1. Write an equation in standard form of the parabola that has the same shape as the graph of f(x) = 2x2, but with the given point as the vertex (5, 3).
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A. f(x) = (2x – 4) + 4 |
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B. f(x) = 2(2x + 8) + 3 |
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C. f(x) = 2(x – 5)2 + 3 |
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D. f(x) = 2(x + 3)2 + 3 |
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2 of 20 |
5.0 Points |
Find the coordinates of the vertex for the parabola defined by the given quadratic function.
f(x) = 2(x – 3)2 + 1
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A. (3, 1) |
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B. (7, 2) |
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C. (6, 5) |
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D. (2, 1)
Find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of the following rational function.
g(x) = x + 3/x(x + 4)
“Y varies directly as the nth power of x” can be modeled by the equation:
40 times a number added to the negative square of that number can be expressed as:
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Solve the following formula for the specified variable:
V = 1/3 lwh for h
7 of 20
Write an equation that expresses each relationship. Then solve the equation for y.
x varies jointly as y and z
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A. x = kz; y = x/k
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B. x = kyz; y = x/kz
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C. x = kzy; y = x/z
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D. x = ky/z; y = x/zk
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8 of 20
8 times a number subtracted from the squared of that number can be expressed as:
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A. P(x) = x + 7x.
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B.P(x) = x2 – 8x.
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C. P(x) = x – x.
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P(x) = x2+ 10x.
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9of 20
Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept.
f(x) = x4 – 9x2
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A. x = 0, x = 3, x = -3; f(x) crosses the x-axis at -3 and 3; f(x) touches the x-axis at 0.
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B. x = 1, x = 2, x = 3; f(x) crosses the x-axis at 2 and 3; f(x) crosses the x-axis at 0.
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C. x = 0, x = -3, x = 5; f(x) touches the x-axis at -3 and 5; f(x) touches the x-axis at 0.
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D. x = 1, x = 2, x = -4; f(x) crosses the x-axis at 2 and -4; f(x) touches the x-axis at 0.
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10 of 20
Find the domain of the following rational function.
f(x) = x + 7/x2 + 49
f(x) = x + 7/x2 + 49
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A. All real numbers < 69
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B. All real numbers > 210
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C. All real numbers ≤ 77
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D. All real numbers
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11 of 20
Write an equation in standard form of the parabola that has the same shape as the graph of f(x) = 3x2 or g(x) = -3x2, but with the given maximum or minimum.
Minimum = 0 at x = 11
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A. f(x) = 6(x – 9)
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B. f(x) = 3(x – 11)2
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C. f(x) = 4(x + 10)
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D. f(x) = 3(x2 – 15)2
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12 of 20
Solve the following polynomial inequality.
3x2 + 10x – 8 ≤ 0
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A. [6, 1/3]
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B. [-4, 2/3]
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C. [-9, 4/5]
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D. [8, 2/7]
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13 of 20
Find the coordinates of the vertex for the parabola defined by the given quadratic function.
f(x) = -2(x + 1)2 + 5
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A. (-1, 5)
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B. (2, 10)
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C. (1, 10)
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D. (-3, 7)
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14 of 20
Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept.
f(x) = -2x4 + 4x3
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A. x = 1, x = 0; f(x) touches the x-axis at 1 and 0
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B. x = -1, x = 3; f(x) crosses the x-axis at -1 and 3
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C. x = 0, x = 2; f(x) crosses the x-axis at 0 and 2
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D. x = 4, x = -3; f(x) crosses the x-axis at 4 and -3
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15 of 20
Find the domain of the following rational function.
f(x) = 5x/x – 4
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A. {x │x ≠ 3}
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B. {x │x = 5}
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C. {x │x = 2}
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D. {x │x ≠ 4}
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16 of 20
Based on the synthetic division shown, the equation of the slant asymptote of f(x) = (3x2 – 7x + 5)/x – 4 is:
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A. y = 3x + 5.
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B. y = 6x + 7.
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C. y = 2x – 5.
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D. y = 3x2 + 7.
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17 of 20
The perimeter of a rectangle is 80 feet. If the length of the rectangle is represented by x, its width can be expressed as:
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A. 80 + x.
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B. 20 – x.
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C. 40 + 4x.
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D. 40 – x.
