1. Let x = total miles the car can travel Answer: x 22 = 18 or x 18 = 22 2. A = 1 2 ah 1 2 bh A = 1 h(a b) 2 2A = h(a b) 2A = h a b Note that when solving for a variable that appears more than once, consider factoring. The variable being solved for should NEVER be on both sides of the equation. Answer: B. If the gas mileage can vary by 2 miles per gallon from 24 miles per gallon, keep in mind that it can to 22 at the lowest and 26 at the highest. Remember that the absolute-value equation is x m = a. x m = a x 24 = 2 x 24 = 2 or x 24 = 2 x = 26 x = 22 If you forget or think that it could be written as x 2 = 24, just solve to check, like shown above. Answer: x 24 = 22 and x = 22,26 4. y + 2 > 5 y > 7 this means y is STRICTLY GREATER than 7 Testing y = 6 Testing y = 6 Testing y = 8 Testing y = 0 6 > 7 7 7 8 7 0 > 7 Note that a good majority of students will think 7 is the answer because they are thinking of the inequality as an equation, y + 2 = 5, and solving for y would make it y = 7. Keep in mind how inequalities behave differently. [ x ] 6 [ ] 7 [ ] 8 [ x ] 0
5. Jasmine total savings > Jasmine s sister s total savings 50 + 10w > 80 + 7w 50 + w > 80 w > 0 w > 10 Note that it would take more than 10 weeks for Jasmine to have more money than her sister. Answer: D 6. Remember that rate of change = slope = y Answer: B = distance x time, and in this case, speed = miles hour 7. Remember that the slope-intercept form is y = mx + b [m = slope and b = y-intercept] Answer: y-intercept 8. When finding a relationship from a table, look for PATTERNS. Since the answers are LINEAR functions, find the changes in y and the changes in x. Substituting x-values into the answers is an option, BUT it should not be relied on because the question may NOT be multiple choice. m = y x = 4 2 = 2 Using (6,9): y y 1 = m(x x 1 ) y 9 = 2(x 6) y 9 = 2x 12 y = 2x f(x) = 2x 9. Remember to use y y 1 = m(x x 1 ) and y = mx + b when dealing with linear relationships. m = y = 7 11 = 4 = 2 x 2 0 2 Using (0,11): y y 1 = m(x x 1 ) y 11 = 2(x 0) y 11 = 2x y = 2x + 11 To find the x-coordinate of the point that intersects the x-axis is the same as finding the x-intercept (so let y = 0). 0 = 2x + 11 2x = 11 x = 11 2 Answer: 11 2
10. 2x 6y = 15 6y = 2x + 15 y = 2 15 x + 6 6 y = 1 x 5 2 So, m = 1 For parallel lines, use the SAME SLOPE. Using (8,1): y y 1 = m(x x 1 ) y 1 = 1 (x 8) y 1 = 1 x 8 y = 1 x 5 To find the x-intercept, let y = 0 0 = 1 x 5 5 = 1 x 5 = x Answer: 5 or (5,0) 11. Slope of AB = y x = 2 0 0 ( 4) = 2 4 = 1 2 For perpendicular lines, use the NEGATIVE RECIPROCAL. 1 2 2 1 or 2 Answer: 2 12. Remember that the slope-intercept form is y = mx + b [m = slope (or rate of change) and b = y-intercept (or CONSTANT)] The constant (or y-intercept), 90, is the initial average temperature at the equator. The slope (or rate of change), 1, is the drop in temperature degree per latitude degree. Answer: A 1. Use x- and y-intercepts to help. x-intercept (let y = 0): y-intercept (let x = 0): 5x + 2y 50 5x + 2y 50 5x + 2(0) 50 5(0) + 2y 50 5x 50 2y 50 x 10 y 25 Note that answer choice D would indicate that the inequality must be GREATER THAN or equal to $50.
