PAP Algebra 2. Unit 4B. Quadratics (Part 2) Name Period

Transcription

1 PAP Algebra Unit 4B Quadratics (Part ) Name Period 1

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3 After Test WS: 4.6 Solve by Factoring PAP Algebra Name Factor. 1. x + 6x x 8x x + 7x x 3x x + 7x x + 10x x 11x x 10x x + 17x 6 Solve by factoring x 1x 3 = x 11x 3 + = 13. = 5 x 15 x 3

4 14. 40x 1x 63 + = x x x = x = 5x x 15x 6x 0 + = 18. 6x 8x 16 + = 19. = + 5 x 6 x 4

5 Solving Quadratics by Factoring Worksheet 4.6 Name Per Solve by factoring. 1. x + 5x + 6 = 0. x x 1 = 0 3. a a = 0 4. t + t 19 = 5 5. x + 15x + 30 = 6 6. d d + 10 = = 0 x x = 15 a a 9. c c = = 0 x x 11. h 7 = t 15t + 6 = d + 10d + 18 = = 3 x a 3a + 17 = = 0 n n = + 3 x x x 18. 6t 15t 36 = 0 Find the dimensions of the rectangles below x +7 x +8 x 50 in x+3 34 ft 5 Page 1 of

6 Write a quadratic equation to represent each situation. Solve by factoring. 1. Thirty-five less than the square of a number is the same as times the number. Find the number.. Find two consecutive positive integers whose product is Find two consecutive odd integers such that the square of the first, added to 3 times the second, is Find three consecutive integers such that the square of the first, increased by the last, is Find three consecutive integers such that the product of the first and third exceeds the product of 8 and the second by Ms. Powers can kick a soccer ball up into the air with an initial velocity of 44 feet per second. Using the formula h = vt 16t, where h = height above the ground, v = initial velocity, and t = time in seconds, find how many seconds it will take for the ball to return to the ground. 7. Suppose you transform a square by increasing one side by 4 units and decreasing the other side by units. If the area of the resulting rectangle equals 16, how many units long was the side of the original square? 8. Suppose you transform a square by increasing one side by 1 units and decreasing the other side by 8 units. If the area of the resulting rectangle equals 44 how many units long was the side of the original square? 9. The length of a rectangle is 4 more than the width. Find the width of the rectangle if the area is The length of a rectangle is 5 times the width. If the length and width are both increased by, the new area would be 85. What is the original area? 6 Page of

7 PAP Algebra II 4.6B Notes GCF greatest common factor EX1) 6xy 9x + 3y EX) a b 10ab 3 EX3) r m 8r m + 1r m EX4) 3z + 4y 4 3 Difference of Squares a b = ( a b)( a + b) EX 5) x 81 EX 6) y + 49 EX 7) 4x 5 EX8) 9a 49b EX9) 16m r EX10) c x 3 3 Sum/Difference of Cubes a b a b a ab b = ( + )( + ) a b = ( a b)( a + ab + b ) EX 5) 3 x 7 EX 6) 3 y + 64 EX 7) 3 8x 1 EX8) 16c x EX9) 3 6 8m r EX10) 6 3 7a 15b 3 3 Factor by Grouping EX 11) 3 x x x EX1 ) 3 6x 3x 4x + 7

8 COMBINED. EX 14) d d + d EX 15) 3 6y 8y + 8y EX 16) g 3 5g EX 17) + 3 4c 500 EX 18) 4 5a 0a EX 19) 3 v v v Ex 0) Find three consecutive integers such that the product of the first and the third is 35 greater than the product of the second and 5. EX 1) Find four consecutive even integers such that the product of the second and fourth is 16 less than the product of -3 and the sum of the first and third. 8

9 PAP Algebra II WS 4.6B Name: Factor the following completely. If it does not factor state not factorable. 1) 4x 1x ) x 0x + 1x 3) m + 7z 4) 3 x x x ) 3 9x 16 x 6) m 11m + 1 7) 3 f f f ) 9 p q 5q 7 9) 3 8x 15 10) 3 0n + 4n 5n 1 9