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18 of 20
Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers.
f(x) = 2x4 – 4x2 + 1; between -1 and 0
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A. f(-1) = -0; f(0) = 2
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B. f(-1) = -1; f(0) = 1
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C. f(-1) = -2; f(0) = 0
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D. f(-1) = -5; f(0) = -3
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19 of 20
Solve the following polynomial inequality.
9x2 – 6x + 1 < 0
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A. (-∞, -3)
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B. (-1, ∞)
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C. [2, 4)
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D. Ø
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20 of 20
Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers.
f(x) = x3 – x – 1; between 1 and 2
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A. f(1) = -1; f(2) = 5
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B. f(1) = -3; f(2) = 7
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C. f(1) = -1; f(2) = 3
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D. f(1) = 2; f(2) = 7
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1. Find the domain of following logarithmic function.
f(x) = ln (x – 2)2
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A. (∞, 2) ∪ (-2, -∞)
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B. (-∞, 2) ∪ (2, ∞)
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C. (-∞, 1) ∪ (3, ∞)
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D. (2, -∞) ∪ (2, ∞)
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2 . Solve the following exponential equation. Express the solution set in terms of natural logarithms or common logarithms to a decimal approximation, of two decimal places, for the solution.
ex = 5.7
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A. {ln 5.7}; ≈1.74
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B. {ln 8.7}; ≈3.74
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C. {ln 6.9}; ≈2.49
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D. {ln 8.9}; ≈3.97
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3. Evaluate the following expression without using a calculator.
Log7 √7
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A. 1/4
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B. 3/5
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C. 1/2
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D. 2/7
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4. Write the following equation in its equivalent logarithmic form.
2-4 = 1/16
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A. Log4 1/16 = 64
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B. Log2 1/24 = -4
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C. Log2 1/16 = -4
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D. Log4 1/16 = 54
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5. Write the following equation in its equivalent exponential form.
4 = log2 16
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A. 2 log4 = 16
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B. 22 = 4
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C. 44 = 256
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D. 24 = 16
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6. The half-life of the radioactive element krypton-91 is 10 seconds. If 16 grams of krypton-91 are initially present, how many grams are present after 10 seconds? 20 seconds?
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A. 10 grams after 10 seconds; 6 grams after 20 seconds
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B. 12 grams after 10 seconds; 7 grams after 20 seconds
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C. 4 grams after 10 seconds; 1 gram after 20 seconds
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D. 8 grams after 10 seconds; 4 grams after 20 seconds
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7. Use properties of logarithms to expand the following logarithmic expression as much as possible.
logb (x2y)
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A. 2 logy x + logx y
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B. 2 logb x + logb y
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C. logx – logb y
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D. logb x – logx y
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8. Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents.
ex+1 = 1/e
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A. {-3}
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B. {-2}
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C. {4}
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D. {12}
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9. Use properties of logarithms to condense the following logarithmic expression. Write the expression as a single logarithm whose coefficient is 1.
log x + 3 log y
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A. log (xy)
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B. log (xy3)
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C. log (xy2)
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D. logy (xy)3
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10. Approximate the following using a calculator; round your answer to three decimal places.
e-0.95
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A. .483
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B. 1.287
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C. .597
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D. .387
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11. Use the exponential growth model, A = A0ekt, to show that the time it takes a population to double (to grow from A0 to 2A0 ) is given by t = ln 2/k.
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A. A0 = A0ekt; ln = ekt; ln 2 = ln ekt; ln 2 = kt; ln 2/k = t
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B. 2A0 = A0e; 2= ekt; ln = ln ekt; ln 2 = kt; ln 2/k = t
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C. 2A0 = A0ekt; 2= ekt; ln 2 = ln ekt; ln 2 = kt; ln 2/k = t
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D. 2A0 = A0ekt; 2 = ekt; ln 1 = ln ekt; ln 2 = kt; ln 2/k = toe
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12. Solve the following logarithmic equation. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, to two decimal places, for the solution.
2 log x = log 25
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A. {12}
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B. {5}
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C. {-3}
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D. {25}
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13. Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents.
31-x = 1/27
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A. {2}
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B. {-7}
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C. {4}
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D. {3}
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14.Find the domain of following logarithmic function.
f(x) = log (2 – x)
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A. (∞, 4)
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B. (∞, -12)
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C. (-∞, 2)
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D. (-∞, -3)
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15.Use properties of logarithms to expand the following logarithmic expression as much as possible.
