Chapter 8 Vocabulary Check


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1 28 CHAPTER 8 Quadratic Equations and Functions d. What is the level of methane emissions for that ear? (Use our rounded answer from part (c).) (Round this answer to 2 decimals places.) Use a graphing calculator to check each eercise. 89. Eercise Eercise 8 9. Eercise Eercise 48 Chapter 8 Vocabular Check Fill in each blank with one of the words or phrases listed below. quadratic formula quadratic discriminant {2b completing the square quadratic inequalit h, k2 0, k2 h, 02 b. The helps us find the number and tpe of solutions of a quadratic equation. 2. If a 2 = b, then a =.. The graph of f 2 = a 2 + b + c, where a is not 0, is a parabola whose verte has value. 4. A is an inequalit that can be written so that one side is a quadratic epression and the other side is 0.. The process of writing a quadratic equation so that one side is a perfect square trinomial is called. 6. The graph of f 2 = 2 + k has verte. 7. The graph of f 2 =  h2 2 has verte. 8. The graph of f 2 =  h2 2 + k has verte. 9. The formula = b { 2b24ac is called the. 0. A equation is one that can be written in the form a 2 + b + c = 0 where a, b, and c are real numbers and a is not 0. Chapter 8 Highlights DEFINITIONS AND CONCEPTS EXAMPLES Section 8. Solving Quadratic Equations b Completing the Square Square root propert If b is a real number and if a 2 = b, then a = {2b. To solve a quadratic equation in b completing the square Step. If the coefficient of 2 is not, divide both sides of the equation b the coefficient of 2. Step 2. Isolate the variable terms. Step. Complete the square b adding the square of half of the coefficient of to both sides. Step 4. Write the resulting trinomial as the square of a binomial. Step. Appl the square root propert and solve for. Solve: = 4. + = {24 =  { 24 Solve: = = = = 2 and 222 = = = = {20 = 2 { 20
2 Chapter 8 Highlights 29 DEFINITIONS AND CONCEPTS EXAMPLES Section 8.2 Solving Quadratic Equations b the Quadratic Formula A quadratic equation written in the form a 2 + b + c = 0 has solutions = b { 2b24ac Solve: = 0. a =, b = , c =  =  2 { # = { 2 2 Section 8. Solving Equations b Using Quadratic Methods Substitution is often helpful in solving an equation that contains a repeated variable epression. Solve: = 0. Let m = 2 +. Then m 2  m + 6 = 0 Let m = 2 +. m  2m  22 = 0 m = or m = = or 2 + = 2 Substitute back. = or = 2 Section 8.4 Nonlinear Inequalities in One Variable To solve a polnomial inequalit Step. Write the inequalit in standard form. Step 2. Solve the related equation. Step. Use solutions from Step 2 to separate the number line into regions. Step 4. Use test points to determine whether values in each region satisf the original inequalit. Step. Write the solution set as the union of regions whose test point value is a solution. Solve: 2 Ú Ú = = 0 = 0 or = 6 A B C 0 6 Test Point Region Value 2 # 6 Result A Ú 622 True B 2 Ú 62 False C Ú 672 True To solve a rational inequalit Step. Solve for values that make all denominators 0. Step 2. Solve the related equation. Step. Use solutions from Steps and 2 to separate the number line into regions. Step 4. Use test points to determine whether values in each region satisf the original inequalit. Step. Write the solution set as the union of regions whose test point value is a solution The solution set is , 0] [6, 2. 6 Solve: = 0 Set denominator equal to = 6  = 2 6 = 22 Multipl b = = 22 = (continued)
