Download 1. Find the period and the equations of the asymptotes of the

Name: __________________
Class:
Date: _____________
1. Find the period and the equations of the asymptotes of the function and
sketch the graph
8. Use the definition of the logarithmic function to find x:
log 9 = 2
y = tan
x
1
x.
4
9. Rationalize the numerator:
n + 9 n
10. Simplify the expression:
______________
2. Use a double or half angle formula to solve the following equation in
the interval [0, 2 ) .
n
1
1
+ m
( n + m)
2
sin 2x + sin x = 0
3. Find the value of cos quadrant I.
from the information: csc 11. Sketch a graph of the rational function r( x ) =
= 5, in
x 2
2
x 3x
Find all intercepts and asymptotes.
4. Find the period of the function
y = cot
2
x +
9
3
Answers: intercepts {x = 2, y none}, asymptotes { x = 0, 3; y = 0}
.
5. The point P is on the unit circle. The x coordinate of P is
33
and P is
7
in quadrant IV. Find the point P(x,y).
6. The graph of y = f (x) is given. What should we do to obtain the graphs of
the functions listed below?
12. Evaluate the following piecewise defined function at the values f ( 5 ), f
( 7 ), and f ( 10 ).
f (x)=
{
2
if x 7
if x > 7
x + 5x
6x
13. Evaluate the expressions.
log ( 8 )
8
( 8)
log 8
8
Match each function in the left column with the corresponding
transformation of the graph in the right column. Sketch a graph of each.
log
8
log
8
9f ( x )
shift the graph down
9 units
f ( x)
stretch the graph vertically
by the factor of 9
f ( x 9 )
reflect the graph
in the y axis
f ( x)
reflect the graph
in the x axis
f ( x) 9
shift the graph 9 units
to the right
7. Rewrite the expression as an algebraic expression in x:
tan (sin
PAGE 1
1
x)
( 8)
1
8
8
14. Find the solution of the exponential equation 11 x = 3 x
four decimal places.
+4
, correct to
15. State the range of the function
h (x) = 6 +
1
9
x
16. Find the midpoint of the segment joining points (6, 7) and (18, 1).
Name: __________________
Class:
17. Simplify the expressions in the left column and match each with the
correct answer in the right column. Write a proof of each.
sin u
27. Perform the indicated operations and simplify:
(9 + x
cos ( u )
2
sin ( u )
cos u
Date: _____________
1
1
+
x + 2
x + 4
sin ( u )
2
) (9 x
2/3
)
28. Solve the inequality. Express the solution in interval form.
sin ( u )
4/3
0
29. Find the value of the trigonometric function cos 1
information: sin = , in quadrant I.
3
18. Factor completely:
3
125x 64
19. The graph of a sine curve is given. Find an equation that represents the
curve.
30. Find sin and cos from the
if x = 2, and y = 1.
31. Simplify the following expression as much as possible:
5
2
tan
+ z
32. Simplify the following trigonometric expression as much as possible:
cot x sec x
csc x
______________
33. Rewrite the expression below as a single logarithm.
20. Find an equation for the line through ( 1,1) with a slope of 5.
21. Find the exact value of the expression:
cos
sin
1
log 18 +
3
2
34. Find all real solutions of the equation
3
x
Enter your answer as a fraction.
22. Find the inverse function of f (x) =
1
log 5 log 3
2
1
.
x + 5
36
+ 3 = 0
x + 6
35. Solve the inequality. Express the answer using interval notation.
23. Find an equation for the line through (3,5) and parallel to the line
passing through (1,2) and ( 3,18).
24. Simplify the following expression by using a double angle formula or a
half angle formula:
2 tan 22x
15 | 4x + 6 | 1
36. Find the quotient and remainder using synthetic division.
x
3
2
+ 16x + 68x + 38
x + 7
37. f (x) = x 5 and g(x) = |x + 2|. Find g (f (x)).
2
1 tan 22x
25. Find all solutions of the following equation in the interval [0, 2 ) .
38. Find the values of the six trigonometric ratios of the angle in the
triangle.
cos x (2 sin x + 1) = 0
26. Find the intercepts and asymptotes of the rational function
r(x) =
x intercept
y intercept
2x + 18
3x + 3
horizontal
asymptote
vertical
asymptote
a = 6, b = 8, c = 10
PAGE 2
Name: __________________
Class:
Date: _____________
2
43. Sketch the graph of the function f(x)= x + 4
sin ( )
cos ( )
tan ( )
cot ( )
csc ( )
sec ( )
39. Find all real solutions of the equation
2
x
= 1
x + 20
40. Use the Laws of Logarithms to rewrite the expression below in a form
with no logarithm of a product, quotient or power.
log
x
a
44. Express the function F(x) = (4 f, g, and h.)
x)
5
in the form f g h . (Find
4
yz
8
41. Find the length of the arc s in the figure if radius R = 14 and angle a =
o
145 .