14. B 15. D 16. C 17. Remember that in order for a relation to be a function, NO x-values can repeat. Answer: D 18. Remember that in order for a relation to be a function, NO x-values can repeat. [ x ] 5 [ ] 1 [ ] 0 [ x ] 2 [ ] 11 [ ] 17 19. This is a DISCRETE graph, so each x- and y-value has to be listed for the domain and range. Answer: B 20. Note that OPENED CIRCLES in a graph CANNOT have: - equal signs in the inequality - brackets in the interval notation Answer: Domain: { 4 < x } or ( 4,] Range: { 5 < y 2} or ( 5,2] 21. Note that arrows continue to infinity. Since the arrow points at the x = 5 or y =, it does not mean it stops there. Note that < x < 2 is REDUNDANT to write and would be INCORRECT because x < 2 already means all real numbers STRICTLY LESS than 2. and can ONLY be used in INTERVAL NOTATION. Answer: Domain: {x < 2} or (, 2) Range: {y < 0} or (, 0)
22. The domain (x-values) for f(x) can only include values greater than or equal to 5. [ x ] f (2 1 ) this one DOES NOT represent the range 2 [ ] f(6) [ ] f(10.5868) [ ] f(100,000) 2. The domain (x-values) for f(x) can only include values STRICTLY greater than 5. [ x ] f(4.8) [ x ] f( 2) [ ] f( 5) [ x ] f(8) [ x ] f ( 1 2 ) [ x ] f(0) [ x ] f(14) [ ] f( 18) 24. Remember that the RANGE is the set of y-values. To find the domain (or x-values) be sure to substitute the range into f(x). f(x) = 6x + 11 f(x) = 6x + 11 f(x) = 6x + 11 f(x) = 6x + 11 7 = 6x + 11 25 = 6x + 11 1 = 6x + 11 1 = 6x + 11 48 = 6x 6 = 6x 24 = 6x 12 = 6x 8 = x 6 = x 4 = x 2 = x [ ] 1 [ x ] 2 [ ] [ x ] 4 [ ] 5 [ x ] 6 [ ] 7 [ x ] 8 25. Consider that the expression for 5 cents per pencil can be written as 5x, where x represents the number of pencils. Also, the equation can be written as y = 5x, where x represents the domain. For 0 pencils: For 1 pencil: For 2 pencils: For 20 pencils: f(0) = 5(0) f(1) = 5(1) f(2) = 5(2) f(20) = 5(20) f(0) = 0 cents f(1) = 5 cents f(2) = 10 cents f(20) = 100 cents Since only 0,1, 2,, or 20 pencils can be purchased, the domain is the integers from 0 to 20. Answer: A 26. Note that x represents the pairs of sunglasses, and C(x) represents the total cost.
27. Note that n is the domain and represents the number of customers, and it goes up to 100 people. Note that M(n) is the range and represents the amount of money the theater takes in on Thursday nights. Answer: D 28. Note that t is the domain and represents the time in weeks. Note that P(t) is the range and represents the population growth. 29. Note that n is the domain and represents the number of tickets. Note that h(n) is the range and represents the amount of money a movie theater receives. 0. Note that the function is supposed to model the population from 1900 (x = 0) to 2000 (x = 10). Answer: D 1. Since the domain of the function in the graph is all real numbers, the functions listed must also have all real numbers. - The domains of f(x) and g(x) cannot have all real numbers because you cannot get a real number if you take the square root of a negative number. - h(x) allows any x-value to be substituted. - The domain of k(x) is restricted only real numbers greater than or equal to 2. 2. In order to find the fastest speed, find the greatest slope between each interval or look for the STEEPEST slope. 0 x 0.5 m = y x = 20 0.5 = 40 0.5 x 2 m = y x = 100 1.5 = 66 2 2 x 2.5 m = y x = 0 (H.O.Y.) 2.5 x 4 m = y x = 20 1.5 = 1 1 Answer: 0.5 x 2