3 0 CHAPTER 8 Quadratic Equations and Functions DEFINITIONS AND CONCEPTS Section 8.4 Nonlinear Inequalities in One Variable (continued) EXAMPLES. A B C 2 4. Onl a test value from region B satisfies the original inequalit.. A B C 2 The solution set is 2, 2. Section 8. Quadratic Functions and Their Graphs Graph of a quadratic function The graph of a quadratic function written in the form f 2 = a  h2 2 + k is a parabola with verte h, k2. If a 7 0, the parabola opens upward; if a 6 0, the parabola opens downward. The ais of smmetr is the line whose equation is = h. (h, k) h f() a( h) 2 k a 0 (h, k) h a 0 Graph g2 = The graph is a parabola with verte (, 4) and ais of smmetr =. Since a = is positive, the graph opens upward (, 4) g() ( ) Section 8.6 Further Graphing of Quadratic Functions The graph of f 2 = a 2 + b + c, where a 0, is a parabola with verte a b, f a b bb Graph f 2 = Find the verte and  and intercepts. b = # = f 2 = = 9 The verte is, = = = 4 or = 2 The intercepts are 4, 02 and  2, 02. f 02 = # 08 = 8 The intercept is 0, 82. ( 2, 0) (0, 8) 9 (4, 0) 2 4 (, 9)
4 Chapter 8 Review Chapter 8 Review (8.) Solve b factoring = a 2 = 29a + 0 Solve b using the square root propert.. 4m 2 = = 2 Solve b completing the square.. z 2 + z + = = 7. If P dollars are originall invested, the formula A = P + r2 2 gives the amount A in an account paing interest rate r compounded annuall after 2 ears. Find the interest rate r such that $200 increases to $277 in 2 ears. Round the result to the nearest hundredth of a percent. 8. Two ships leave a port at the same time and travel at the same speed. One ship is traveling due north and the other due east. In a few hours, the ships are 0 miles apart. How man miles has each ship traveled? Give an eact answer and a onedecimalplace approimation. 20. The hpotenuse of an isosceles right triangle is 6 centimeters longer than either of the legs. Find the length of the legs. (8.) Solve each equation for the variable. 2. = = b. Find the time it takes the hat to hit the ground. Give an eact time and a onedecimalplace approimation = = /  6 / + = = a 6  a 2 = a 4 ?? 0 mi (8.2) If the discriminant of a quadratic equation has the given value, determine the number and tpe of solutions of the equation Solve b using the quadratic formula = = = = = = 9. Cadets graduating from militar school usuall toss their hats high into the air at the end of the ceremon. One cadet threw his hat so that its distance d(t) in feet above the ground t seconds after it was thrown was dt2 = 6t 2 + 0t = Two postal workers, Jerome Grant and Tim Bozik, can sort a stack of mail in hours. Working alone, Tim can sort the mail in hour less time than Jerome can. Find the time that each postal worker can sort the mail alone. Round the result to one decimal place. 0. A negative number decreased b its reciprocal is Find the number. (8.4) Solve each inequalit for. Write each solution set in interval notation a. Find the distance above the ground of the hat second after it was thrown
5 2 CHAPTER 8 Quadratic Equations and Functions (8.) Sketch the graph of each function. Label the verte and the ais of smmetr. 4. f 2 = g2 = H2 = h2 = F 2 = G2 = f 2 = f 2 = (8.6) Sketch the graph of each function. Find the verte and the intercepts. 49. f 2 = f 2 = f 2 = f 2 = Find the verte of the graph of f 2 = Determine whether the graph opens upward or downward, find the intercept, approimate the intercepts to one decimal place, and sketch the graph. 4. The function ht2 = 6t t + 00 gives the height in feet of a projectile fired from the top of a building after t seconds. a. When will the object reach a height of 0 feet? Round our answer to one decimal place. b. Eplain wh part (a) has two answers.. Find two numbers whose product is as large as possible, given that their sum is Write an equation of a quadratic function whose graph is a parabola that has verte , 72. Let the value of a be MIXED REVIEW Solve each equation or inequalit = = = n = = = = = 0 6. a a = = >  6 > = Ú Ú 0 7. The busiest airport in the world is the Hartsfield International Airport in Atlanta, Georgia. The total amount of passenger traffic through Atlanta during the period 2000 through 200 can be modeled b the equation = ,62, where is the number of passengers enplaned and deplaned in thousands, and is the number of ears after (Source: Based on data from Airports Council International) a. Estimate the passenger traffic at Atlanta s Hartsfield International Airport in 20. b. According to this model, in what ear will the passenger traffic at Atlanta s Hartsfield International Airport first reach 99,000 thousand passengers? Chapter 8 Test Solve each equation = = 0. m 2  m + 8 = 0 4. u 26u + 2 = = = = = = = 0
