Reference no: EM131260812

Part 1

Question 1

"Y varies directly as the nth power of x" can be modeled by the equation:

A. y = kxn.

B. y = kx/n.

C. y = kx*n.

D. y = knx.

Question 2

Find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of the following rational function.

f(x) = x/x + 4

A. Vertical asymptote: x = -4; no holes

B. Vertical asymptote: x = -4; holes at 3x

C. Vertical asymptote: x = -4; holes at 2x

D. Vertical asymptote: x = -4; holes at 4x

Question 3

The graph of f(x) = -x2 __________ to the left and __________ to the right.

A. falls; rises

B. rises; rises

C. falls; falls

D. rises; rises

Question 4

All rational functions can be expressed as f(x) = p(x)/q(x), where p and q are __________ functions and q(x) ≠ 0.

A. horizontal asymptotes

B. polynomial

C. vertical asymptotes

D. slant asymptotes

Question 5

Determine the degree and the leading coefficient of the polynomial function f(x) = -2x3 (x - 1)(x + 5).

A. 5; -2

B. 7; -4

C. 2; -5

D. 1; -9

Question 6

The perimeter of a rectangle is 80 feet. If the length of the rectangle is represented by x, its width can be expressed as:

A. 80 + x.

B. 20 - x.

C. 40 + 4x.

D. 40 - x.

Question 7

Find the coordinates of the vertex for the parabola defined by the given quadratic function.

f(x) = -2(x + 1)2 + 5

A. (-1, 5)

B. (2, 10)

C. (1, 10)

D. (-3, 7)

Question 8

Solve the following polynomial inequality.

9x2 - 6x + 1 < 0

A. (-∞, -3)

B. (-1, ∞)

C. [2, 4)

D. Ø

Question 10

Based on the synthetic division shown, the equation of the slant asymptote of f(x) = (3x2 - 7x + 5)/x - 4 is:

A. y = 3x + 5.

B. y = 6x + 7.

C. y = 2x - 5.

D. y = 3x2 + 7.

Question 11

Write an equation that expresses each relationship. Then solve the equation for y.

x varies jointly as y and z

A. x = kz; y = x/k

B. x = kyz; y = x/kz

C. x = kzy; y = x/z

D. x = ky/z; y = x/zk

Question 12

Solve the following polynomial inequality.

3x2 + 10x - 8 ≤ 0

A. [6, 1/3]

B. [-4, 2/3]

C. [-9, 4/5]

D. [8, 2/7]

Question 13

Find the domain of the following rational function.

f(x) = 5x/x - 4

A. {x ¦x ≠ 3}

B. {x ¦x = 5}

C. {x ¦x = 2}

D. {x ¦x ≠ 4}

Question 14

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) = x2(x - 1)3(x + 2)

A. x = -1, x = 2, x = 3 ; f(x) crosses the x-axis at 2 and 3; f(x) touches the x-axis at -1

B. x = -6, x = 3, x = 2 ; f(x) crosses the x-axis at -6 and 3; f(x) touches the x-axis at 2.

C. x = 7, x = 2, x = 0 ; f(x) crosses the x-axis at 7 and 2; f(x) touches the x-axis at 0.

D. x = -2, x = 0, x = 1 ; f(x) crosses the x-axis at -2 and 1; f(x) touches the x-axis at 0.

Question 15

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.

Maximum = 4 at x = -2

A. f(x) = 4(x + 6)2 - 4

B. f(x) = -5(x + 8)2 + 1

C. f(x) = 3(x + 7)2 - 7

D. f(x) = -3(x + 2)2 + 4

Question 16

Find the coordinates of the vertex for the parabola defined by the given quadratic function.

f(x) = 2(x - 3)2 + 1

A. (3, 1)

B. (7, 2)

C. (6, 5)

D. (2, 1)

Question 17

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

A. f(1) = -1; f(2) = 5

B. f(1) = -3; f(2) = 7

C. f(1) = -1; f(2) = 3

D. f(1) = 2; f(2) = 7

Question 18

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

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.

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.

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.

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.

Question 19

Find the domain of the following rational function.

g(x) = 3x2/((x - 5)(x + 4))

A. {x¦ x ≠ 3, x ≠ 4}

B. {x¦ x ≠ 4, x ≠ -4}

C. {x¦ x ≠ 5, x ≠ -4}

D. {x¦ x ≠ -3, x ≠ 4}

Question 20

If f is a polynomial function of degree n, then the graph of f has at most __________ turning points.

A. n - 3

B. n - f

C. n - 1

D. n + f

Part 2

Question 1

An artifact originally had 16 grams of carbon-14 present. The decay model A = 16e -0.000121t describes the amount of carbon-14 present after t years. How many grams of carbon-14 will be present in 5715 years?

A. Approximately 7 grams

B. Approximately 8 grams

C. Approximately 23 grams

D. Approximately 4 grams

Question 2

Write the following equation in its equivalent exponential form.

