1. Explain how the graph of y-5 = (x-3)2 can be obtained from the graph of y = x2.
Shift the graph of y = x2 left 3 units and down 5 units
Shift the graph of y = x2 left 3 units and up 5 units
Shift the graph of y = x2 right 3 units and down 5 units
Shift the graph of y = x2 right 3 units and up 5 units
2. Give the equation for the circle with center C(3, -2) and radius 4.
x2+y2=52
(x-3)2+(y-2)2=16
(x+3)2+(y-2)2=42
(x-3)2+(y+2)2=16
3. Given f(x) = 5x + 7 and g(x) = x2 + 7, find (gf)(x).
(gf)(x)= 5x2 + 7
(gf)(x)= 5x2 + 42
(gf)(x)= (5x)2 + 14
(gf)(x)= 25x2 + 70x + 56
4. Find the point on the positive y-axis that is a distance 5 from the point P(3, 4).
A(0, 6)
B(0,8)
C(6,0)
D(8,0)
5. Give the center of the circle with equation x2+2x+y2-10y+22=0.
A(2, 4)
B(1, 5)
C(-1, 5)
D(-2, 4)
6. An object is projected upward from the top of a tower. Its distance in feet above the ground after t seconds is given by s(t)=-16t2+64t+80. How many seconds will it take to reach ground level?
1 second
4 seconds
5 seconds
8 seconds
7. The figure shows the graphs of y = f(x) and y = g(x). Express the function g in terms of f.
g(x) = f(x - 2)
g(x) = -f(x + 2)
g(x) = 2 - f(x)
g(x) = 2 - f(x - 2)
8. From a square piece of cardboard with width x inches, a square of width x - 3 inches is removed from the center. Write the area of the remaining piece as a function of x.
f(x) = 6x - 9
f(x) = 6x + 9
f(x) = 2x2 - 9
f(x) = 2x2 - 6x – 9
9. Find the midpoint of the line segment from A(-2, 9) to B(4, 5).
C(1, 7)
D(3, 7)
P(4, 9)
Q(5, 9)
10. The figure shows the graph of a function that is ____.
even
odd
both even and odd
neither even nor odd
11. If P(4, -5) is a point on the graph of the function y = f(x), find the corresponding point on the graph of y = 2f(x - 6).
A(1, 8)
B(2, -5)
C(6, 8)
D(10,-10)
12. If f(x) = x(x - 1)(x - 4)2, use interval notation to give all values of x where f(x) > 0.
(-,0)(4,)
(-,1)(4,)
(-,1)(4,)
(-,0)(1,4)(4,)
13. A rectangle is placed under the parabolic arch given by f(x) = 27 - 3x2 by using a point (x, y) on the parabola, as shown in the figure. Write a formula for the function A(x) that gives the area of the rectangle as a function of the x-coordinate of the point chosen.
f(x) = 6(27 - 3x2)
f(x) = 27x - 3x3
f(x) = 54x - 6x3
f(x) = 162x - 6x3
14. If f(x) = x(x + 3)(x - 1), use interval notation to give all values of x where f(x) > 0.
(-3, 1)
(-3, 0) (1,)
(1, 3)
(0, 1)(3,)
15. Find all roots of the polynomial x3 - x2 + 16x - 16.
1, 4, -4
-1, 4, -4
-1, 4i, -4i
1, 4i, -4i
16. Find a polynomial with leading coefficient 1 and degree 3 that has -1, 1, and 3 as roots.
x3 - 3x2 - x + 3
x3 - 3x2 + x - 3
x3 + 3x2 - x - 3
x3 + 3x2 + x + 3
17. Express the following statement as a formula with the value of the constant of proportionality determined with the given conditions: w varies directly as x and inversely as the square of y. If x = 15 and y = 5, then w = 36.
y2
w / =12 / x
y2
w / =36 / x
y2
w / =60 / x
y2
18. Find the third degree polynomial whose graph is shown in the figure.
f(x) = x3 - x2 -2x + 2
4 / x3 / - / 1
2 / x2 / - / x+2
f(x) = / 1
4 / x3 / - / 1
4 / x2 / + / 2x+2
f(x) = / 1
2 / x3 / - / 1
2 / x2 / - / x+2
19. The period of a simple pendulum is directly proportional to the square root of its length. If a pendulum has a length of 6 feet and a period of 2 seconds, to what length should it be shortened to achieve a 1 second period?
1 foot
1.5 feet
2 feet
3 feet
20. The figure shows the graph of:
f(x) = / 6x-102x-a
Find the value of a.
