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Review for 2nd Semester Final – Answer Key 3. If the x in the rational parent function is 9. Graph . If you translate the replace with x + 8, how is the graph changed? graph 3 units to the left and 3 units up, write the domain and range for the new function. The graph is shifted 8 units to the left, and the vertical asymptote shifts from x = 0 to 2 16 y= +3 x (5 3) 16 x = –8. See graph below. 14 14 12 2 →y= +3 12 10 10 x 2 8 8 VA: x = 2 6 6 1 4 HA: y = 3 1 f(x) = h(x) = 4 x 2 x+8 2 25 20 15 10 5 5 10 15 20 25 20 15 10 5 5 10 15 20 25 HA: y = 0 2 2 2 y= 4 x 5 4 6 6 VA: x = 5 8 8 10 10 12 6. An automobile’s velocity starting from a 12 Domain: All real numbers, x 2 Range: All real numbers, y 2 complete stop is , where v is measured in feet per second. What happens 11. Solve and check your answer. to the auto’s velocity as time increases? The automobile’s velocity would accelerate rapidly for about 2.5 seconds then start to slow until it reached just 65 under 28 feet per second. 60 55 50 , both sides by (x + 2) 45 40 , 2 from both sides 35 , by 4 30 Check 25 20 15 140∙x 10 r(x) = 5∙x + 3 5 This solution does not work, because it 40 30 20 10 10 20 30 40 50 makes the denominator of the original 60 70 80 90 100 5 problem = 0 NO SOLUTION! 10 15 18 16 13. Graph . 14 18 21. has moved the parent function 16 in what way? 12 Locate where the graph of14 Solve for y y 10 b – 5 = x x+3 12 y= is > y = 0 x 7 10 x+3 8 You will find 2 distinct parts 8 y= x 7 6 where x ≤ -3 and where x > 0 6 4 4 HA: y = 1 2 y = 2x 2 20 15 10 5 5 10 15 20 25 30 35 40 A 2 y=0 4 20 15 10 5 5 10 15 6 VA: x = 7 2 HA: y = 0 8 10 y = 2x 5 4 12 6 From the denominator 14 16 and the HA: y = -5 numerator x = 3 and x = 7 8 23. Suppose that the wealth of a business owner 10 Factors -3 7 is increasing exponentially. In 1993, he had $20 million. In 2001, he had $35 million. For Test # -4 0 8 the domain of 1993 to 2010, what is a x+3 reasonable range for this situation? x–7 1) Create a table with the information given to determine what the x value will be in 2010, if value >0 <0 >0 x = 0 is the year 1993. 15. Carl can do a particular job in 4 hours. It takes Domain Range Mike 6.5 hours to do the same job. Write an year x y equation that shows how long it will take the boys 1993 0 40,000,000 to complete the job if they work together. 1994 1 1995 2 1996 3 1997 4 1998 5 1999 6 Possible variations: 2000 7 2001 8 55,000,000 2002 9 2003 10 18. The cost of per person to rent a chartered bus 2004 11 varies inversely to the number of people who 2005 12 ride the bus. If 40 people pay $9.50 each to 2006 13 ride the chartered bus, what is the cost per 2007 14 person if only 25 people go? 2008 15 2009 16 2010 17 $78,696,103 So, 2) Put the data into the “STAT” list in the calculator. Then STAT > CALC menu > k = 40(9.50) = 380 0:ExpReg() > Enter > Enter 3) To put the equation into Y=: Y= > CLEAR > VARS > 5:Statistics >EQ menu > 1:RegEQ Then 4) Go to the TABLE > Scroll to x = 17 y =$15.20 per person Range: 40,000,000 < x < 78,696,103 25. Solve. 