45. Match each expression in the left column with the equivalent expression
in the right column. Write a proof of each.
2 sin 6x cos 6x
sin 12x
2
2
cos 12x
2
2
1
cos 6x + sin 6x
cos 6x sin 6x
46. Solve the logarithmic equation for x:
log 3( 5 x ) = 9
47. Simplify the expression:
x
42. Find the function of the form f (x) = Ca whose graph is given.
(x
10 5 4
y z
)
10
9
Eliminate any negative exponents. Assume that all variables are positive
numbers.
48. The graph of p = h ( r) is given in the blue graph (between the graphs of 1
and 4).
Match each equation with its graph.
PAGE 3
Name: __________________
Class:
2v ( k+6 )
58. Simplify the expression:
3
( 2u 9v 6) 4 ( 3u 2v 5) 3
1
v (k)
3
4
v ( k+4 )
2
v ( k 4 )
1
v ( k ) +3
5
Eliminate any negative exponents.
59. Find the inverse function of f (x) = 3 +
60. Find the degree measure of the angle:
49. Find the domain and range of the function that is graphed below.
50. Express the inequality 6 < x < 1 in interval notation.
51. Find the terminal point P (x, y) on the unit circle determined by
25
t =
. Give exact values.
3
Please enter your answer as an ordered pair in the form (x, y).
52. Simplify the following trigonometric expression as much as possible:
sec x cos x
sec x
3
2
53. Given that x = 5 is zero of P (x) = x 3 x 18 x + 40, find all other
zeros of P (x). Express P in factored form.
54. Evaluate the expression | | 12 | | 8 | |.
______________
55. Which of the points A(7, 8) or B(6, 9) is closer to the origin?
______________
56. What is the period and phase shift of the function y = cos (3x + ) ?
2
57. For the function f (x) = 4 x + 1 find the following:
f ( a + h) and
PAGE 4
Date: _____________
f ( a + h) h
f (a)
where h 0.
3
x .
rad .
18
ANSWER KEY
Name: __________________
Class:
1. 4,2+4k
2
4
2. 0 , , , 3
3
24
3.
5
9
4.
2
33 4
,
5.
7
7
shift the graph down
6. f ( x ) 9 ,
9 units
shift the graph 9 units
f ( x 9 ) ,
to the right
stretch the graph vertically
9f ( x ) ,
by the factor of 9
reflect the graph
f ( x) ,
in the x axis
reflect the graph
f ( x) in the y axis
x
7.
2
1 x
8. x=3
9
9.
( n+9 + n )
10.
( n+m )
( n m )
Date: _____________
8
3
1
1
30.
,
5
5
31. cot ( z )
32. 1
33. log ( 6 5 )
34. 2,3
35. ( , 5 2,
29.
)
2
36. x +9x+5,3
37. g ( f ( x ) ) = x 3
4
3
4
3
5
5
38. sin ( ) = , cos ( ) = , tan ( ) = , cot ( ) = , csc ( ) = , sec ( ) =
5
5
3
4
4
3
39. 5, 4
40. 4loga ( x ) loga ( y ) 8loga ( z )
41. 52.53
x
42. 5 3
3
43.
5
44. f ( x ) =x ,g ( x ) =4 x,h ( x ) = x
2
2
2
2
45. cos 6x sin 6x cos 12x ,
2 sin 6x cos 6x sin 12x ,
cos 6x + sin 6x 46. 19678
11.
12. 0,84,60
( 8)
13. log8 ( 8 ) =1 , log8 8 =8, log8 ( 8 ) =
14. 3.3822
15. ( 6, )
16. ( 12, 3)
17. sin u
2
sin ( u ) cos u
2
cos ( u ) ,
sin ( u ) ,
(
sin ( u )
)
2
18. ( 5x 4 ) 25x +20x+16
19. f ( x ) =2sin
( x 3)
4
20. y=5x+6
1
21.
2
1
22. 5+
x
23. y= 4x+17
24. tan ( 44x )
3 7 11
25. ,
,
,
2 2 6
6
26. ( 9,0 ) , ( 0,6 ) , y= 0.666667, x=1
2
4
3
2
3
27. 81 x +9x 9x
28. ( , 4 ) 3, 2 )
PAGE 1
1
, log8
2
1
8
8
= 8
47.
1
100
9
x
50
9
40
9
y z
48. v ( k 4 ) 3,
v ( k ) +3 1 ,
1
v ( k ) 4,
3
v ( k+4 ) 5,
2v ( k+6 ) 2
49. 7,7 , 5,5
50. ( 6, 1 )
51. ( 0.5,0.87)
2
52. ( sin ( x ) )
53. x=2,x= 4, ( x 5) ( x 2 ) ( x+4 )
54. 4
55. A
2 56.
,
3 3
f ( a+h ) f ( a )
2
2
57. f ( a+h ) =4h +4a +8h a+1,
=4h+8a
h
30
58.
9
16u v
27
59. ( x 3)
60. 10
3