. There are two criteria: i. Obtaining a greater y-intercept from the table. ii. Obtaining a perpendicular slope from the equation. i. m = y x = 8 4 = 2 1 = 2 Using ( 1, 6): y y 1 = m(x x 1 ) y ( 6) = 2(x ( 1)) y + 6 = 2(x + 1) y + 6 = 2x + 2 y = 2x 4 Use a number greater than the y-intercept of 4. ii. y + 1 4 x = 2 y = 1 4 x + 2 So, m = 1 4 For perpendicular lines, use the NEGATIVE RECIPROCAL. Since the slope of the equation is 1, then the slope to use is 4. 4 Therefore, you can make up a linear function, such as y = 4x. Answer: 4. Note that average rate of growth refers to average rate of change (or slope). Average rate of growth = slope = y = meters = 1 meter = 0.5 meter/year x 6 years 2 years Answer: B
5. Remember that average rate of change refers to slope. To find the average rate of change (or slope), first identify the (x, y) coordinates. For x = 9 For x = 21 f(x) = 2 x 5 + f(x) = 2 x 5 + f(9) = 2 (9) 5 + f(21) = 2 (21) 5 + f(9) = 2 4 + f(21) = 2 16 + f(9) = 2(2) + f(21) = 2(4) + f(9) = 4 + f(21) = 8 + f(9) = 7 f(21) = 11 (9, 7) is one point (21, 11) is another point So, the average rate of change = slope = y x = 11 7 21 9 = 4 12 = 1 6. Remember that (average) rate of change refers to slope. To find the (average) rate of change (or slope), first identify the (x, y) coordinates. For a radius of 4 inches For a radius of 12 inches A(r) = πr 2 A(r) = πr 2 A(4) = π(4) 2 A(12) = π(12) 2 A(4) = π(16) A(12) = π(144) A(4) = 16π A(12) = 144π (4, 16π) is one point (12, 144π) is another point So, the rate of change = slope = y = 144π 16π = 128π = 16π = 16π 50. x 12 4 8 1 Answer: 16π or 50. 7. Remember that average rate of change refers to slope. To find the average rate of change (or slope), first identify the (x, y) coordinates. For t = 1.5 seconds For t = 1.75 seconds (1.5, 12) is one point (1.75, 7) is another point So, the average rate of change = slope = y = 7 12 = 5 = 20 x 1.75 1.5 0.25 Answer: A
9. Remember that slope-intercept is y = mx + b m = slope (or SPEED) and b = y-intercept (or starting point) For d = 1 15 t + 1, Monika s speed is 1 mile 15 minutes Note that if Mark s speed is as fast as Monika s speed, then he is slower than Monika. 4 So, 1 = 1 Mark s speed 4 15 20 a. d = 1 20 t b. Note that running one time around the park would indicate a 7-mile run, so the range would be from 0 to 7 miles. c. Domain: {0 x 140} or [0,140] 40. Remember that the x-intercepts are when y = 0, so this refers to ( 1,0) and (,0) from the table. x y 2 2.5 1 0 0 1.5 1 2 2 1.5 0 4 2.5 Find the x-intercepts of each answer choice to find which has the same as f(x). g(x) = 2x + 2 h(x) = 1 x + 2 j(x) = x2 + 2x k(x) = x 1 2 0 = 2x + 2 0 = 1 x + 2 0 = x2 + 2x 0 = x 1 2 2x = 2 1 x = 2 0 = (x + )(x 1) 2 = x 1 x = 1 x = 6 x =, 1 x = 1, Answer: D
41. A way to determine which function matches the graph, select one point on the graph to substitute into each function. I suggest using ( 2, 4): g(x) = x 2 + 4 g(x) = x + 2 4 g(x) = x + 4 2 g(x) = x 4 + 2 4 = ( 2) 2 + 4 4 = ( 2) + 2 4 4 = ( 2) + 4 2 4 = ( 2) 4 + 2 4 4 + 4 4 = 0 4 4 2 2 4 6 + 2 4 = 0 4 4 = 4 ALTERNATIVE: Sketch what f(x) = x looks like and see which transformations match. For g(x) = x 2 + 4, this is the same as g(x) = f(x 2) + 4, which shifts right 2 units and up 4 units. For g(x) = x + 2 4, this is the same as g(x) = f(x + 2) 4, which shifts left 2 units and down 4 units. For g(x) = x + 4 2, this is the same as g(x) = f(x + 4) 2, which shifts left 4 units and down 2 units. For g(x) = x 4 + 2, this is the same as g(x) = f(x 4) + 2, which shifts right 4 units and up 2 units. Answer: B 42. Note that f(x) = x is the parent function. Consider that: - f(x + 2) means the graph shifts left 2 units. - f(x) + 2 means the graph shifts up 2 units. - 2f(x) means the graph vertically stretches by a factor of 2. - 1 2 f(x) means the graph vertically shrinks by a factor of 2. 4. Substitute f(x) = 5x + 2 y = f(x) 4 y = (5x + 2) 4 y = 5x + 2 4 y = 5x 2 Answer: y = 5x 2
