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Precalculus
Midterm Exam – Study Guide
Name: ___________________________________________________________________
Consider the functions
f x   2 x  1 and g x    x 2
Find:
1.
f
Consider the functions
f x   x 2  4 and g x   2 x
Find:
 g 0
4.
0
1
2.
g  g  3
f
5.   x 
g
x2  4
,x  0
2x
 81
3.
 f  g 2
 f  g x
 2x 1
2
6.
 f  g x
4x 2  4
Precalculus
Midterm Exam – Study Guide
Name: ___________________________________________________________________
Determine whether the graph of the
equation has symmetry with respect to the
x-axis, y-axis, and/or origin.
7. xy  x 2  1
Symmetry with respect to the origin
8. y 2  9  x
Symmetry with respect to the x-axis
9. x 2  3 y  16
Symmetry with respect to the y-axis
Classify the function as even, odd, or
neither.
10. f x   x3  1
Neither
11. g x   x5  x
Odd
12. hx   x 2  4
Even
Precalculus
Midterm Exam – Study Guide
Name: ___________________________________________________________________
Find an equation of the graph, the parent function, and the transformations of the graph, in correct order.
13.
15.
y
y


x



x







Equation: f x   3x  2  4
Equation: f x   4 x  5  6
Parent Function: f x   x 2
Parent Function: f x   x
Transformations: horizontal translation 2 units
right; vertical stetch factor 3; vertical translation 4
units down
14.
Transformations: horizontal translation 5 units
left; vertical stetch factor 4; reflection with respect
to the x-axis; vertical translation 6 units up
16.
2
y
y


x



x






Equation: f x   
5
3
x2
1
Parent Function: f x  
x
Transformations: horizontal translation 2 units
left; vertical stetch factor 5; reflection with respect
to the x-axis; vertical translation 3 units down

Equation: f x   2x  6  5
3
Parent Function: f x   x 3
Transformations: horizontal translation 6 units
right; vertical stetch factor 2; vertical translation 5
units up
Precalculus
Midterm Exam – Study Guide
Name: ___________________________________________________________________
Sketch the graph of each function.
17. ax   5x  2  5
19. cx   2 x  3  6
2
y
y


x


x




20. d x   
18. bx   3 x  2
4
3
x 1
y
y


x
x






Precalculus
Midterm Exam – Study Guide
Name: ___________________________________________________________________
Make tables of values for both original and the inverse, and graph both relations.
21. f x   x  1  2
3
f x 
f 1 x 
x
-3
-2
-1
0
1
x
-6
1
2
3
10
y
-6
1
2
3
10
y
-3
-2
-1
0
1
y

x



22. f x   2 x  8  2
f x 
f 1 x 
x
-8
-7
-4
1
8
x
-2
0
2
4
6
y
-2
0
2
4
6
y
-8
-7
-4
1
8
y

x



Precalculus
Midterm Exam – Study Guide
Name: ___________________________________________________________________
Find the inverse of the function algebraically.
23. f ( x) 
3
x2
3
f 1 ( x)  27 x 3  2
24. g x  
2
x 5
3
f 1 ( x) 
3
15
x
2
2
25. g x   2 x3  8
f 1 ( x)  3
x 8
2
Precalculus
Midterm Exam – Study Guide
Name: ___________________________________________________________________
26. Consider the following function:
f x   3 x  1  4
a. Find the coordinates of the vertex.
1,4
b. Sketch the graph.
c. State the domain and range.
Domain: 1,  
y

x

Range:  4,  


d. State the domain and range of the
inverse function.
Domain:  4,  
Range: 1,  
e. Find the inverse function algebraically.
 x4
f x   
 1
 3 
2
1
f. Write down your final answer.
 x4
f x   
 1
 3 
2
1
x  4
Precalculus
Midterm Exam – Study Guide
Name: ___________________________________________________________________
27. Consider the following function:
f x   4 x  2  1
a. Find the coordinates of the vertex.
 2,1
b. Sketch the graph.
y

