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Honors Pre-Calculus
Midterm Exam Review
Name:
January 2013
Chapter 1: Functions and Their Graphs
1.
Evaluate the function at each specified value of the independent variable and simplify.
f ( x)  x 2  1
f (3)
a.
b.
f ( x  1)
c.
f (b 3 )
1a.)_____________
1b.)_____________
1c.)_____________
2.
Evaluate the function at each specified value of the independent variable and simplify.
2 x  1
f ( x)   2
x  2
a.
x  1
x  1
f (2)
b.
f (1)
c.
f ( 0)
d.
f (2) 2a.)_____________
2b.)_____________
2c.)_____________
2d.)_____________
3.
Find the domain of the given function. Use interval notation!
a.
f ( x) 
x 1
x2
b.
g ( x) 
x2
x2 1
c.
f ( x)  x 2  16
3a.)_____________
3b.)_____________
3c.)_____________
4.
Use a graphing utility to graph the function and determine the open intervals on which the function is increasing,
decreasing, or constant (use interval notation!). Identify any relative minimum or relative maximum values of the
function.
a.
f ( x)  x 3  3 x
b.
f ( x)  x 2  9
4a.)
increasing:____________
decreasing:____________
constant:______________
rel. min.:______________
rel. max.:______________
4b.)
increasing:____________
decreasing:____________
constant:______________
rel. min.:______________
rel. max.:______________
5.
Determine algebraically if the function is even, odd, or neither. Verify using a graphing utility.
a.
f ( x)  x 2  6
5a.)________________
b.
f ( x)  x 2  6 x  4
5b.)________________
c.
f ( x)  x 3  3 x
5c.)________________
6.
f ( x)  x 2 , g ( x)  x and h( x)  x . Describe the sequence of transformations from
f ( x), g ( x), or h(x) to j (x) .
1
j ( x)   x  5  7
j ( x)  2 x  3
a.
c.
j ( x)  ( x  2) 2  6 b.
2
Given
a.)________________________________________________________________________________________
b.)________________________________________________________________________________________
c.)________________________________________________________________________________________
7.
Given f ( x )  3  2 x , g ( x ) 
a.
( f  g )( 4)
b.
x and h( x)  3x 2  2 . Find the indicated values.
( f  h)(5)
c.
( fh)(1)
7a.)________________
7b.)________________
7c.)________________
d.
g
 1
h
e.
( g  f )( 2)
f.
( f  h)( 4)
7d.)________________
7e.)________________
7f.)________________
8.
Find the inverse of the function algebraically.
a.
f ( x) 
7x  3
8
b.
f ( x)  4 x 3  3
c.
f ( x) 
x  10
8a.)________________
8b.)________________
8c.)________________
9.
Show that f (x ) and g (x ) are inverses.
f ( x)  x  1 , g ( x)  x 2  1, x  0
10.
Given the graph y  f (x) at the right, sketch the following:
a. y 
1
f ( x)
2
b. y   f (x )
c. y  f ( x  2)
d.
y  f (x)
Chapter 2: Polynomials and Rational Functions
11.
Express each complex number in standard form
a.
 12   48
b.
3  2i 2
a  bi .
c.
1
4  7i
11a.)_______________
11b.)_______________
11c.)_______________
d.
3  2i
e. i
3  2i
23
f.
43  4i   5i1  i 
11d.)_______________
11e.)_______________
11f.)_______________
12.
Solve the following equations for x.
a. 12 x  12  25 x
b. x  4 x  7
2
2
c. x  6 x  18  0
2
12a.)_______________
12b.)_______________
12c.)_______________
13.
If
a.
P( x)  4 x 3  5 x 2  1, use synthetic substitution to find:
1

2
b. P
P ( 2)
c. P (i )
13a.)_______________
13b.)_______________
13c.)_______________
14.
If -2 is a zero of the polynomial
15.
A polynomial P (x ) is divided by
P( x)  2 x 3  x  k , find the value of k.
x  2 .
14.)________________
The quotient is 2 x  x  2 and the remainder is -1. Find P (x ) .
2
15.)______________________
16.
Two roots of the equation x  x  5 x  x  6  0 are x
4
3
2
 2 and x  3 . Find the remaining roots.
16.)________________
17.
Find all real zeros of
f ( x)  2 x 4  7 x 3  4 x 2  27 x  18 . Use the rational root theorem and synthetic division.
17.)________________
18.
Sketch the graph of the equation (without a graphing calculator). Use the zeros of the function and the end
behavior of the function.
f ( x)   x 2 ( x  1)( x  2)
19.
A rectangular enclosure, sub-divided into three congruent pens as shown, is to be made using a barn as one side
and 120 meters of fencing for the rest of the enclosure. Find the value of x that gives the maximum area for the
enclosure.
19.)________________
BARN
20.
Solve for x. 2 x  2 x  4 x  4  0
21.
Find all real and imaginary roots of 2 x  3 x  7 x  7 x  5  0
21.)________________
22.
Find a cubic equation with integral coefficients having the solutions 1  i, and 3 .
22.)________________
23.
 x  2 if x  -1 
 2