44. Remember that translating ANY function 8 units to the right can be written as f(x 8). f(x) = 1 2 x f(x 8) = 1 (x 8) 2 f(x 8) = 1 x + 4 2 f(x 8) = 1 2 x + 1 Answer: y = 1 2 x + 1 45. Remember that translating ANY function 2 units to the left can be written as f(x + 2). f(x) = x 2 4x f(x + 2) = (x + 2) 2 4(x + 2) f(x + 2) = (x + 2)(x + 2) 4x 8 f(x + 2) = x 2 + 4x + 4 4x 8 f(x + 2) = x 2 4 Answer: y = x 2 4 46. Remember that the area of a rectangle is A = bh. A = bh A = (x + 4)(2x + 1) A = 2x 2 + x + 8x + 4 A = 2x 2 + 9x + 4 47. Since x + 8 is the height and the height is 20 cm, set x + 8 = 20, which x = 12. Use x = 12 to substitute to the other dimensions. a. For (x + 1) dimension For (x ) dimension (12) + 1 = 1 (12) = 9 Answer: 1 cm and 9 cm b. Since the volume of the rectangular pyramid is represented by 1 (x + 1)(x )(x + 8), then 1 (1)(9)(20) = 780 Answer: 780 cm
48. f(x) + g(x) f(x) g(x) (x 2 x 2) + (x 2 + x 6) (x 2 x 2) (x 2 + x 6) be sure to use PARENTHESIS!!! x 2 x 2 + x 2 + x 6 x 2 x 2 x 2 x + 6 2x 2 8 2x + 4 Answer: Quadratic Answer: Linear f(x) = x2 x 2 = (x 2)(x+1) = x+1 f(x) g(x) g(x) x 2 +x 6 (x 2)(x+) x+ (x 2 x 2)(x 2 + x 6) Answer: Neither x 4 + x 6x 2 x x 2 + 6x 2x 2 2x + 12 x 4 9x 2 + 4x + 12 Answer: Neither 49. To find the base of the shaded region, take (x + 10) (x + 4), which is 2x + 6. So, the base of the shaded region = 2x + 6 and the height of the shaded region = x + 5 a. Perimeter = (2x + 6) + (2x + 6) + (x + 5) + (x + 5) = 2(2x + 6) + 2(x + 5) = 4x + 12 + 2x + 10 = 6x + 22 Answer: (6x + 22) units b. Area = (2x + 6)(x + 5) = 2x 2 + 10x + 6x + 0 = 2x 2 + 16x + 0 Answer: (2x 2 + 16x + 0) units 2 50. Area of shaded region = area of bigger rectangle area of smaller rectangle 24 = (x 2)(x + ) (6)(x) 24 = (x 2 + 9x 2x 6) 6x 24 = x 2 + x 6 0 = x 2 + x 0 0 = (x + 10)(x ) x = 10, Remember that there cannot be negative dimensions, so ignore 10. Answer: B 51. Remember to SUBTRACT exponents when dividing the same bases. x 18 y 12 +x 9 y 8 x y 4 = x18 y 12 x y 4 + x9 y 8 x y 4 = x15 y 8 + x 6 y 4
52. Completely factor the numerator. x 2 27 x Answer: A = (x2 9) x = (x+)(x ) x = (x+) = (x + ) 1 5. Remember that y = f(x). The answer choices are comparing the RANGE (or y-values), since they are given as f( 2), f( 1), g( 2), g(0), It is highly recommended to sketch the graphs of g(x) and f(x) by using the given vertex and zeros (or x-intercepts). To write f(x) as a function: - Write the zeros x = 2 and x = 4 as factors (x + 2) and (x 4). - y = (x + 2)(x 4) could give the equation y = x 2 2x + 8 - But the vertex of this equation would be (1, 9) - Multiplying the equation by 1 (or reflecting over x-axis) would give y = 1(x 2 2x + 8), which then has the same given vertex, (1,9), from the question So, f(x) = x 2 + 2x 8 and g(x) = x + 2 Then the x-values can be substituted to check which has the greater y-value. [ ] f( 2) is greater than g( 2) [ ] f( 1) is greater than g(0) [ x ] f(0) is greater than g(0) [ ] f(1) is less than g(1) [ x ] f(2) is greater than g(2)
54. C 55. Since g(x) = k f(x), then finding the value of k can be written as k = g(x) f(x). Write the linear functions for g(x) and f(x): - g(x) = 6x - f(x) = 2x + 1 So, k = g(x) f(x) = 6x 2x+1 = (2x 1) ( 1)(2x 1) = Answer: k = 56. Remember that finding the y-intercept is setting x = 0 (or t = 0 in this case). y = 16t 2 + 40t + y = 16(0) 2 + 40(0) + y = this is the initial height Note that y represents the height in feet, and t represents the seconds after the ball is thrown. 57. Note h(t) represents the height and t represents the time. 58. B