32. Which equation is the inverse of the function Change 729 to base 9 raised to a power shown in the graph? This is the parent exponential function which Since the base is the same, then the is given in choice A. The inverse of an exponents must have the same inequality. exponential equation is it’s logarithm equation a–5<3 + 5 to both sides after exchanging x and y. a<8 x = 2y log2x = y 27. The weight of a particular bacteria in a culture NOTE: Equation colors match their graph colors is tripling every 20 minutes. The weight of the A. bacteria was originally 17 grams. B. A. Write an equation which expresses the This is the inverse equation of A after weight, w, in grams after t minutes. exchanging the x and y then converting to a log This is an exponential equations in the C. This is the log form of A: x form of y = Abx and y have not been exchanged – same A, Initial amount = 17 grams graph b, the overall rate = 3 D. t, time in minutes, and The exponential form of this equation is t/20 = the intervals xy = 2! See graph below!?!?! Equation: 8 D B. Calculate the weight of the bacteria after 2 6 hours. 4 2 hours = 120 minutes So, = 12,393 grams 2 A&C 10 5 5 10 29. Merlin Industries bought a laptop for $2100. It 2 is expected to depreciate at a rate of 14% per year. What will the value of the laptop be in 5 4 years? Round to the nearest dollar. B6 This problem uses the growth formula y = A(1 – r)t 8 A = $2100 r = 14% 0.14 t = 5 years 33. What is the domain and asymptote of Substitute into the formula, ? V(t) = 2100(1-.14)5 = $947.90 See graph B in # 32. Domain: x > 0 40. Solve: ? 52. The x-intercepts of a parabola are at (5, 0) and (9, 0). What is the equation of the axis of For logbX - logbY, when subtracting logs, symmetry? What are the roots of the you can write it as a single log by dividing equation? X/Y Since a parabola is symmetric about a midline, the the mid-point of the 2 x- intercepts would give the point through Convert to exponential form which the axis of symmetry would pass 20 = 1 So, both sides by (3x – 2) Axis of Symmetry is x = 7 8 3x – 2 = x Solve for x The roots are the x values of the x- intercepts, x = 5, 9 x=1 6 Check using change of base formula 4 x=7 2 47. What conic section is formed when the plane (7,0) intersects the cone an angle to the base but does not intersect the base? 5 (5,0) 5 (9,0) 10 15 2 A circle 4 51. Graph: 6 2 From the denominator of the x ratio, a = 5 54. Identify the conic section modeled by the From the denominator of the y2 ratio, b = 3 Center is at (0, 2) equation: 8 . Since y leads the equations, the graph will open Hyperbola. up and down. 10 58. Complete the square to find the coordinates 8 of the center of the following conic section. 6 BTW this is a circle 4 Add ½ of 4, squared and ½ of 6, squared 2 to both sides 5 3 (x2 + 4x + 22) + (y2 + 6y +32) = 0 + 22 + 32 10 5 5 10 Factor each ()s 2 (x + 2)2 + (y + 3)2 = 13 Center: (–2, –3) 4 6 8 60. Sonja made a pot of hot tea and recorded its 68 The area of a rectangle is 348 ft2. The length degrees above room temperature for one is 5 ft longer than twice the width. Which hour. Find the most appropriate regression system of equations can be solved to find the given the following data. length, L, and the width, W, of the rectangle? Enter data into the STAT List The length is 5 ft longer than twice the STAT > Edit menu >1:Edit width L = 2W + 5 Turn on Plot 1 The area of a rectangle is 348 ft2 2nd > Y= > 1: Plot1 > Enter (L)(W) = 348 Graph the data: ZOOM > 9:ZoomStat Which is choice A What kind of function does this look like – Exponential Decay! 