Name:________________________________________Period:_______
PreCal PreAP Fall Final Exam Review
Write the equation for the function that has the graph with the given characteristics.
1. Cubic translated 4 units right, reflected over the x-axis, and moved 2 units up.
2. Rational translated 5 units left, reflected over the x-axis, and translated 7 units up.
3. Quadratic reflected over the y-axis, stretched vertically by a factor of 3and shifted 1 unit down.
4. Square Root with a horizontal reflection, and shifted down 3 units.
5. Rational Function stretched horizontally by a factor of 2, translated 4 units left and 2 units up.
6. Absolute value translated right 1 units, reflected over the x-axis and stretched vertically by a factor of 5.
Determine if the following are Odd/Even/Neither. Also, determine if the function is symmetric about the origin/yaxis/neither.
7.
f ( x)  3x 2  4
11. sin( x )
8.
1
f ( x)  x 3  x 2  3
3
12. csc(x)
13. sec(x)
9.
14. tan(x)
f ( x0  x 3  3 x
10. cos( x)
15. cot(x)
Graph the following functions.
16.
1
f ( x)  ( x  3) 2
2
17. f ( x)   x  3  2
20. Find the domain and range of
f ( x) 
18.
19 .
3 x 3
x 3
21. determine the natural domain of the function f ( x) 
22.find all intervals on which
f ( x)  2( x  2)  1
x4
x  x  20
2
f ( x)  x  2  x  2 is increasing (hint: draw the graph)
23. The graphs of f and g are shown in the figure below. What is (gof)(2)?
24. which, if any, of the following statements are true?
f ( x) 
2
1
x5
Name:________________________________________Period:_______
A. if f and g are even, then f-g is even.
B. if f and g are odd, then f*g if even.
C. if f is not even, then it is odd.
Simplify the following for the functions below. List any and all restrictions.
Let: f ( x)  x 2  4
g ( x)  x 2  4
g
 ( x)
 j
h( x )  3 x  2
25. (h  j)(x)
26. 
27. (g  f )(x)
31. (g +f) (3)
32(h/j)(-4)
33. h- g(-1)
For 34- 39 Let: f ( x)  2 x 2  1
34. f(g(x))
35. g(f(x))
Algebra of Functions:
29. (f  g  h)(x)
28. (fg)(x)
h( x ) 
g ( x0  x  5
36. h(h(x))
f(x) = 2x + 5
37. f g( (4))
g ( x) 
40.Find f/g(x) and its domain
x4
x5
j ( x)  x  4
1
x4
2
38. h f (2)
h(x) = x 2  4
30. f ( -2)
j(x) =
39. g h( ( 4))
2x  3
41. Find h(j(x)) and its domain
Graph the following Piecewise functions:
42.
 x  3 if x<-2
f ( x)  
2 x  1 if x  -2
43.
 x 2 if x  0
f ( x)  
 x  1 if x>0
 x  1  3 if x  2


 x  1 if x>2
44. f ( x)  
2 x  3 if x<-1

45. f ( x)  3  x
if -1  x  1

if x>1
 x
46. Write a polynomial with lowest degree, if 3  2i and –4 are two of the roots of the polynomial.
47.Given that f  x   20  12 x  3x 2  2 x3 has –2 as one root. Use synthetic division to find other roots.
48.Given that x  2 is a factor of the polynomial p  x   2 x3  3x 2  kx  30 , find the value of k.
49.Find the slant asymptote of the rational function r  x  
50.Solve the inequality 2 x3  9 x 2  3x  14  0 .
51.Find the interval(s) where the rational function r  x  
x3  3x  3
.
x2  x  1
 2 x  1 x  3
2
 x  1
is non-negative.
Name:________________________________________Period:_______
For each function find the following, if possible: domain, range, vertical asymptotes, hole, horizontal
asymptotes, y-int., zeros, and slant asymptote. State the end behavior, and if possible the behavior
around a hole or an asymptote. Then GRAPH.
52. f ( x) 
x2  6 x  7
x2 1
53. f ( x) 
x2  4 x  3
x2
54. f ( x) 
x2
x  2x  3
2
55. Define a power function and give three examples.
56. Give three examples of functions that are not power functions.
57. if  is in standard position and has radian measure 15.878, in what quadrant does  lie?
58. convert the angel measure -306.77  to DM ' S " form.
59. convert the angle measure 0.24 to DM ' S " form.
60. find two angle, one positive and one negative, that are coterminal with the given angle.
a. -35 
b.
5
6
61. a. convert 165  to radians. Given answer in terms of 
b. convert 208  to radians. Given answer to the nearest hundredth of a radian.
62. a. convert 
7
to degrees.
6
b. convert 1.8 radians to nearest tenth of a degree.
63. if  is a third quadrant angle and cos(  )=-2/3, find the other five trig ratios.
64. Express each of the following in terms of a reference angle.
a. cos(236  ) b. sin (485  )
c. sin (-62  )
65. EXACT TRIG VALUES!!!!!!
a) sin 45 
b) sin 210 
c) sin (-60  )
g) cos (-240  ) h) sin (-135  ) i) sec 120 
m) cos