59. It is required to solve this quadratic equation by completing the square. Moving to the LEFT side of the equal sign: Moving 7 to the RIGHT side of the equal sign: 4x 2 24x + 7 = 4x 2 24x + 7 = 4x 2 24x + 4 = 0 4x 2 24x = 4 4(x 2 6x) + 4 = 0 4(x 2 6x) = 4 4(x 2 6x + 9) 6 + 4 = 0 4(x 2 6x + 9) = 4 + 6 4(x )(x ) 2 = 0 4(x )(x ) = 2 4(x ) 2 2 = 0 4(x ) 2 = 2 4(x ) 2 = 2 4(x ) 2 = 2 (x ) 2 = 8 (x ) 2 = ± 8 x = ± 8 x = ± 8 also note that it can be written as x = ± 2 2 Step: 4(x [ ]) 2 = [ 2 ] Solution: x = [ ] ± [ 8 ] 60. It is required to solve this quadratic equation by completing the square. Moving 4 to the LEFT side of the equal sign: Moving 7 to the RIGHT side of the equal sign: x 2 12x + 7 = 4 x 2 12x + 7 = 4 x 2 12x + = 0 x 2 12x = (x 2 4x) + = 0 should not factor by GCF (x 2 4x) = should not divide by on both sides (x 2 4x + 4) 12 + = 0 (x 2 4x + 4) = + 12 (x 2)(x 2) 9 = 0 (x 2)(x 2) = 9 (x 2) 2 9 = 0 (x 2) 2 = 9 (x 2) 2 = 9 (x 2) 2 = 9 (x 2) 2 = (x 2) 2 = ± x 2 = ± x = 2 ± Step: (x [ 2 ]) 2 = [ 9 ] Solution: x = [ 2 ] ± [ ] 61. Note that C represents the production cost, and since the cost needs to be at or below $115,000, the constraint of range (or y-values) must be less than or equal to $115,000. Note that x represents the number of balls produced in one day, and it cannot be negative. Answer: D
62. Recall that finding the zeros is finding the x-intercepts (or setting y = 0 or f(x) = 0) f(x) = (x 2 + 2x 8)(x 6) 0 = (x 2 + 2x 8)(x 6) 0 = (x + 4)(x 2)(x 6) x + 4 = 0 x 2 = 0 x 6 = 0 x = 4 x = 2 x = 6 ( 4,0) (2,0) (6,0) [ x ] (2,0) [ x ] (6,0) [ ] (0, 8) [ x ] ( 4,0) [ ] ( 6,0) [ ] (0,2) [ ] (0,8) 6. Recall that the formula to find the axis of symmetry is x = b So, x = b 2a = 6 2() = 1 Answer: B 2a 64. Note that t represents the time in minutes, and to find the time for the water to DRAIN refers to finding the xintercepts. Answer: D Note that the PROPERTY the question is referring to is the ZERO PRODUCT PROPERTY, which is setting the factors equal to zero. Also, note that although answer choice C does find the zeros of the function, it is NOT using the Zero Product Property to find the x-intercepts. When solving the function by completing the square, it is finished by using square roots. 0 = 4t 2 2t + 6 0 = 4t 2 2t + 6 0 = (2t 7)(2t 9) 0 = 4(t 2 8t) + 6 0 = 4(t 2 8t + 16) 64 + 6 The Zero Product Property allows: 0 = 4(t 4)(t 4) 1 2t 7 = 0 2t 9 = 0 0 = 4(t 4) 2 1 would then be solved by using square roots Answer: B
65. Remember that exponential growth function is y = a(1 + r) t, and when 1 + r is greater than 1, it is a growth. [ x ] f(t) = 1.25 t [ ] f(t) = 2(0.9) 0.5t [ x ] f(t) = (1.07) t [ ] f(t) = 18(0.85) t [ x ] f(t) = 0.5(1.05) t [ x ] f(t) = (1.71) 5t [ ] f(t) = 0.68 2t [ x ] f(t) = 8(1.56) 1.4t 66. Remember that the exponential function is y = a b x [a = y-intercept (or initial value) and b = common ratio] 67. Remember that the exponential function is y = a b x, [a = y-intercept (or initial value) and b = common ratio] Note that t represents the time in years, and y represents the estimated number of elephants. If the base years is worked back to t = 0, that would be the y-intercept (or initial value). Year Base Year Estimated Number of Elephants 1995 0 2,649 1996 1 1997 2 1998,218 2000 5,628 2002 7,721 2004 9,571 Answer: B 68. Remember that the exponential growth function is y = a(1 + r) t, and when 1 + r is greater than 1, it is a growth. Note that 44,000 is the initial value. Note that 1.045 = 1 + r, so r = 0.045. Then changing this decimal to a percent is 4.5%. [ x ] Johanna initially earns $44,000 per year. [ ] Johanna initially earns $45,980 per year. [ ] Johanna s salary increases by 1.045% per year. [ x ] Johanna s salary increases by 4.5% per year. [ ] Johanna s salary increases by 104.5% per year