69. Simone went to an outlet mall and found a blowout sale. Sweaters were $5, shirts were Check out your choices: $3 and a pair of socks was $1 each. She A This is a quadratic function spent $70 and bought 26 items. She bought one more shirt that she did sweaters. B This is a linear function Ignoring tax, set up the system of equations C This is a cubic function ( notice x3) that can be used to find the number of D This is the only function that is an sweaters, shirts and pairs of socks Simone exponential function-where x is the bought. Solve the system. exponent Let x = the number of sweaters And y = the number of shirts You can calculate the exponential And z = the number of pairs of socks function from the data. x + y + z = 26 STAT > CALC menu > 5x + 3y + z = 70 0:ExpReg>Enter > Enter y = x + 1 standard form x – y = –1 ExpReg Use the MATRIX in the calculator to solve. Y = ab^x Using 3x4 matrix a=145.4910691 2nd > x1 > EDIT menu >[A] > 3 > ENTER > 4 > b=.9380039207 ENTER Enter data 64. The function y = 64(x – 2.50)2 + 400 models 1 1 1 26 a store’s profits, in dollars, on potato chips 5 3 1 70 where x is the price of a bag of potato chips. 1 1 0 1 What should the store charge for a bag of nd nd 1 2 > MODE > 2 > x > MATH menu > B:rref( potato chips to maximized its profits? What is >2nd > NAMES menu > 1:[A] > ENTER the maximum profit earned? 1 0 0 7 0 1 0 8 The vertex of this parabola is all we need. 0 0 1 11 The maximum charge, x, is the x of the Last column is your answers: x = 7, y = 8, z = 11 vertex, and maximum profit is the y of the vertex. So from the equation the vertex is 72. Solve the following system of equations by an (2.50, 400) appropriate method. (2, 4) or (4, 2) So, maximum charge is $2.50 and the xy = 8 st maximum profit is $400. y = x – 2 Substitute this into the 1 equation x(x – 2) = 8 2 x – 2x = 8 8 from both sides 2 x – 2x – 8 = 0 Factor the quadratic (x + 2)(x – 4) = 0 Set each factor = 0 and solve x + 2 = 0 x = –2 x–4=0 x=4 Substitute –2 into xy = 8 (–2)y = 8 y = –4 Check: (–2)( –4) = 8 TRUE –4 = –2 – 2 TRUE Substitute 4 into xy = 8 (4)y = 8 y = 2 Check: (4)(2) = 8 TRUE 2 = 4 2 TRUE 74. The length of a rectangle is 3 inches longer 2a. Solve: 0 = 3x2 95x 984 than the width. The area is 54 in2. In solving this problem, Aaron factors the resulting Using the quadratic formula equation and gives 9 and 6 as answers for the width. Is he correct? Why or why not? x= This is a “no brainer” – distance can = 8.22 or 39.89 not be negative! 3a. Surface area of a sphere is: SA = 4r2. Find 4 additional problems the radius for a sphere having a surface area of 462 in2. 1a. Bailey is building a rectangular pen for animals using the side of a barn as one side. Substitute the given into the equation: He has 200 feet of fencing to use for the other 3 sides. What is the maximum area that he 462 = 4r2 4 both sides then can enclose? r = 6.06 in BARN 4a. Write a system of equations that describe the W W graph. L 10 2W + L = 200 L = 200 – 2W Area = LW A = (200 – 2W)(W) 8 Set = 0 and solve for W: (200 – 2W)(W) = 0 200 – 2W = 0 2W = 200 6 W = 100 and W = 0 these are the zeroes 4 x-intercepts are (0,0) and (100,0) The midpoint of these will give the x value of 2 the vertex of the parabola and the y value of that vertex is the maximum 10 5 5 10 15 Midpoint = 2 Substitute x = 50 2 4 (200 – 2*50)(50) = 5000 ft Identify the zeroes of the parabola (2,0) and (5,0) the maximum area that can be enclosed x=2 x–2=0 6 OR The most the 2W could = is 100 (x – 2) is one of the factors of the quadratic 8 So maximum W = 100/2 = 50 ft And x = 5 x – 5 = 0 this leaves L = 100 ft (x – 5) is the other factor Area = 50(100) = 5000 ft 2 So, (x – 2)(x – 5) = 0 is the factor form of the equation Is the maximum area FOILing x – 7x + 10 = y one equation 2 Identify 2 points on the line. (–2, 5) and (1, 3) Substitute into the slope formula: Then substitute into the point-slope form y–3= (x – 1) Simplify and solve for y y–3= x+ y= x+ the other equation

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