2
3
4
6
21. The figure shows the graphs of f(x) = x3 and g(x) = ax3. What can you conclude about the value of a?
a < –1
–1 < a < 0
0 < a < 1
1 < a
22. Find the horizontal asymptote of the rational function:
f(x)= / 8x-124x-2
y = 1/2
y = 3/2
y = 2
y = 4
23. Find the quotient and remainder of f(x) = x4 - 2 divided by p(x) = x - 1.
x3 + x2 + 1; -1
x3 + x2 + x + 1; -1
x3 + x + 1; -1
x3 - x2 - x - 1; -1
24. Identify the rational function whose graph is shown in the figure.
x+1
f(x) = / x+5
x+3
f(x) = / 3x-5
x-1
f(x) = / x+5
x+1
25. Find the polynomial f(x) of degree three that has zeroes at 1, 2, and 4 such that f(0) = -16.
f(x)=x3-7x2+14x-16
f(x)=2x3-14x2+28x-16
f(x)=2x3-14x2+14x-16
f(x)=2x3+7x2+14x+16
26. Find the vertical asymptote of the rational function:
f(x) = / 3x-124x-2
x = 1/2
x = 3/4
x = 2
x = 4
27. The figure shows the graph of y = (x - 3)(x - 5)(x - a). Determine the value of a.
3
4
5
7
28. The table shows several values of the function f(x) = -x3 + x2 - x + 2. Complete the missing values in this table, and then use these values and the intermediate value theorem to determine (an) interval(s) where the function must have a zero.
x / –2 / –1 / 0 / 1 / 2f(x) / 16 / –4
(0, 1)
(1, 2)
(0,1)(2,)
(-)(2,)
29. For the following equation, find the interval(s) where f(x) 0.
f(x) = / 1x2-2x-8
(-4, 2)
(-2, 4)
(2, 4)
(2, 8)
30. Find the quotient and remainder of f(x) = x3 - 4x2 + 5x + 5 divided by p(x) = x - 1.
x2 + 2x + 2; 7
x2 - 3x + 3; -5
x2 - 3x + 2; 7
x2 - 2x + 3; -5
31. The figure shows the graph of the polynomial function y = f(x). For which of the values k = 0, 1, 2, or 3 will the equation f(x) = k have complex roots?
0
1
2
3
32. The polynomial f(x) divided x - 3 results in a quotient of x2+3x-5 with a remainder of 2. Find f(3).
-5
-2
2
3
33. Let f(x) = x3 - 8x2 + 17x - 9. Use the factor theorem to find other solutions to f(x) - f(1) = 0, besides x = 1.
-2, 5
2, -3
2, 5
2, 10
34. The electrical resistance R of a wire varies directly as its length L and inversely as the square of its diameter. A wire 20 meters long and 0.6 centimeters in diameter made from a certain alloy has a resistance of 36 ohms. What is the resistance of a piece of wire 60 meters long and 1.2 centimeters in diameter made from the same material?
24 ohms
27 ohms
30 ohms
48 ohms
35. The degree three polynomial f(x) with real coefficients and leading coefficient 1, has 4 and 3 + i among its roots. Express f(x) as a product of linear and quadratic polynomials with real coefficients.
f(x)=(x+4)(x2+6x+10)
f(x)=(x-4)(x2-6x-9)
f(x)=(x-4)(x2-6x+10)
f(x)=(x-4)(x2-6x+9)
36. Given that (3x - a)(x - 2)(x - 7) = 3x3 - 32x2 + 81x - 70, determine the value of a.
1
3
5
7
37. Identify the exponential function of the form f(x)=a(2x)+b whose graph is shown in the figure.
f(x)=3(2x)
f(x)=2x-3
f(x)=2(2x)-4
f(x)=2x-2
38. From the information in the table providing values of f(x) and g(x), evaluate (f g)-1(3)
f(x) / 5 / 3 / 5 / 1 / 2
g(x) / 4 / 5 / 1 / 3 / 2
1
2
4
5
39. For the function f(x) shown,
find the domain and range of f -1(x).
