Formula. a b. 1 c d Gaetano shoots a basketball from a height of 6. ft with an initial vertical velocit of 17 ft/s. The equation 16t 1 17t 1 6. can be used to determine the time t in seconds at which the ball will have a height of ft, the same height as the basket. a. Solve the equation b using the Quadratic Formula. e f b. Attend to precision. To the nearest tenth of a second, when will the ball have a height of ft? g h c. Eplain how ou can check that our answers to part b are reasonable College Board. All rights reserved. SpringBoard Algebra, Unit Practice

7 LESSON Write the discriminant of the quadratic equation a 1 b 1 c 0. Eplain how it is used. 47. For each equation, compute the value of the discriminant and describe the solutions without solving the equation. a b LESSON Which equation does the graph represent? 1 c d The discriminant of a quadratic equation is less than 0. What is the nature of the solutions of the equation? A. one rational solution B. two comple conjugate solutions C. two irrational D. two rational solutions solutions 49. A quadratic equation has one real, rational solution. a. What is the value of the discriminant? A. ( 1) 1 3 B. ( 1) 3 C. ( 1 1) 1 3 D. ( 1 1) 3. A parabola has a focus of (, 3) and a directri of 1. Answer each question about the parabola, and eplain our reasoning. a. What is the ais of smmetr? b. Reason abstractl. Give an eample of a quadratic equation that has one real, rational solution. 0. a. Under what circumstances will the radicand in the Quadratic Formula be positive? b. What is the verte? b. What does this tell us about the solutions? c. When will the solutions be rational? c. In which direction does the parabola open? 7 01 College Board. All rights reserved. SpringBoard Algebra, Unit Practice

8 3. Reason quantitativel. Use the given information to write the equation of each parabola. a. verte: (, 3); directri: LESSON - 6. Write the equation of the quadratic function whose graph passes through each set of points. a. (1, 1), (1, 4), (, 3) b. focus: (, 4); directri: c. ais of smmetr: 0; verte: (0, 0); directri: 4 b. (0, 1), (, 7), (3, 14) d. focus: (3, ); verte: (1, ) c. (, 11), (1, ), (3, 6) 1 4. The equation of a parabola is Identif the verte, ais of smmetr, focus, and directri of the parabola.. Graph the parabola given b the equation 1 ( 13) 1. ( ). d. (, 3), (0, 1), (1, 6) 7. The table below shows the first few terms of a quadratic function. Write a quadratic equation in standard form that describes the function f ( ) Which equation describes the parabola that passes through the three points (0, 14), (3, 4), and (, 4)? A B C D College Board. All rights reserved. SpringBoard Algebra, Unit Practice

9 9. Graph the quadratic function that passes through the points (0, ), (, 11), and (, 3). 6. The tables show time and height data for two rockets. Rocket A Time (s) Height (m) Rocket B Time (s) Height (m) a. Use appropriate tools strategicall. Use a graphing calculator to perform a quadratic regression for each data set. Write the equations of the quadratic models. Round the coefficients and constants to the nearest tenth. 60. Reason quantitativel. The graph of a quadratic function passes through the point (, ). The verte of the graph is (1, 3). a. Use smmetr to identif another point on the graph of the function. Eplain how ou determined our answer. b. Write the equation of the quadratic function in standard form. LESSON Tell whether a linear model or a quadratic model is a better fit for each data set. Justif our answer. a b. Use our models to predict which rocket had a greater maimum height. Eplain. c. Use our models to predict which rocket hit the ground first and how much sooner. 63. Which quadratic model is the best fit for the data in the table? Use our calculator A B C D What is the least number of points that are needed to perform a quadratic regression on a graphing calculator? Eplain. b College Board. All rights reserved. SpringBoard Algebra, Unit Practice

10 6. The Qualit Shoe Compan tests different prices of a new tpe of shoe at different stores. The table shows the relationship between the selling price of a pair of shoes and the monthl revenue per store the compan made from selling the shoes. Selling Price ($) Monthl Revenue per Store ($) , ,60 300, LESSON Describe each function as a transformation of f(). Then use the information to graph each function on the coordinate grid. a. f() 3 f () a. Use a graphing calculator to determine the equation of a quadratic model that can be used to predict, the monthl revenue per store in dollars when the selling price for each pair of shoes is dollars. Round values to the nearest hundredth. b. f() ( 3) f () b. Is a quadratic model a good model for the data set? Eplain. c. Use our model to determine the price at which the compan should sell the shoes to generate the greatest revenue. 01 College Board. All rights reserved. SpringBoard Algebra, Unit Practice

11 c. f() ( 1 ) 1 3 f () 67. Each function graphed below is a transformation of f(). Describe each transformation and write the equation of the transformed function. a. g() b. h() d. f() ( ) 3 f() c. j() College Board. All rights reserved. SpringBoard Algebra, Unit Practice