if - 1  x  1
Graph f ( x)  x
1

if x  1


3
2
20.)________________
4
3
2
24.
Sketch the graph of the rational function by hand. Identify any x-intercepts, the y-intercept, horizontal
asymptotes, vertical asymptotes, and slant asymptotes.
a.
f ( x) 
x3
x2
x-intercept(s) = _______
y-intercept = _________
H.A. = ______________
V.A. = ______________
S.A. = ______________
b.
f ( x) 
2x 2
x2  4
x-intercept(s) = _______
y-intercept = _________
H.A. = ______________
V.A. = ______________
S.A. = ______________
c.
f ( x) 
x2  x 1
x3
x-intercept(s) = _______
y-intercept = _________
H.A. = ______________
V.A. = ______________
S.A. = ______________
25.
Determine whether x  xy 
Circle the correct answer (s)
2
a. the x-axis
26.
y 2  6 has symmetry in:
b. the y-axis
c. the line y=x
d. the origin
Sketch the graph of the quadratic function. Identify the vertex & intercept(s).
f ( x)  ( x  4) 2  4
vertex = _________________
x-intercept(s) = ___________
y-intercept = _____________
Chapter 3 and Exponents Packet (Ch 5 from other textbook): Exponential Functions and Logarithmic Functions
27.
Evaluate the following exponential expressions:
a.
3 5  310
32
b.
4
2
 40

1
c.
95
32
d.
2 4  4 3
2 3
27a.)_______________
27b.)_______________
27c.)_______________
27d.)_______________
28.
Solve each exponential equation:
a. 3
8 x
 27 2 x
b.
4 32  2 3 x
28a.)_______________
28b.)_______________
29.
A gallon of gasoline cost $3.09 two years ago. Now it costs $1.78. To the nearest percent,
what has been the annual rate of decrease in cost?
29.)________________
x
30.
1
Graph y  3 and y    on the same set of axes.
3
x
How are the graphs related?
31.
Suppose $1,200 is invested at an interest rate of 9.6%. how much is the investment worth
after 2 years if interest is compounded:
a. Monthly?
b. Continuously?
31a.)_______________
31b.)_______________
32.
Given log 40  1.6021 , using rules of logarithms, find the value of: {without a calculator!}
a. log 4
b. log .25
32a.)_______________
32b.)_______________
33.
Find the exact value of: without a calculator!
33a.)_______________
a.
log 2 16
b.
c. 5
log 27 3
log4 64
33b.)_______________
33c.)_______________
34.
Express the following in terms of
a.
log b
3
log b M and log b N
M3
N2
3
b. log b M N
2
34a.)__________________________________
34b.)__________________________________
log 2 x  2  log 2 x  5  1.
35.
Solve
36.
Solve 3  31 to the nearest hundredth.
37.
If 250 mg of ibuprofen has a half-life of 3 hours, then how much ibuprofen is in a
person’s bloodstream after (a) 7 hours? (b) after t hours?
x
35.)________________
36.)________________
37a.)_______________
37b.)_______________
38.
Simplify:
a.
35  33
3 2
38a.)_______________
b.
3
81
8 7
38b.)_______________
c.
6x
4
3
 2x
2x
1
3
5
3
38c.)_______________
39.
40.
Solve for x:
a.
log 2 x  4
b.
log 3 x 2  5  log 3 4 x 2  2 x
c.
log 5 25 5  x
39c.)_______________
d.
log 6 4  log 6 3  log 6 2  x
39d.)_______________