Domain = [0, 6], Range [ 2, 5]
Domain = [0, 5], Range [ 2, 6]
Domain = [2, 5], Range [ 0, 6]
Domain = [2, 6], Range [ 0, 5]
40. Write the expression loga(y+5)+2loga(x+1) as one logarithm.
loga(y+2x+7)
loga(y+x2+7)
loga[2(y+5)(x+1)]
loga[(y+5)(x+1)2]
41. Solve loga(8x+5)=loga(4x+29)
4
5
6
8
42. For the function defined by f(x)=2-x2, 0 x, use a sketch to help find a formula for f-1(x).
f-1(x) = x2-2, x 2
2-x2 / , 0 x
f-1(x) = - 2 + x , 0 x
f-1(x) = (2-x) , x 2
43. The figure shows the entire graph of the function f(x). If the graph of f -1(x) was sketched in the same figure, which of the following would give the best description?
The graph of f -1(x) decreases from 5 to 2
The graph of f -1(x) decreases from 6 to 0
The graph of f -1(x) increases from 2 to 5
The graph of f -1(x) increases from 0 to 6
44. The amount A in an account after t years from an initial principle P invested at an annual rate r compounded continuously is given by A = Pert where r is expressed as a decimal. How many years will it take an initial investment of $1,000 to grow to $1,700 at the rate of 4.42% compounded continuously?
10 years
11 years
12 years
13 years
45. The population P of a certain culture is expected to be given by a model p=100ert where r is a constant to be determined and t is a number of days since the original population of 100 was established. Find the value of r if the population is expected to reach 200 in 3 days.
0.231
0.549
1.098
1.50
46. The figure shows the graph of g(x)=ex and a second exponential function f(x). Identify the second function.
f(x)=2+e-x
f(x)=2-ex
f(x)=-2+ex
f(x)=2+ex
47. A bacteria culture started with a count of 480 at 8:00 A.M. and after t hours is expected to grow to f(t)=480(3/2)t. Estimate the number of bacteria in the culture at noon the same day.
810
1920
2430
4800
48. The amount of a radioactive tracer remaining after t days is given by A = Ao e-0.058t, where Ao is the starting amount at the beginning of the time period. How many days will it take for one half of the original amount to decay?
10 days
11 days
12 days
13 days
49. Solve the equation 42x+1=23x+6.
-5
2
4
5
50. If a piece of real estate purchased for $75,000 in 1998 appreciates at the rate of 6% per year, then its value t years after the purchase will be f(t)=75,000(1.06t). According to this model, by how much will the value of this piece of property increase between the years 2005 and 2008?
$14,300
$21,500
$37,800
$59,300
51. The amount A in an account after t years of an initial principle P invested at an annual rate r compounded continuously is given by A = Pert where r is expressed as a decimal. What is the amount in the account if $500 is invested for 10 years at the annual rate of 5% compounded continuously?
$750.00
$800.00
$814.45
$824.36
52. Find the number:
log5 / ( / 15 / )
-5
-1
0.2
1
53. Given that loga(x)= 3.58 and loga(y)=4.79, find loga(y/x).
1.21
1.34
8.37
17.1
54. The decibel level of sound is given by:
D= / 10 / log / ( / I10-12 / )
where I is the sound intensity measured in watts per square meter. Find the decible level of a whisper at an intensity of 5.4 x 10-10 watts per square meter.
2.73 decibel
3.73 decibels
27.3 decibels
37.3 decibels
55. The amount of a radioactive tracer remaining after t days is given by A = Ao e-0.18t, where Ao is the starting amount at the beginning of the time period. How much should be acquired now to have 40 grams remaining after 3 days?
47.9 gm
48.8
61.6 gm
68.6 gm
56. Find the exact solution to the equation 3x+5=9x.
5/3
5/2
5
6
57. For the function defined by f(x) =5x - 4, find a formula for f -1(x).
f-1(x) = -5x+4
5x-4
f -1 / (x)= / x+4
5
f -1(x)= / x
5 / +4
58. Solve the equation ln(x + 5) - ln(3) = ln(x - 3).
2.5 4.5
5
7
59. The figure shows the graph of g(x) = log2 (x) and a second function f(x). Identify the function f(x).
log2 (x + 2)
2 log2 (x)
2 + log2 (x)
log2 (2x)
60. The amount A in an account after t years from an initial principle P invested at an annual rate r compounded continuously is given by A = Pert where r is expressed as a decimal. Solve this formula for t in terms of A, P, and r.
r / )
t = ln / ( / A
rP / )
t =r ln / ( / A
P / )
t= / 1
r / ln / ( / A
P / )
61. Find an exponential function of the form f(x)=bax+c with y-intercept 2, horizontal asymptote y=-2, that passes through the point P(1,4).
f(x)=-2(2x)
f(x)=2(2x) -2
f(x)=2(1.5x)-2
f(x)=4(1.5x)-2