10 11) Find two consecutive odd integers such that the square of the larger diminished by three times the smaller equals 34. 1) One number is 10 less than 8 times another number. If their product is -3, find the numbers. 13) If the sides of a square garden are increased by 4 meters, the area of the new larger square garden is 56 square meters more than the area of the original smaller garden. How long was the side of the original smaller garden? 14) One leg of a right triangle is 3 cm shorter than the other leg. If the hypotenuse is 15 cm, what is the length of the longer leg? 15) A path of uniform width is constructed around a 0 X 5 meter rectangular garden. If 196 square meters of brick are used to construct the path, what is the width of the path? 10

11 PAP Algebra II Notes 4.7 Complex Roots, Solve by Graphing, Solve by Quadratic Formula Complex numbers 3 + 4i 10i 11 Irrational Numbers Real Numbers e π 3 7 can not be written as a fraction 1 8 Rational Numbers can be written as a fraction. 5. e π 11 7 Integers Today we are going to work with some imaginary (i.e. complex) numbers! The base for an imaginary number is i, which is the square root of negative 1. i = 1 This is imaginary because it does not exist. Put 1 what does it say? Complex numbers have a real part, and an imaginary part. a and b is the imaginary part. in your calculator and EX1) Identify the real and imaginary part of these complex numbers: a) 3+ i real: imaginary: b) 7 9i real: imaginary: c) 5i 3 real: imaginary: d) 4i real: imaginary: e) 8 real: imaginary: + bi is the form of a complex number. a is the real part, This applies to quadratics because sometimes we will have roots that are not real. 11

12 An equation of the form ax + bx + c = 0 can be solved by using the quadratic formula: x = Use this formula to solve each of the following. Remember to set equal to zero first! 1. 8x x - 1 = 0. x + 3x - 1 = 0 b ± b 4ac a 3. x x + 1 = 0 4. x 4x + 13 = x + 4x = x +7x - 4 = x +3 In the quadratic formula, the expression b 4ac is called the discriminant. We can use this to determine how many solutions an equation has and if they are real or imaginary. Value of discriminant b 4ac>0 b 4ac=0 b 4ac<0 Number and type of solutions Example Graph Find the discriminant of the following equations and give the number and type of solutions of the equation. Remember to set equal to zero! 1. 7x x + 5 = 0. 3 x + 1x + 1 = x 5x + 1 = 6 7x 1

13 PAP Algebra II WS 4.7 Name: Use the quadratic formula to find the roots for the following quadratics. Tell whether the roots are real or imaginary. Put all complex roots in the form a+bi. 1) y x x = ) y = x 5x 3 3) y = x x 3 Roots: Roots: Roots: real imaginary real imaginary real imaginary 4) y x x = ) y x x = 5 7 6) y x x = + 7 Roots: Roots: Roots: real imaginary real imaginary real imaginary Find the value of the discriminant. Describe the number and type of roots: 7) y x x = ) y x x = 4 5 9) y = x x

14 10) A quadratic equation has real rational roots. Which of the following could be the value of its discriminant? a. -10 b. 10 c. 5 d. 37 1) Factor: 3 8m 7 13) Solve by factoring: 16x 5 = 0 Tell what intervals the following quadratics are increasing or decreasing. 14) y = x 7x + 5 inc: dec: 15) y = 3x 11x 4 inc: dec: 16) What is the b value for the quadratic y = 3x + bx 1 if the roots are x=1 and x=-4? 14

15 Pre-AP Algebra 4.8 Write Quadratic Equations Name: Ex 1) Write an equation for the parabola with the following roots in standard form. Graph your answer to verify. a) x = 3 and x = -8 b) x = only c) x = 3i and x = -3i d) x = 4+i and x = 4 i e) x = 3+i and x = 3 i f) 1 x =, x = 3 15

16 INCREASING: From left to right the graph is going up. DECREASING: From left to right the graph is going down. Ex ) Write the intervals where f(x) is increasing and decreasing. g) y = x 9x + 1 inc: dec: h) y = x + 7x 4 inc: dec: Ex 3) Find the value of b in the equation y = 4x + bx 1 if the function has roots of x = and = 6 1 x. Ex 4) Over what interval(s) is the parabola in Ex 3 positive? 16