39a.)_______________


39b.)_______________
Solve for x:
a.
e 5 x  2 x 3
40a.)_______________
b.
5 x 1  2 3 x  6
40b.)_______________
c.
e 2 x  3e x  2  0
40c.)_______________
d.
9 x 3
 81
27 x
40d.)_______________
Chapter 4 [4.1-4.6]: Trigonometry
41.
42.
43.
A circle has a radius of 7 inches. Find the length of the arc intercepted by
a central angle of 240 .
41.)________________
The circular blade on a saw rotates at 2400 revolutions per minute.
a.
Find the angular speed in radians per second.
42a.)_______________
b.
The blade has a radius of 4 inches. Find the linear speed of a blade tip
in inches per second.
42b.)_______________
A satellite in circular orbit 1125 km above a planet makes one complete revolution
every 120 minutes. Assuming that he planet is a sphere of radius 6400 km, find the
linear speed of the satellite in kilometers per minute. Round your answer to the nearest
whole number.
43.)________________
44.
45.
46.
A truck is moving at a rate of 90 km per hour and the diameter of its wheels is 1.25 meters.
Find the angular speed of the wheels in radians per minute.
44.)________________
Evaluate (if possible) the six trigonometric functions if
 
2
.
3
45.
 2 
sin  

 3 
 2 
cos 

 3 
 2 
csc 

 3 
 2 
sec 

 3 
 2 
tan  

 3 
 2 
cot  

 3 
Evaluate the trigonometric function.
a.
 3 
sin  

 4 
b.
 7 
csc

 6 
c.
 5 
tan  
 3 
46a.)_______________
46b.)_______________
46c.)_______________
d.
sec 4 
e.
 5 
cot  
 2 
f.
 13 
cos

 4 
46d.)_______________
46e.)_______________
46f.)_______________
47.
Find the exact values of the six trigonometric functions of the angle

shown in the figure.
18

12
48.
47.
sin   
csc  
cos  
sec  
tan   
cot   
Simplify each expression.
a.
sin x cos x
1  cos 2 x
b.
sin x  cos x 2  sin x  cos x 2
48a.)_______________
48b.)_______________
c.
tan 2 x
1
sec x  1
d.
sec x  sin x tan x
48c.)_______________
48d.)_______________
49.
Find two solutions of the equation. Give your answers in degrees 0    360 and radians 0    2 
without using a calculator.
a.
cot   
3
3
b.
sec   2
49a.)_____________________
49b.)_____________________
50.
51.
52.
53.
John stands 150 meters from a water tower and sights the top at an
angle of elevation of 36 . How tall is the tower?
50.)________________
The Aerial run in Snowbird, Utah has an angle of elevation of 20.2 .
Its vertical drop is 2900 feet. Estimate the length of the run.
51.)________________
In a sightseeing boat near the base of the Horseshoe Falls at Niagara Falls, a
passenger estimates the angle of elevation to the top of the falls to be 30 .
If the Horseshoe Falls are 173 feet high, what is the distance from the boat
to the base of the falls?
52.)________________
Use the given value and the trigonometric identities to find the remaining trigonometric functions of the angle.
3
cos    , sin   0
7
54.
53.
sin   
csc  
cos  
sec  
tan   
cot   
The point is on the terminal side of an angle in standard position. Determine the exact values of the six
trigonometric functions of the angle.
8,  15
54.
sin   
csc  
cos  
sec  
tan   
cot   
Answer the questions regarding the amplitude, period, phase shift and vertical shift of each trigonometric
function y  a cosb  c   d or y  a sin b  c  d . Then graph two full periods of the trigonometric function.
Be sure to label your axes accurately!
55.
56.
57.
y  cos 4  2
 
y  3 sin    1
2


y  3 cos    4
2

a=
b=
c=
d=
amplitude =
period =
phase shift =
vertical shift =
a=
b=
c=
d=
amplitude =
period =
phase shift =
vertical shift =
a=
b=
c=
d=
amplitude =
period =
phase shift =
vertical shift =
58.
 x  
y  4 cos    3 a =
 2 2
b=
c=
d=
period =
phase shift =
vertical shift =
a=
b=
c=
d=
amplitude =
period =
phase shift =
vertical shift =
a=
b=
c=
d=
amplitude =
period =
phase shift =
vertical shift =
amplitude =
Graph two periods of the trigonometric function.
59.
60.
y  3 tan 2
y
1
cot   1
2
Answer Key
1a.
2b.
Midterm Review Packet January 2013
x 2  2x  2
1b.
2c.
3b.
 , 
3c.
 ,  4  4, 
4a.
decreasing:
4a.
constant: none
4a.
4a.
rel. max.:
4b.
increasing:
4b.
4b.
5a.
6a.
constant: none
4b.
4b.
rel. max.: none
even
5b.
neither
5c.
odd
f (x) has a horizontal shift 2 units left, reflects over the x-axis and has a vertical shift 6 units down
6b.
6c.
7a.
3, 
rel. min.:  3, 0; 3, 0
4a.
2a.
3a.
-3
 ,  2   2, 
increasing:  ,  1  1, 
rel. min.: 1,  2
decreasing:  ,  3
6
f (x) has a vertical stretch of 2 and a vertical shift up 3 units
f (x) has a horiz. shift 5 units right, reflects over the x-axis, a vertical shrink of ½ and a vertical shift 7 units down
-7
7e.
8b.
 1, 1
 1, 2
2
1c.
2d.
b6  1
10
-1
7
f 1 ( x)  3
x 3
4
7b.
70
7c.
5
7f.
-97
8a.
f
8c.
f
1
( x)  x 2  10, x  0
7d.
1
( x) 
8x  3
7
1
5
9.
show that f g ( x)  g  f ( x)  x
10.
See graphs
11a.
 2i 3
11b.
5  12i
11c.
11d.