Logb (√xy3 / z3)
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A. 1/2 logb x – 6 logb y + 3 logb z
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B. 1/2 logb x – 9 logb y – 3 logb z
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C. 1/2 logb x + 3 logb y + 6 logb z
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D. 1/2 logb x + 3 logb y – 3 logb z
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16. Solve the following exponential equation. Express the solution set in terms of natural logarithms or common logarithms to a decimal approximation, of two decimal places, for the solution.
32x + 3x – 2 = 0
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A. {1}
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B. {-2}
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C. {5}
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D. {0}
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17. You have $10,000 to invest. One bank pays 5% interest compounded quarterly and a second bank pays 4.5% interest compounded monthly. Use the formula for compound interest to write a function for the balance in each bank at any time t.
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A. A = 20,000(1 + (0.06/4))4t; A = 10,000(1 + (0.044/14))12t
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B. A = 15,000(1 + (0.07/4))4t; A = 10,000(1 + (0.025/12))12t
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C. A = 10,000(1 + (0.05/4))4t; A = 10,000(1 + (0.045/12))12t
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D. A = 25,000(1 + (0.05/4))4t; A = 10,000(1 + (0.032/14))12t
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18. The exponential function f with base b is defined by f(x) = __________, b > 0 and b ≠ 1. Using interval notation, the domain of this function is __________ and the range is __________.
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A. bx; (∞, -∞); (1, ∞)
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B. bx; (-∞, -∞); (2, ∞)
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C. bx; (-∞, ∞); (0, ∞)
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D. bx; (-∞, -∞); (-1, ∞)
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19.
| Write the following equation in its equivalent exponential form. log6 216 = y
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20. Write the following equation in its equivalent exponential form.
5 = logb 32
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A. b5 = 32
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B. y5 = 32
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C. Blog5 = 32
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D. Logb = 32
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21. Perform the long division and write the partial fraction decomposition of the remainder term.
x5 + 2/x2 – 1
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A. x2 + x – 1/2(x + 1) + 4/2(x – 1)
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B. x3 + x – 1/2(x + 1) + 3/2(x – 1)
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C. x3 + x – 1/6(x – 2) + 3/2(x + 1)
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D. x2 + x – 1/2(x + 1) + 4/2(x – 1)
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22. Solve each equation by the substitution method.
| x + y = 1 x2 + xy – y2 = -5 |
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A. {(4, -3), (-1, 2)}
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B. {(2, -3), (-1, 6)}
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C. {(-4, -3), (-1, 3)}
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D. {(2, -3), (-1, -2)}
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23. A television manufacturer makes rear-projection and plasma televisions. The profit per unit is $125 for the rear-projection televisions and $200 for the plasma televisions.
Let x = the number of rear-projection televisions manufactured in a month and let y = the number of plasma televisions manufactured in a month. Write the objective function that models the total monthly profit.
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A. z = 200x + 125y
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B. z = 125x + 200y
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C. z = 130x + 225y
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D. z = -125x + 200y
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24. Solve the following system by the substitution method.
{x + 3y = 8
{y = 2x – 9
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A. {(5, 1)}
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B. {(4, 3)}
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C. {(7, 2)}
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D. {(4, 3)}
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25. Solve the following system by the addition method.
{4x + 3y = 15
{2x – 5y = 1
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A. {(4, 0)}
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B. {(2, 1)}
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C. {(6, 1)}
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D. {(3, 1)}
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26. Many elevators have a capacity of 2000 pounds.
If a child averages 50 pounds and an adult 150 pounds, write an inequality that describes when x children and y adults will cause the elevator to be overloaded.
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A. 50x + 150y > 2000
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B. 100x + 150y > 1000
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C. 70x + 250y > 2000
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D. 55x + 150y > 3000
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27.Find the quadratic function y = ax2 + bx + c whose graph passes through the given points.