17 PAP Algebra II 4.8P Quadratics Practice Name: 1. Graph the function. y = -3(x +1) +4 Domain: Range: Axis of Symmetry:. Given the factored form of the equation: y = -4(x -5) (x +3) a) identify the root/s b) convert to standard form c) identify the y-intercept 3. Consider the graphs of the following equations. List the equations in order from narrowest graph to widest graph. a) y = -4x b) y = 1 3 x c) y = 5 3 x d) y = -x b= 4. Find the value of b in the equation y = x + bx 3 if the vertex is (3,-1). 5. Write the equation for this parabola in a) vertex form: b) factored form: c) standard form: 17

18 6. Write equation for the path of the golf ball in: a) vertex form: b) factored form: c) standard form: 7. A rocket is shot straight up into the air from the top of a 30 foot cliff with an initial velocity of 3 ft / sec. The equation for the path of the rocket is a) What was the maximum height of the rocket? b) When did the rocket reach it s maximum height? c) When did the rocket land on the ground? d) How long after it was shot did it land on the ground? e) What is a realistic domain for this situation? h t t = Graph the following quadratic inequalities. Be sure to label at least 3 points. a) y 1 < 3( x 5) + 4 b) y ( ) 4 x 3 18

19 9. Find the vertex of the following parabolas. V= a) y = x + 6x 9 V= b) y = 3x + 4x A football is kicked from the ground at the 10 yard line, and it lands on the 50 yard line. If the maximum height of the football is 80 yards, what is the equation for the path of the football? F(x)= V(x)= S(x)= 11. Find all roots of the following parabolas. Simplify completely! Put all complex roots in the form a+bi. a) y = 3x + x b) y = 4x + x 3 Roots: Roots: 1. Write the equation in standard form for the parabola with roots: a) x= -9 only b) x = ± 7i c) x = ± 5i 19

20 13. Find the best fit quadratic equation to fit through the following data points. (round to 3 decimal places) TIME(s) HEIGHT(m) a) Best fit quadratic: b) What is the maximum height of the object? c) When does the object hit the ground? 14. A golf ball is hit from the top of a hill that is 16 feet high. It reaches it s maximum height of 87 feet 9 seconds after it was hit. a) What is the equation for the path of the golf ball? (round to 3 decimals) V(x)= S(x)= b) When does the golf ball hit the ground? 15. Use interval notation to identify what intervals the parabola y = 3x 4x + 5 is increasing and decreasing. Increasing: Decreasing: 16. Factor the following quadratics completely and find the roots: a) y = x 16 factored: roots: b) y x x = factored: roots: c) y x x = factored: roots: 0

21 d) y x x = factored: roots: e) y x x = factored: roots: f) y x x = factored: roots: 17. Write the equation and graph the area of the set of all right triangles whose base is three more units than the height. Area = What is a reasonable domain and range for this situation? Domain: Range: 18. Determine whether each equation has no solution, one real solution, two real solutions or two complex solutions. I. 0 = x + 5x + 1 a) no solution b) one real solution c) two real solutions d) two complex solutions II. 0 = x + 6x + 9 a) no solution b) one real solution c) two real solutions d) two complex solutions III. 0 = x + 3x + 5 a) no solution b) one real solution c) two real solutions d) two complex solutions 19. Fill in the table for g(x) given that g( x) = f ( x 3) + 3. x f(x) x g(x)

22 Pre-AP Algebra WS 4.8 Writing Quadratics Equations Name: 1. Write the equation of a quadratic with solutions at -3 and 15.. Write the equation of a quadratic with an a value of 3 and solutions at -3 and Write a quadratic equation of the form ax bx c + + = 0that has roots 8 and Write the equation of a quadratic given the following graph. 5. A rectangular enclosure at a zoo is 35 feet long by 18 feet wide. The zoo wants to double the area of the enclosure by adding the same distance, x, to the length and the width. Write and solve an equation to find the value of x. What are the new dimensions of the enclosure? 6. Write the equation of the quadratic with solutions at 3± 6i. 7. Write the equation of the quadratic with solutions at 4 ± 4i. 8. Write the equation of a quadratic whose discriminant is -16 with an a value of 1 and a b value of -18.

