Lesson 3
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
Lesson 4
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 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.
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)}