6 Chapter 8 Cumulative Review Solve b completing the square = = 4a Solve each inequalit for. Write the solution set in interval notation Ú Graph each function. Label the verte. 7. f 2 = 2 8. G2 = Graph each function. Find and label the verte, intercept, and intercepts (if an). 9. h2 = F 2 = Dave and Sand Hartranft can paint a room together in 4 hours. Working alone, Dave can paint the room in 2 hours less time than Sand can. Find how long it takes Sand to paint the room alone. 22. A stone is thrown upward from a bridge. The stone s height in feet, s(t), above the water t seconds after the stone is thrown is a function given b the equation st2 = 6t 2 + 2t a. Find the maimum height of the stone. b. Find the time it takes the stone to hit the water. Round the answer to two decimal places. 26 ft 2. Given the diagram shown, approimate to the nearest tenth of a foot how man feet of walking distance a person saves b cutting across the lawn instead of walking on the sidewalk ft Chapter 8 Cumulative Review. Write each sentence using mathematical smbols. a. The sum of and is greater than or equal to 7. b. is not equal to z. c. 20 is less than the difference of and twice. 2. Solve = .. Find the slope of the line containing the points (0, ) and (2, ). Graph the line. 4. Use the elimination method to solve the sstem. e 6 + = 42 = 6. Use the elimination method to solve the sstem:  = 2 e  + = 4 6. Simplif. Use positive eponents to write each answer. a. a 2 bc 2  b. a a4 b 22 b c c. a a8 b 2 b b Multipl. a b Multipl. a. 4a  27a  22 b. + b  b2 9. Factor. a b.  2z 4 c Factor. a b c Factor the polnomials in Eercises through b + b 2
7 4 CHAPTER 8 Quadratic Equations and Functions 4. Factor a 2. Solve = Solve = a Solve = Find the verte and an intercepts of f 2 = Simplif Simplif Add 22. Subtract a + a 26a a 2 2. Simplif Simplif 2 + b  a  + 2b Divide Divide b If P2 = a. Find P(2) b substitution. b. Use snthetic division to find the remainder when P() is divided b If P2 = , a. Find P 22 b substitution. b. Use snthetic division to find the remainder when P() is divided b Solve = Solve = If a certain number is subtracted from the numerator and added to the denominator of 9, the new fraction is equivalent to. Find the number. 2. Mr. Brile can roof his house in 24 hours. His son can roof the same house in 40 hours. If the work together, how long will it take to roof the house?. Suppose that varies directl as. If is when is 0, find the constant of variation and the direct variation equation. 4. Suppose that varies inversel as. If is 8 when is 4, find the constant of variation and the inverse variation equation.. Simplif. a b. 2 2 c d e f g Simplif. a b. 2 2 c. 2 4 a d e Use rational eponents to simplif. Assume that variables represent positive numbers. a b c. 2 4 r 2 s 6 8. Use rational eponents to simplif. Assume that variables represent positive numbers. a b. 2 c Use the product rule to simplif. a. 22 b c z 40. Use the product rule to simplif. Assume that variables represent positive numbers. a. 264a b. 2 24a 7 b 9 c Rationalize the denominator of each epression. 2 a. 2 c. A 2 b Multipl. Simplif if possible. a b c. + b2  b2 4. Solve =. 44. Solve 22 = Divide. Write in the form a + bi. a. 2 + i b. 7  i i 46. Write each product in the form of a + bi. a. i  2i2 b. 6  i2 2 c i222i2 47. Use the square root propert to solve = Use the square root propert to solve = Solve = Use the quadratic formula to solve m 2 = 4m + 8.
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