23 9. Write the equation of the quadratic in the graph shown. 10. Find two consecutive odd integers whose product is 79 greater than their sum. 11. The length of a rectangular picture frame is 6 inches less than twice the width. Find the width of the frame if the area of the picture frame is 80 square inches. 1. Suppose you transform a square corn field by increasing one side by units and decreasing the other side by 3 units. If the area of the resulting corn field equals 4, what was the length of the original corn field? 13. A rectangle is 5 inches wide and 1 inches long. How much should be added to the width of the rectangle to increase the diagonal by 7 inches? 14. Find three consecutive even integers such that the product of the first and third exceeds the product of the second and 8 by 16. 3

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25 Pre-AP Algebra Notes 4.9 Solve and Graph Quadratic Inequalities Name: 1. a. Given ( ) y x + 1 4, what are the transformations being applied? b. Would you use a solid or dotted line to graph? c. Sketch the parabola on the graph to the right. b. Using the inequality y ( x + 1) 4, test the following ordered pairs to determine where shading of the solution set should occur. ( 3,0) ( 1, 1) (3, ) c. What ordered pair is usually an excellent test point? Why? When would this point not work to check for the solution region?. a. Given y x x + 1, where are the roots? The vertex? b. Would you use a solid or dotted line to graph? Why? c. Sketch the parabola on the graph to the right. d. Using the inequality y x + x 1, test the following ordered pairs to determine where shading of the solution set should occur. (0, 15) (0,0) (,0) Sketching the Graph of a Quadratic Inequality: 1) Sketch the graph of the parabola y = ax + bx + c. Use a solid line for and a dashed line for > <. ) Test one point inside the U-shaped and one outside. 3) Only one of the test points will be the solution. Shade that region. Ex. 1: Sketch the solution for y 3x 6x 5

26 Ex. : Sketch the solution for y -x + x + 4 Word Problems: 1. The path of a football kicked from the ground can be modeled by h =.0x + 1. x where h is the height (in yards) and x is the horizontal distance (in yards) from where the ball is kicked. The crossbar on a field goal post is 10 feet above the ground. a. Write an inequality to find the values of x where the ball is high enough to go over the crossbar. b. Solve the inequality. c. A player attempts to kick a field goal from 5 yards away. Will the ball have enough height to go over the crossbar from this distance?. The arch of the Sydney Harbor Bridge in Sydney, Australia, can be modeled by y =.0011x x where x is the distance in meters from the left pylons and y is the height in meter of the arch above the water. If the road is 5 meters above the water, for what distances x is the arch above the road? 6

27 Pre-AP Algebra WS 4.9 Solve and Graph Quadratic Inequalities Name: Using your graphing calculator, plot at least three accurate points to form the boundary line. Shade the correct region and then list ordered pairs that could be solutions. 1. y ( x ) + 3 y > 3( x + 1) + 6 y y x x 3.Which of the points given would satisfy the inequality? There can be more than one answer. y x + x 1 a. (1, 3) b. (, 0) c. (5, -) d. (-, 4) Application Problems 4) You decide to start at the left base of a mountain and show off your hiking skills. The mountain might be represented by the function y = -.006(x 310) You need to use oxygen above 00 feet. You are asked to analyze the part of the path that is safe to hike without oxygen? a. Write the inequality b. Solve for reasonable solutions. c. Identify the vertex of the mountain d. Describe the window used on your graphing calculator. X min= X max= Y min= Y max= 5) The Arc-de-Triomphe in Paris might use the equation y = -.444x x +.1, to make the arch at the top. A row of decorative stone is 85 feet above the ground. Painters need you to analyze the area above the stone so they can paint that area. a. Write the inequality b. Solve for reasonable solutions. c. Identify the vertex of the arch 7