12 d. k() LESSON Describe the graph of each function as a transformation of the graph of f(). Then use the information to graph each function on the coordinate plane. a. f() 1 3 f () 68. Make use of structure. p() ( 1 1) is a transformation of f(). Which is a description of the transformation? A. translation 1 unit to the right and units down B. translation 1 unit to the left and units down C. translation units to the right and 1 unit up D. translation units to the left and 1 unit up 69. What is the verte of the function g() ( 1 ) 4? Justif our answer in terms of a translation of f(). b. f() 1 f () 70. What is the ais of smmetr of the function h() ( 3) 1 1? Justif our answer in terms of a translation of f() College Board. All rights reserved. SpringBoard Algebra, Unit Practice

13 1 c. f() 3 f () 7. Each function graphed below is a transformation of f(). Describe the transformation and write the equation of the transformed function below each graph. a. g() d. f() f () b. h() College Board. All rights reserved. SpringBoard Algebra, Unit Practice

14 c. j() f() is translated 1 unit down, shrunk b a factor of 1, and reflected over the -ais. Which is the equation of the transformation? A. g() ( 1 ) 1 B. g() ( 1 ) 1 C. g() 1 1 D. g() Without graphing, determine the verte of the graph of h() 3( 1 1) 1. Eplain how ou found our answer. LESSON 11-3 d. k() 76. Write each quadratic function in verte form. Then describe the transformation(s) from the parent function and use the description to graph the function. a. g() g() 73. Make use of structure. Describe how the graph of g() 1 differs from the graph of 1 h() College Board. All rights reserved. SpringBoard Algebra, Unit Practice

15 b. h() h() d. k() k() 77. What is the verte of the graph of the function f() ( 1 3) 1? Eplain our answer. c. j() j () 78. Write each function in verte form. Then identif the verte and ais of smmetr of the function s graph, and tell in which direction the graph opens. a. h() 1 4 verte form: verte: ais of smmetr: graph opens: b. h() verte form: verte: ais of smmetr: graph opens: 1 01 College Board. All rights reserved. SpringBoard Algebra, Unit Practice

16 c. h() verte form: verte: ais of smmetr: graph opens: 8. Construct viable arguments. Suppose ou are asked to find the verte of the graph of f() Eplain our answer. Find the verte and eplain how ou found the verte using our chosen method. d. h() verte form: verte: ais of smmetr: graph opens: 83. Use the formula for the -coordinate of the verte to find the verte of each function. a. f() Which function has an ais of smmetr to the left of the -ais? A. f() B. f() 6 1 C. f() 1 3 D. f() b. f() Epress regularit in repeated reasoning. Sal is writing f() 3 6 in verte from. What number should he write in the first bo below to complete the square inside the parentheses? What number should he write in the second bo to keep the epression on the right side of the equation balanced? Eplain. f() 3( 1 ) c. f() LESSON The graph of a quadratic function f() opens downward, and its verte is (3, ). For what values of does the value of f() increase? For what values of does the value of f() decrease? Eplain our answers. d. f() College Board. All rights reserved. SpringBoard Algebra, Unit Practice

17 84. Which is the verte of f() ? A. (0, 30) B. (, 0) C. (0, ) D. (, 0) LESSON Identif the - and -intercepts of each function. a. f() Mrs. Miller would like to create a small vegetable garden adjacent to her house. She has 0 ft of fencing to put around three sides of the garden. a. Let be the width of the garden. Write the standard form of a quadratic function G() that gives the area of the garden in square feet in terms of. -intercepts: -intercept: b. f() b. Graph G() and label the aes. G() -intercepts: -intercept: c. f() 7 1 -intercepts: -intercept: c. What is the verte of the graph of G()? What do the coordinates of the verte represent in this situation? d. f() d. Reason quantitativel. What are the dimensions of the garden that ield the maimum area? Eplain our answer. -intercepts: -intercept: College Board. All rights reserved. SpringBoard Algebra, Unit Practice

18 87. What is the -intercept of a quadratic function? How man -intercepts can a quadratic function have? c. Give the reasonable domain and range of T(), assuming that the tour compan does not want to lose mone b selling the tickets. Eplain how ou determined the reasonable domain and range. 88. Which of the following are the -intercepts of f() 6 4? A. and B. and 0 C. 1 and 3 D. and 3 d. Make sense of problems. What selling price for the tickets would maimize the tour compan s profit? Eplain our answer. 89. When does the graph of a quadratic function have onl one -intercept? 90. You can bu a 4-hour ticket for the Hop-On Hop-Off bus tour in London for 0. (The basic unit of mone in the United Kingdom is the pound,.) The tour compan is considering increasing the cost of the ticket to increase the profit. If the tickets are too epensive, the will not have an customers. The function T() 1 0 models the profit the tour compan makes b selling tickets for pounds each. a. What is the -intercept of the graph of T(), and what is its significance? LESSON For each function, identif the verte, -intercept, -intercept(s), and ais of smmetr. Graph the function. Identif whether the function has a maimum or minimum and give its value. a. f() 7 1 verte: -intercept: -intercept(s): ais of smmetr: ma or min: f() b. What are the -intercepts of the graph of T(), and what is their significance? College Board. All rights reserved. SpringBoard Algebra, Unit Practice