3
s) sin (5pi/6)
n) sin (3 pi/4)
o) cos(-7pi/4)
t) cot(7pi/4)
u) sec (pi/2)
d) cos 225 
e) sin 150 
f) cos 90 
j) tan 120 
k) cot 225 
l) sin ( 
p) sin (11 pi/6) q) cos (3pi/4)

)
4
r) tan (pi)
66.
67. if sec x  5 and   x  2 , find exact value for the other five trigonometric functions of x.
68. Determine the sinusoidal function of sine with amplitutde 1/3, period  , and translation 2 units up.
69. a 4/9 clockwise rotation would terminate in which quadrant and yield what angel measurement?
Name:________________________________________Period:_______

8
70. express the function sin( ) in terms of its cofunction.
71.
72. find sinx, cosx, and tanx for the point (x,y) of the angle in standard positions.
a) (2,5)
b) (4,4)
c) (3, -10)
D) (-1, -4)
73. evaluate the following with a calculator. Round to three decimal places. Be sure to be in the correct mode.
a) sin 40 
b) tan 129 
c) cos 236 
d) sec 23 
g) cos pi/12
h) tan 2.3
i) sec 2 pi/7
j) cot 7pi/8
e) csc 145 
f) sin 3pi/5
74. a sector of a circle with radius 5 has an arc length of 4. Find the measure of the central angle in radians.
75. a sector of a circle with radius 12 has an arc length of 17. Find the measure of the central angle in radians.
Name:________________________________________Period:_______
76. determine the arc length of a circle of radius 9cm intercepted by a central angle of 2pi/3.
77. determine the areas of a sector with the following information:
a) r =5,
  120
b) r=8.4

2
3
Applications of trig functions: write the equation that expresses the following situations.
78. as you ride a Ferris wheel, your distance from the ground varies sinusoidally with time. You are in the last seat
filled and the Ferris wheel starts immediately. Let t be the number of seconds that have elapsed since the Ferris
wheel started. You find that it takes 5 seconds to reach the top, 57 feet above the ground, and that it makes a
revolution every 10 seconds. The diameter of the wheel is 54 feet.
79. Han notices that Luke is not feeling well; he decides to take his temperature. His body temperature varies
sinusoidally with time. 10 minutes after he started timing, it reached a high of 102  F. 3 minutes later, it reaches
its next low, 99  F.
80. virologists find that the number of people sick with a certain disease varies periodically. Assume that
the number of sick people varies sinusoidally with time. Records stat being kept when t=0 years. A
minimum number, 50 people have the disease when t = 3.5 years. The next maximum, 250 people,
occurred at t =7.5 years.
81. a 2.3 inch diameter pulley on an electric motor that runs at 1293rev/min is connected by a belt to a
5 inch diameter pulley on a saw arbor. Find the angular speed of the motor pulley, the angular speed of
the saw pulley in radians per minute, and the speed of the saw in revolution/minute.
82. a gear makes 6.2 rotations about its axis. What is the angular displacement in radians of a point on
the gear?
83. what is the angular velocity in radians/min of a notch on a wheel that makes 24 rotations per second
about its axis?
84. the minute hand of a watch is 1.3cm long. What is the linear velocity of the tip of the hand?
Ch.3: Exponential and Logarithmic Functions
85.Consider the two functions y  e x and y  ln x . Find the domain, range, x and y-intercepts and
asymptotes of these two functions.
y  e x : domain all real numbers, range all positive real numbers, y-int at  0,1 , horiz asy: x-axis
y  ln x : domain positive real numbers, range all real numbers, x-int at 1,0  , vert. asy: y-axis
Name:________________________________________Period:_______
86.Write the logarithm expression log3
3x 
1
 x in the corresponding exponential form.
27
1
, and x  3
27
87.Write the exponential expression a x  y in the corresponding logarithm form.
log a y  x
88.Find the inverse function of y  10x .
y  log x
 5x2 y3 
89.Expand the logarithm expression log 2 
.
 z 
1
log 2 5  2log 2 x  3log 2 y  log 2 z
2
1
90.Condense the logarithm expression 3ln x  ln  x  1  2ln  x  1 .
2
3
x x 1
ln
2
 x  1
91.Use Base Change Formula to express log 2 5 using: (a) common log; (b) natural log; (c) log base 3.
log5 ln 5 log3 5
,
,
log 2 ln 2 log 3 2
92.Solve the equation 22 x  2x  6  0 .
ln 3
x  log 2 3 
ln 2
93.Solve the equation log5  x  1  log5  x  3  1 .
x4
94.[Calc] A bank offers an annual interest rate of 5% compounded quarterly, then how much money is
needed to deposit in order to receive $25000 in 8 years? What if the interest rate was 5% compounded
continuously?
16799.60,16758.00
95.[Calc] Half life of a radioactive element is 5 years. If 55 grams of this element is found today, then
how much will it remain after 12 years? How many years ago was the mass 100 grams?
10.421 grams; 4.312 years
96.[Calc] The population of a certain bacteria grows exponentially. If population takes 10 days to grow
from 200 to 900, then how many days will it take to grow from 900 to 3000?
8.005 days
Name:________________________________________Period:_______
97.
Be sure to study optimization and the graph of all trig functions and inverse trig functions with their
domains and range.