28 6) The Eiffel tower in Paris has four arches at the base. Each one has an equation similar to y = -.0x + 3x.5. If they are starting at the right and left bases and painting up the arch, what domain will get painted if the need to stay under 65 feet? a. Write the inequality b. Write the domain of the arch that gets painted on both sides? 7) The Grand Canyon has valleys shaped like parabolas. The inequality y.006x represents the part of a canyon that can be seen from an observation tower. 10.8x 400 a. Find a good fit window. b. Write the domain of the valley that is below sea level? (allowing negative values) c. Explain how you decided on a good fit window. 8) A plane is flying over the Grand Canyon. It follows the function f(x) = -.5(x 1000) , where x is minutes and f(x) is height in feet. For what times, x, is the plane above 0,000 feet? a. Write the inequality b. Write the times of the plane? 9) A picture is 1 inches by 14 inches. You are going to add a frame that is x inches thick. You want the resulting framed picture to have an area less than or equal to 500 square inches. What is a reasonable solution for the thickness of the frame? (Assume the frame must be at least 1 inch thick) a. Write the inequality b. What is a reasonable solution? 10) Twenty-eight less than the square of some number is 3 times the same number. Find the number. Is there more than one possibility? 8

29 Name: Solving Linear Quadratic Systems Algebraically Algebra 1 Date: In this lesson we will begin to work with solving linear-quadratic systems of equations. Recall that to x, y that satisfy all equations in the system. We solve a system we must find the set of all points ( ) will review this concept with an example from linear systems. Exercise #1: Consider the linear system shown to the right. y = x + 5 x + y = 15 (a) Solve this system algebraically using the substitution method. (b) Explain, in graphical terms, what the ordered pair from (a) represents. The substitution method was used above because it is the only method that we can use to solve linear quadratic systems algebraically. Solving such systems requires solving a quadratic equation. Since we are working with quadratics, it is natural to expect more than one answer. This has a graphical connection as Exercise # will illustrate. Exercise #: Consider the sketch of a line and a parabola shown at the right. (a) What is the maximum number of intersection points that a line and a parabola could have? Illustrate with a picture. (b) What is the minimum number of intersection points that a line and a parabola could have? Illustrate with a picture. (c) Is it possible for a line and a parabola to intersect in only one point? If so, illustrate with a picture. Algebra 1, Unit #6 Quadratic Algebra L16 The Arlington Algebra Project, LaGrangeville, NY

30 Exercise #3: Solve each of the following systems of equations algebraically and check using STORE on your calculator. In each case the substitution method should be used to begin the process. (a) y = x + 4x 1 y = 7x + 9 (b) y = x + x + 7 y = 6x + 3 (c) y = x + x 6 3x + y = 1 (d) y 10x = 5 y = x + 7x + 5 Algebra 1, Unit #6 Quadratic Algebra L16 The Arlington Algebra Project, LaGrangeville, NY

31 Name: Skills Solving Linear Quadratic Systems Algebraically Algebra 1 Homework Date: 1. Which of the following is a solution to the system of equations shown to the right? (1) ( 4, 7 ) (3) ( 3, 0 ) y = x 9 y = x + 3 () ( 4, 1) (4) (, 5 ). Mateo produced the following table on his calculator to find the solutions to a linear-quadratic system of equations. Based on this table, which of the following sets gives the x-values that solve this system? (1) { 4, } (3) {3, 6} () { 4, 3} (4) {,1} 3. Which of the following is not a possible number of solutions to a linear-quadratic system? (1) 1 (3) 3 () (4) 0 Solve each of the following linear quadratic systems of equations algebraically and check using STORE on your calculator. 4. y = x + 5x y = x 5. y = x 3x + 3 y 3x = 6 Algebra 1, Unit #6 Quadratic Algebra L16 The Arlington Algebra Project, LaGrangeville, NY

32 6. y = x + x 8 4x y = 5 7. x y = 10 y = x x Applications 8. The price C, in dollars per share, of a high-tech stock has fluctuated over a twelve-year period according to the equation C = x x, where x is in years. The price C, in dollars per share, of a second high-tech stock has shown a steady increase during the same time period according to the relationship C = x (a) For what values are the two stock prices the same? (Only an algebraic solution will be accepted.) (b) Determine the values of x for which the quadratic stock price is greater than the linear stock price. State your answer as an inequality. (Hint: You should be able to answer this almost immediately based upon your analysis in part (a) above.) Reasoning 9. Which value below for b would result in the linear-quadratic system y = x + 3x + 1 and y = x b having only one intersection point? Justify your answer algebraically, graphically or with a table. (1) 1 () (3) 3 (4) 4 Algebra 1, Unit #6 Quadratic Algebra L16 The Arlington Algebra Project, LaGrangeville, NY