69. Remember that the exponential growth function is y = a(1 + r) t, and when 1 + r is greater than 1, it is a growth. a. Note that r = 0.025, so changing this decimal to a percent is 2.5%. Answer: 2.5% b. Increasing by 0.4% would be 2.5% + 0.4% = 2.9%, and changing this percent to a decimal is 0.029. f(x) = 2,400(1 + 0.029) x Answer: f(x) = 2,400(1.029) x 70. Recall that the compound interest function is A = P (1 + r n )nt, but compounding ANNUALLY is the SAME as the formula used for exponential growth. A = P (1 + r 1 )1t annually means that n = 1 A = P(1 + r) t y = a(1 + r) t y = a(1 + r) t y = 1000(1 + 0.08) t V(t) = 1000(1.08) t Answer: D simplified same as the exponential growth function 71. Remember that the exponential decay function is y = a(1 r) t. Since the value of the car DEPRECIATES, it is referring to a DECREASE, so use the exponential growth function. y = a(1 r) t y = 60,000(1 0.15) t y = 60,000(0.85) t f(x) = 60,000(0.85) x Answer: D 72. Note that if Bill plans to charge DOUBLE the amount every week, the ratio is 2 (r = 2). Note that this should NOT be confused with exponential growth or decay. Also, note that this should NOT be confused with the exponential function, y = a b x. To test that this DOES NOT work, use x = 1 for one week versus x = 2 for two weeks. Instead, this requires the geometric sequence formula, a n = a 1 r n 1.
7. Note that the salesperson starts with a monthly salary of $500, which acts as the initial value (or y-intercept) on the graph. Note that since the salesperson also gets an additional percentage based on what he sells, it refers to the average rate of change (or slope). Answer: A 74. Note that in order to have a linear relationship (or straight line) on the graph between two points, the slope must have a CONSTANT rate of change. When there is a horizontal slope, the rate of change is zero. H 0 zero slope Y Answer: 75. Since both functions f(x) and g(x) are equal to each other ONLY WHEN x = 1 and x =, they also both share the SAME y-values at x = 1 and at x =. If there at least two ordered pairs, g(x) can be written as a linear function. In order to get the ordered pairs, using x = 1 and x = in f(x) will obtain the y-values. For x = 1 For x = f(x) = 2 x f(x) = 2 x f(1) = 2 1 f() = 2 f(1) = 2 f() = 8 (1,2) is one point (,8) is another point To create a linear function for g(x): m = y = 8 2 = 6 = x 1 2 Using (1,2): y y 1 = m(x x 1 ) y 2 = (x 1) y 2 = x y = x 1 Answer: g(x) = x 1
76. Since both functions f(x) and g(x) are equal to each other ONLY WHEN x = 0 and x = 2, they also both share the SAME y-values at x = 0 and at x = 2. If there at least two ordered pairs, g(x) can be written as an exponential function. In order to get the ordered pairs, using x = 0 and x = 2 in f(x) will obtain the y-values. For x = 0 For x = 2 f(x) = 2 x + 4 f(x) = 2 x + 4 f(0) = 2 (0) + 4 f(2) = 2 (2) + 4 f(0) = 4 f(2) = + 4 f(2) = 1 (0,4) is one point (2,1) is another point To create an exponential function for g(x): x f(x) 0 4 1 2 1 Note that the pattern is 1 4, and since there are 2 jumps, take the square root, so 1 4 = 1 2 = r. Remember that the exponential function is y = a b x. y = a b x y = 4 ( 1 2 )x Answer: g(x) = 4 ( 1 2 )x