5 = logb 32

A. b5 = 32

B. y5 = 32

C. Blog5 = 32

D. Logb = 32

Question 3

Use properties of logarithms to condense the following logarithmic expression. Write the expression as a single logarithm whose coefficient is 1.

3 ln x - 1/3 ln y

A. ln (x / y1/2)

B. lnx (x6 / y1/3)

C. ln (x3 / y1/3)

D. ln (x-3 / y1/4)

Question 4

Consider the model for exponential growth or decay given by A = A0ekt. If k __________, the function models the amount, or size, of a growing entity. If k __________, the function models the amount, or size, of a decaying entity.

A. > 0; < 0

B. = 0; ≠ 0

C. ≥ 0; < 0

D. < 0; ≤ 0

Question 5

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

A. log (xy)

B. log (xy3)

C. log (xy2)

D. logy (xy)3

Question 6

Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents.

ex+1 = 1/e

A. {-3}

B. {-2}

C. {4}

D. {12}

Question 7

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

A. {1}

B. {-2}

C. {5}

D. {0}

Question 8

Use properties of logarithms to expand the following logarithmic expression as much as possible.

Logb (√xy3 / z3)

A. 1/2 logb x - 6 logb y + 3 logb z

B. 1/2 logb x - 9 logb y - 3 logb z

C. 1/2 logb x + 3 logb y + 6 logb z

D. 1/2 logb x + 3 logb y - 3 logb z

Question 9

Use properties of logarithms to expand the following logarithmic expression as much as possible.

logb (x2y)

A. 2 logy x + logx y

B. 2 logb x + logb y

C. logx - logb y

D. logb x - logx y

Question 10

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

A. {ln 5.7}; ≈1.74

B. {ln 8.7}; ≈3.74

C. {ln 6.9}; ≈2.49

D. {ln 8.9}; ≈3.97

Question 11

The graph of the exponential function f with base b approaches, but does not touch, the __________-axis. This axis, whose equation is __________, is a __________ asymptote.

A. x; y = 0; horizontal

B. x; y = 1; vertical

C. -x; y = 0; horizontal

D. x; y = -1; vertical

Question 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

A. {12}

B. {5}

C. {-3}

D. {25}

Question 13

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.

A. A = 20,000(1 + (0.06/4))4t; A = 10,000(1 + (0.044/14))12t

B. A = 15,000(1 + (0.07/4))4t; A = 10,000(1 + (0.025/12))12t

C. A = 10,000(1 + (0.05/4))4t; A = 10,000(1 + (0.045/12))12t

D. A = 25,000(1 + (0.05/4))4t; A = 10,000(1 + (0.032/14))12t

Question 14

Evaluate the following expression without using a calculator.

8log8 19

A. 17

B. 38

C. 24

D. 19

Question 15

Write the following equation in its equivalent logarithmic form.

3√8 = 2

A. Log2 3 = 1/8

B. Log8 2 = 1/3

C. Log2 8 = 1/2

D. Log3 2 = 1/8

Question 16

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 __________.

A. bx; (∞, -∞); (1, ∞)

B. bx; (-∞, -∞); (2, ∞)

C. bx; (-∞, ∞); (0, ∞)

D. bx; (-∞, -∞); (-1, ∞)

Question 17

Find the domain of following logarithmic function.

f(x) = log (2 - x)

A. (∞, 4)

B. (∞, -12)

C. (-∞, 2)

D. (-∞, -3)

Question 18

Evaluate the following expression without using a calculator.

Log7 √7

A. 1/4

B. 3/5

C. 1/2

D. 2/7

Question 19

Use properties of logarithms to expand the following logarithmic expression as much as possible.

logb (x2 y) / z2

A. 2 logb x + logb y - 2 logb z

B. 4 logb x - logb y - 2 logb z

C. 2 logb x + 2 logb y + 2 logb z

D. logb x - logb y + 2 logb z

Question 20

Write the following equation in its equivalent exponential form.

4 = log2 16

A. 2 log4 = 16

B. 22 = 4

C. 44 = 256

D. 24 = 16

Question 21

Solve the following system by the addition method.

{2x + 3y = 6

{2x - 3y = 6

A. {(4, 1)}

B. {(5, 0)}

C. {(2, 1)}

D. {(3, 0)}

Question 22

A television manufacturer makes rear-projection and plasma televisions. The pro?t 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.

A. z = 200x + 125y

B. z = 125x + 200y

C. z = 130x + 225y

D. z = -125x + 200y

Question 23

Find the quadratic function y = ax2 + bx + c whose graph passes through the given points.

(-1, 6), (1, 4), (2, 9)

A. y = 2x2 - x + 3

B. y = 2x2 + x2 + 9

C. y = 3x2 - x - 4

D. y = 2x2 + 2x + 4

Question 24

Write the form of the partial fraction decomposition of the rational expression.