19 b. f() verte: -intercept: -intercept(s): ais of smmetr: 9. Make sense of problems. Consider the London bus tour compan function T() 1 0 whose graph is below T() ma or min: f() a. Based on the model, what selling price(s) would result in a profit of 300? Eplain how ou determined our answer. b. Could the tour compan make 00? Eplain. c. If the tour compan sells the tickets for each, how much profit can it epect to make? Eplain how ou determined our answer. 93. Eplain how to find the -intercept of the quadratic function f() without graphing the function. 94. If a parabola opens down, then the -coordinate of the verte is the A. minimum value B. ais of smmetr C. -intercept D. maimum value 9. Suppose ou are given the verte of a parabola. How can ou find the ais of smmetr? College Board. All rights reserved. SpringBoard Algebra, Unit Practice

20 LESSON What does the discriminant of a quadratic function tell ou about the -intercepts of the graph of the function? 97. The discriminant of a quadratic equation is greater than zero but not a perfect square. What is the nature of the solutions? How man -intercepts will the graph of the equation have? b value of the discriminant: nature of the solutions: -intercepts: 98. For each equation, find the value of the discriminant and describe the nature of the solutions. Then graph the related function and find the -intercepts if the eist. a value of the discriminant: nature of the solutions: -intercepts: c value of the discriminant: nature of the solutions: -intercepts: 0 01 College Board. All rights reserved. SpringBoard Algebra, Unit Practice

21 d value of the discriminant: nature of the solutions: -intercepts: LESSON 1-1. Solve each quadratic inequalit b graphing. a. $ b If the discriminant is zero, what is true about the solution(s) of the quadratic equation? A. There are two comple conjugate solutions. B. There are two real solutions. C. There is one, rational solution. D. There are no solutions. 0. Construct viable arguments. A quadratic equation has two comple conjugate solutions. What can ou conclude about the value of the discriminant of the equation? 1 01 College Board. All rights reserved. SpringBoard Algebra, Unit Practice

22 c. # Which inequalit s solutions are shown in the graph? A. # B., C. $ D d., Which of the following is NOT a solution of the inequalit $ 3 1? A. (0, ) B. (3, 3) C. (, 1) D. (1, 9) 01 College Board. All rights reserved. SpringBoard Algebra, Unit Practice

23 4. Graph the quadratic inequalit, Then state whether each ordered pair is a solution of the inequalit. LESSON Graph each sstem. Write the solution(s) if an. a a. (0, ) b. (3, 4) c. (3, 6) d. (4, 1). Model with mathematics. Foresters use a measure known as diameter at breast height (DBH) to measure trees for logging. To find the DBH, the use diameter tape or a caliper to measure the tree at a height of 4. feet (breast height) from the ground to determine the diameter. A tree of a certain species should have a cross-sectional area of at least square feet at breast height for it to be logged. Suppose that mature trees for this species do not have a cross-sectional area of more than 30 square feet at this height. b a. Write the function for the DBH in terms of the radius, r. Is this a linear function or a quadratic function? Eplain. b. Write the function for the area which is acceptable for logging in terms of the radius, r. Is this a linear function or a quadratic function? What is the reasonable domain and range of the function? 3 01 College Board. All rights reserved. SpringBoard Algebra, Unit Practice

24 c. 1 ( 4) 1 7. Critique the reasoning of others. Mari claims that a sstem of a linear equation and a quadratic equation can have one solution. Zell sas that the sstem has to have two solutions. Who is correct? Eplain using a sstem and a graph as an eample. d. 1 ( 3) The demand function for a product is f() 3 1. The suppl function is g() Use a graphing calculator to determine the solution(s) to the sstem. A. (64.9, 13.3) and (0., 8.3) B. (13.3, 64.9) and (8.3, 0.) C. (13.3, 64.9) and (8.3, 0.) D. (64.9, 13.3) and (0., 8.3) 4 01 College Board. All rights reserved. SpringBoard Algebra, Unit Practice

25 9. Aaron sells T-shirts at the Jazz Festival in New Orleans. He decides to lower the price of the T-shirts. a. How might this affect the demand for the T-shirts? LESSON Epress regularit in repeated reasoning. Write a sstem of equations that consists of one linear equation and one quadratic equation. Eplain how ou would solve the sstem algebraicall. b. Will he realize an increase in profit? Eplain. 11. Find the real solutions of each sstem algebraicall. a b c. What will be the break-even point, the point where revenue from sales covers the cost? Eplain. c d Use appropriate tools strategicall. Cathie wrote the following sstem of equations to model a problem in her research project. f( ) g ( ) Sketch a graph of the sstem and identif the solution(s) How man real solutions does the following sstem have? A. none B. one C. two D. infinitel man 114. Describe the solutions of the sstem of equations from Item 11b. 11. Confirm the solutions to Item 11c b graphing. Describe the solution(s). 01 College Board. All rights reserved. SpringBoard Algebra, Unit Practice

Ready To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions

Ready To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions Read To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions Find these vocabular words in Lesson 5-1 and the Multilingual Glossar. Vocabular quadratic function parabola verte