77. To know what numbers to use, create 20 empty spaces. i. Before attempting to create a list of numbers from the box plot, FIRST identify the FIVE-NUMBER SUMMARY. Minimum: 58 First quartile (Q1): 62 Median: 65 Third quartile (Q): 70 Maximum: 76 ii. Then place the five-number summary around the list of 20 spaces. iii. Then consider how the 10 other points already listed can be placed in the spaces. iv. Lastly, select remaining numbers that will properly fit in the list. Sample Answer: 78. C 79. Remember than if the correlation coefficient is: - close to 1 or 1, then it is a STRONG correlation - close to 0, then it is a WEAK correlation. Negative correlation: 1 r < 0 Positive correlation: 0 < r 1 Since 0.4 is closer to 0 than 1, then it is a weak correlation. Note that a WEAK correlation means that the scattered points are NOT close to the LINE OF BEST FIT. Answer: B
80. Note that a residual plot is created from a line of fit through a scattered plot, and the points should be evenly distributed above and below the horizontal line of the residual plot. Note that this residual plot starts with points above the line, and for the other half, it ends with points below the line. Thus, this would NOT be a good model. [ ] It would be a good model because residual plot indicates a strong linear trend. [ ] It would be a good model because the residual values are allowed to form a linear pattern. [ ] It would be a good model because there seems to be an equal number of points above and below the x-axis. [ x ] It would not be a good model because the points on the residual plot have a linear pattern. [ x ] It would not be a good model because the points on the residual plot are not randomly distributed. [ ] It would not be a good model because the points are too far from 0. [ x ] It would not be a good model because the residual values should be randomly distributed and have values close to 0. Px + Qy = R 81. Note that since (, 1) is a solution to {, any manipulation can be done to either equation, AS LONG AS Fx + Gy = H is it done evenly to the equations (PROPERTIES OF EQUALITY). (P + F)x + (Q + G)y = R + H [ x ] { Fx + Gy = H (P + F)x + Qy = R + H [ ] { Fx + (G + Q)y = H Px + Qy = R [ ] { (P + F)x + (Q + G)y = H + R Px + Qy = R [ x ] { (F 2P)x + (G 2Q)y = H 2R Px + Qy = R [ x ] { 5Fx + 5Gy = 5H the 2 nd equation is added to the 1 st equation cannot add a PART of one equation to another equation (must use entire equation) the 1 st equation that is added to the 2 nd equation is not FULLY multiplied by the 2 nd equation is subtracted by the 1 st equation that is multiplied by 2 the 2 nd equation is multiplied by 5 ax + by = c 82. Note that since (10, ) works for {, any manipulation can be done to either equation, AS LONG AS is it fx + gy = h done evenly to the equations (PROPERTIES OF EQUALITY). (a + f)x + (b + g)y = c + h [ x ] { fx + gy = h (a + f)x + (b + g)y = c + h [ ] { (b + f)x + (a + g)y = c + h ax + by = c [ x ] { (a + f)x + (b + g)y = c + h fx + gy = h [ x ] { (f a)x + (g b)y = h c (a f)x + (b g)y = c h [ x ] { ax + by = c [ x ] { ax2 + bxy = xc 5fx 5gy = 5h the 2 nd equation is added to the 1 st equation cannot mix coefficients of different terms the 1 st equation is added to the 2 nd equation the 2 nd equation is subtracted by the 1 st equation the 1 st equation is multiplied by and then subtracted by the 2 nd equation the 1 st equation is multiplied by x, and the 2 nd equation is multiplied by 5 (5a f)x + (b 5g)y = c 5h [ ] { the 1 st equation is NOT multiplied by the same number (same with the 2 nd equation) 2fx + 2gy = 2h ax + by = c [ ] { ax + by = c the 2 nd equation is missing