x



c. Use your graph to restrict the domain
of the function and make it a one-toone function. Write down a simpler
equation for the new function. State the
new domain and range.
f x   4 x  9
Domain:  2,  
Range:  ,1
d. State the domain and range of the
inverse function.
Domain:  ,1
Range:  2,  
e. Find the inverse function algebraically.
f 1 x   
x9
4
f. Write down your final answer.
f 1 x   
x  1
x9
4
Precalculus
Midterm Exam – Study Guide
Name: ___________________________________________________________________
28. Consider the following function:
f x   2x  5  8
2
a. Find the coordinates of the vertex.
 5,8
c. Use your graph to restrict the domain
of the function and make it a one-toone function. State the new domain
and range.
b. Sketch the graph.
y

x


Domain:  5, 
Range:  ,8

d. State the domain and range of the
inverse function.
Domain:  ,8
Range:  5, 
e. Find the inverse function algebraically.
f 1 x    
x 8
5
2
f. Use a random value from the domain
of the inverse to test whether the value
of the inverse function belongs to the
range.
 
68
 5  4
2
 
68
 5  6
2
g. Write down your final answer.
f 1 x   
x8
x 8
5
2
Precalculus
Midterm Exam – Study Guide
Name: ___________________________________________________________________
Convert to radians.
29. 150

5
6
Sketch the angle in the standard position,
determine the quadrant in which the
terminal side lies, and find the reference
angle.
33.  295
y
I
x
65
30. 420
7
3
34.
5
4
y
x
45
Convert to degrees.
7
31.
4
315
III
35. 

6
y
32. 
5
6
150
x
30
IV
Precalculus
Midterm Exam – Study Guide
Name: ___________________________________________________________________
Find the six trigonometric ratios for  .
36.
37

35
sin   
35
37
sec  
37
12
cos   
12
37
csc  
37
3
tan  
35
12
cot   
12
35
37.
6.1
4.6

sin   0.602
sec   1.252
cos   0.798
csc   1.661
tan   0.754
cot    1.326
Precalculus
Midterm Exam – Study Guide
Name: ___________________________________________________________________
Find the values of x and y.
38.
40.
x
8 
4
y
22
60
x
x  11 3
x 8
y  11
y 8 2
y
41.
39.
45
5
y
33

6
x
x
x
y
5 2
2
5 2
2
x  22 3
y  11 3
y
Precalculus
Midterm Exam – Study Guide
Name: ___________________________________________________________________
Find the values of the trigonometric ratios.
 5 
42. cos 

 6 

3
2
43. sin 135
2
2
44. tan90
Undefined
 
45. csc  
 3

2 3
3
Precalculus
Midterm Exam – Study Guide
Name: ___________________________________________________________________
Find the exact value of the trigonometric expression, given a point on its terminal side.
46. 2,3 ; sec 
13
2
47. 20,5 ; cot  
4
Precalculus
Midterm Exam – Study Guide
Name: ___________________________________________________________________
Find the value of each variable. Round your answers to three decimal places.
8.342
48.
h
50.
c
j
4.7
5.5
b
k
a
70
a  8.877
h  2.857
b  3.036
j  58.710
c  20
k  31.290
65
49.
16.622
51.
d
n
10.03
l
g
m
f
8.21
d  7.025
l  12.962
f  15.065
m  50.698
g  25
n  39.302
Precalculus
Midterm Exam – Study Guide
Name: ___________________________________________________________________
52. The height of an inaccessible cliff needs to be calculated. From a boat, which is 150m out from the
bottom of the cliff, the angle of elevation to the top of the cliff is 54˚. Find the height of the cliff.
The height of the cliff is 206.457 meters.
53. A truck travels 8km up a mountain road. The change in height from the bottom to the top is 1400m.
Find the angle of inclination of the road.
The angle of inclination of the road is 10.079 degrees.
54. A 6.1 meter ladder leans against a wall. The angle formed by the ladder and the ground is 71.
a. How far is the foot of the ladder from the wall?
b. How high up the wall does the ladder reach?
a. The foot of the ladder is 1.986 meters away from the wall.
b. The ladder reaches 5.768 meters up the wall?