7x - 4/x2 - x - 12

A. 24/7(x - 2) + 26/7(x + 5)

B. 14/7(x - 3) + 20/7(x2 + 3)

C. 24/7(x - 4) + 25/7(x + 3)

D. 22/8(x - 2) + 25/6(x + 4)

Question 25

Write the partial fraction decomposition for the following rational expression.

ax +b/(x - c)2 (c ≠ 0)

A. a/a - c +ac + b/(x - c)2

B. a/b - c +ac + b/(x - c)

C. a/a - b +ac + c/(x - c)2

D. a/a - b +ac + b/(x - c)

Question 26

Solve each equation by either substitution or addition method.

x2 + 4y2 = 20

x + 2y = 6

A. {(5, 2), (-4, 1)}

B. {(4, 2), (3, 1)}

C. {(2, 2), (4, 1)}

D. {(6, 2), (7, 1)}

Question 27

Solve the following system by the addition method.

{4x + 3y = 15

{2x - 5y = 1

A. {(4, 0)}

B. {(2, 1)}

C. {(6, 1)}

D. {(3, 1)}

Question 28

Write the partial fraction decomposition for the following rational expression.

x2 - 6x + 3/(x - 2)3

A. 1/x - 4 - 2/(x - 2)2 - 6/(x - 2)

B. 1/x - 2 - 4/(x - 2)2 - 5/(x - 1)3

C. 1/x - 3 - 2/(x - 3)2 - 5/(x - 2)

D. 1/x - 2 - 2/(x - 2)2 - 5/(x - 2)3

Question 29

Solve each equation by the substitution method.

x2 - 4y2 = -7

3x2 + y2 = 31

A. {(2, 2), (3, -2), (-1, 2), (-4, -2)}

B. {(7, 2), (3, -2), (-4, 2), (-3, -1)}

C. {(4, 2), (3, -2), (-5, 2), (-2, -2)}

D. {(3, 2), (3, -2), (-3, 2), (-3, -2)}

Question 30

Perform the long division and write the partial fraction decomposition of the remainder term.

x4 - x2 + 2/x3 - x2

A. x + 3 - 2/x - 1/x2 + 4x - 1

B. 2x + 1 - 2/x - 2/x + 2/x + 1

C. 2x + 1 - 2/x2 - 2/x + 5/x - 1

D. x + 1 - 2/x - 2/x2 + 2/x - 1

Question 31

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.

A. 50x + 150y > 2000

B. 100x + 150y > 1000

C. 70x + 250y > 2000

D. 55x + 150y > 3000

Question 32

Solve each equation by the addition method.

x2 + y2 = 25

(x - 8)2 + y2 = 41

A. {(3, 5), (3, -2)}

B. {(3, 4), (3, -4)}

C. {(2, 4), (1, -4)}

D. {(3, 6), (3, -7)}

Question 33

Write the partial fraction decomposition for the following rational expression.

x + 4/x2(x + 4)

A. 1/3x + 1/x2 - x + 5/4(x2 + 4)

B. 1/5x + 1/x2 - x + 4/4(x2 + 6)

C. 1/4x + 1/x2 - x + 4/4(x2 + 4)

D. 1/3x + 1/x2 - x + 3/4(x2 + 5)

Question 34

Write the partial fraction decomposition for the following rational expression.

4/2x2 - 5x - 3

A. 4/6(x - 2) - 8/7(4x + 1)

B. 4/7(x - 3) - 8/7(2x + 1)

C. 4/7(x - 2) - 8/7(3x + 1)

D. 4/6(x - 2) - 8/7(3x + 1)

Question 35

Let x represent one number and let y represent the other number. Use the given conditions to write a system of equations. Solve the system and find the numbers.

The sum of two numbers is 7. If one number is subtracted from the other, their difference is -1. Find the numbers.

A. x + y = 7; x - y = -1; 3 and 4

B. x + y = 7; x - y = -1; 5 and 6

C. x + y = 7; x - y = -1; 3 and 6

D. x + y = 7; x - y = -1; 2 and 3

Question 36

Solve the following system by the substitution method.

{x + y = 4

{y = 3x

A. {(1, 4)}

B. {(3, 3)}

C. {(1, 3)}

D. {(6, 1)}

Question 37

Solve the following system.

3(2x+y) + 5z = -1

2(x - 3y + 4z) = -9

4(1 + x) = -3(z - 3y)

A. {(1, 1/3, 0)}

B. {(1/4, 1/3, -2)}

C. {(1/3, 1/5, -1)}

D. {(1/2, 1/3, -1)}

Question 38

Solve the following system.

2x + y = 2

x + y - z = 4

3x + 2y + z = 0

A. {(2, 1, 4)}

B. {(1, 0, -3)}

C. {(0, 0, -2)}

D. {(3, 2, -1)}

Question 39 Solve the following system.

2x + 4y + 3z = 2

x + 2y - z = 0

4x + y - z = 6

A. {(-3, 2, 6)}

B. {(4, 8, -3)}

C. {(3, 1, 5)}

D. {(1, 4, -1)}

Question 40

Solve each equation by the substitution method.

x + y = 1

x2 + xy - y2 = -5

A. {(4, -3), (-1, 2)}

B. {(2, -3), (-1, 6)}

C. {(-4, -3), (-1, 3)}

D. {(2, -3), (-1, -2)}