1 4 3

i
7
7
11e.
i
11f.
17  11i
12a.
12b.
 2  11
12c.
3  3i
13a.
13
13b.
13c.
6  4i
14.
18
15.
P( x )  2 x 3  3 x 2  4 x  3
16.
i
17.
 1, 2,  3, 
3
2
18.
See graph
20.
 1,  2
21.
22.
23.
See graph
1
1,  ,  1  2i
2
x-intercept: 3,

19.
4
7

i
65 65
3 4
,
4 3
1
4
15 meters
f ( x)  x 3  5 x 2  8 x  6
0 ; y-intercept:
24b.
24c.
 3  ; H.A.: y = 1; V.A.: x = 2; S.A.: none
 0, 
 2
x-intercept: 0, 0 ; y-intercept: 0, 0 ; H.A.: y = 2; V.A.: x = 2 and x = -2; S.A.: none
x-intercept: none; y-intercept: 
1  ; H.A.: none; V.A.: x = 3; S.A.: x + 2
25.
c and d
24a.
 0,  
3

26.
vertex: 4,  4 ; x-intercept: 2, 0; 6, 0 ; y-intercept:
0, 12
27c.
729
27d.
5
8
29.
24.1%
30.
See graph
$1454.00
32a.
0.6021
32b.
-0.6021
33b.
1
3
33c.
125
34a.
log b M 
35.
 3  57
2
36.
3.13
37a.
 49.61
 1 3
y  250 
2
38a.
81
38b.
64
38c.
3  x3
x
39a.
1
16
39b.
40a.
-0.483
40b.
41.
28 inches
3
42a.
44.
2400 rad/min
45.
27a.
27
27b.
28a.
-1
28b.
31a.
$1452.89
31b.
33a.
4
34b.
3 log b M  2 log b N
37b.
16
17
3
2
t
5
, 1
3
40c.
5
39d.
2
0, 0.693 (ln 2) 40d.
42b.
 1005 in/sec
39c.
 5.412
80 rad/sec
3
 2 
sin  
 
2
 3 
1
 2 
cos 
 
2
 3 
2 3
 2 
csc 
 
3
 3 
 2 
sec 
 2
 3 
 2 
tan  

 3 
3
 2 
cot  

3
 3 
3
2
log b N
3
1
-2
43.
394 km/min
46a.

2
2
46b.
46f.
-2

2
2
48c.
sec x
49b.
45, 315,
53.
55.
 7
4
,
47.
3 13
sin   
csc  
13
13
3
cos  
2 13
sec  
13
13
2
tan   
3
2
50.
46d.
cot   
48a.
1
cot x
2
3
cos x
108.98 meters
49a.
120, 300,
51.
8398.54 feet
46e.
0
48b.
2
2 5
,
3
3
52.
299.64 feet
4
2 10
7 10
csc   
7
20
cos   
3
7
sec   
2 10
3
cot   
a=1
 3
48d.
sin    
tan   
46c.
b=4
54.
7
3
sin    
15
17
csc   
17
15
cos  
sec  
8
17
tan    
3 10
20
c=0
d = -2
55. – 60. See graphs
17
8
15
8
cot    
8
15
Amp = 1
Per = 
Phase Shift = 0
Vert. Shift = -2
Vert. Shift = 1
2
1
2
56.
a = -3
b=
57.
a=3
b =1
58.
a=4
b= 
d=1
Amp = 3
Per = 4
Phase Shift = 0
c=  d=4
Amp = 3
Per = 2
Phase Shift =  
Vert. Shift = 4
c = -1
d = -3
Amp = 4
Per = 4
Phase Shift = -1
Vert. Shift = -3
c=0
d=0
Amp = none
Per = 
Phase Shift = none
Vert. Shift = none
Phase Shift = none
Vert. Shift = 1
c=0
2
2
2
59.
a = -3
b=2
2
60.
a= 1
2
b=1
c=0
d=1
Amp = none
Per =