(-1, -4), (1, -2), (2, 5)
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A. y = 2x2 + x – 6
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B. y = 2x2 + 2x – 4
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C. y = 2x2 + 2x + 3
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D. y = 2x2 + x – 5
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28.Write the partial fraction decomposition for the following rational expression.
x + 4/x2(x + 4)
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A. 1/3x + 1/x2 – x + 5/4(x2 + 4)
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B. 1/5x + 1/x2 – x + 4/4(x2 + 6)
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C. 1/4x + 1/x2 – x + 4/4(x2 + 4)
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D. 1/3x + 1/x2 – x + 3/4(x2 + 5)
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29. Find the quadratic function y = ax2 + bx + c whose graph passes through the given points.
(-1, 6), (1, 4), (2, 9)
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A. y = 2x2 – x + 3
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B. y = 2x2 + x2 + 9
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C. y = 3x2 – x – 4
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D. y = 2x2 + 2x + 4
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30. Solve the following system.
| 3(2x+y) + 5z = -1 2(x – 3y + 4z) = -9 4(1 + x) = -3(z – 3y) |
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A. {(1, 1/3, 0)}
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B. {(1/4, 1/3, -2)}
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C. {(1/3, 1/5, -1)}
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D. {(1/2, 1/3, -1)}
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31. On your next vacation, you will divide lodging between large resorts and small inns. Let x represent the number of nights spent in large resorts. Let y represent the number of nights spent in small inns.
Write a system of inequalities that models the following conditions:
You want to stay at least 5 nights. At least one night should be spent at a large resort. Large resorts average $200 per night and small inns average $100 per night. Your budget permits no more than $700 for lodging.
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A.
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B.
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C.
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D.
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32. Solve the following system.
| 2x + 4y + 3z = 2 x + 2y – z = 0 4x + y – z = 6 |
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A. {(-3, 2, 6)}
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B. {(4, 8, -3)}
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C. {(3, 1, 5)}
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D. {(1, 4, -1)}
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33. Solve the following system.
| x = y + 4 3x + 7y = -18 |
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A. {(2, -1)}
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B. {(1, 4)}
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C. {(2, -5)}
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D. {(1, -3)}
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34. Write the partial fraction decomposition for the following rational expression.
4/2x2 – 5x – 3
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A. 4/6(x – 2) – 8/7(4x + 1)
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B. 4/7(x – 3) – 8/7(2x + 1)
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C. 4/7(x – 2) – 8/7(3x + 1)
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D. 4/6(x – 2) – 8/7(3x + 1)
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35. Solve each equation by either substitution or addition method.
| x2 + 4y2 = 20 x + 2y = 6 |
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A. {(5, 2), (-4, 1)}
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B. {(4, 2), (3, 1)}
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C. {(2, 2), (4, 1)}
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D. {(6, 2), (7, 1)}
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36. Solve each equation by either substitution or addition method.
| x2 + 4y2 = 20 x + 2y = 6 |
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A. {(5, 2), (-4, 1)}
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B. {(4, 2), (3, 1)}
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C. {(2, 2), (4, 1)}
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D. {(6, 2), (7, 1)}
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37. Write the partial fraction decomposition for the following rational expression.
1/x2 – c2 (c ≠ 0)
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A. 1/4c/x – c – 1/2c/x + c
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B. 1/2c/x – c – 1/2c/x + c
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C. 1/3c/x – c – 1/2c/x + c
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D. 1/2c/x – c – 1/3c/x + c
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38. Solve each equation by the substitution method.
| x2 – 4y2 = -7 3x2 + y2 = 31 |
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A. {(2, 2), (3, -2), (-1, 2), (-4, -2)}
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B. {(7, 2), (3, -2), (-4, 2), (-3, -1)}
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C. {(4, 2), (3, -2), (-5, 2), (-2, -2)}
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D. {(3, 2), (3, -2), (-3, 2), (-3, -2)}
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39. Solve each equation by the substitution method.
| y2 = x2 – 9 2y = x – 3 |
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A. {(-6, -4), (2, 0)}
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B. {(-4, -4), (1, 0)}
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C. {(-3, -4), (2, 0)}
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D. {(-5, -4), (3, 0)}
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40. Write the partial fraction decomposition for the following rational expression.
ax +b/(x – c)2 (c ≠ 0)
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A. a/a – c +ac + b/(x – c)2
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B. a/b – c +ac + b/(x – c)
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C. a/a – b +ac + c/(x – c)2
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D. a/a – b +ac + b/(x – c)
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