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Formula Sheet
Reciprocal Identities:
csc x
1
= -.-
secx=--
SlUX
Quotient Identities:
Pythagorean
tanx=--
Identities:
1
cosx
cosx
SlUX
=
I
-I+ cor' x
tan ' x + I = see" x
cos2x
sin2x = 2sinxcosx
tan2x =
Logarithms:
2 tan x
2
I-tan
x
Y = log, x
is equivalent to
m
log, n
m - log, n
Product property:
Quotient property:
= log,
Power property:
= log, n , then m
Property of equality:
If log, m
Change of base formula:
log, n
1og n=--
log, a
a
Slope-intercept
form: y
Point-slope form: y - Yl
Standard form:
= mx + b
= m(x - Xl)
Ax +
By + C
=a
tan x
cosx
cotx=-.-
SlUX
sin" x+ cos" x
Double Angle Identities:
1
cotx=--
=n
=
csc'
= cos" x - sin" x
= 1- Zsirr' x
= Zcos" x-I
x=a-r
X
lc:Jx= 2
2. Solve
3. Rewrite using a single logarithmic value for 08';35 + 1Qiglli
4. Rewrite using a single logarithmic value for lng: 5 - In
3
/j
Change the base to the common loqarithm for
5.
·'aBf
=
6. ~aa;J.'f
=
Complex Fractions
When simplifying complex fractions, multiply by a fraction equal to 1 which has a numerator and
denominator composed of the common denominator of all the denominators in the complex fraction.
Example:
_ 7 __ 6_
x+l
5
_7 __ 6_
=
x+l
-2
x
3x
x-4
-+-1
5---
x-4
x+l
----,=--'----'--=-
x+l
g--
5
x+l
.r +I
-2
x
3x
x-4
-+--
=
1
5--x-4
-7x-7-6
x(x- 4)
g --'----'-- =
x(x- 4)
5
=
-7x-13
5
-2(x-4)+3x(x)
5(x)(x - 4) -lex)
=
-2x+ 8 + 3x2
5x2 - 20x-x
=
3x2 -2x+ 8
5x2
-
2lx
Simplify each of the following.
25
--a
7. --"a,,--_
5+a
4
4-~
2-8.
x+2
9.
5+~
5+--
2x-3
x+2
x
1
x+l
x
x
1
x+l
x
----
10.
--+-
2x-3
15
11.
1-~
3x-4
32
x+-3x-4
Functions
To evaluate a function for a given value, simply plug the value into the function for x.
Recall: (f og JX)
= f(g(x))
OR f[g(x)]
read "fof gof
x'
Means to plug the inside function (in this
case g(x) ) in for x in the outside function (in this case, f(x».
Given f(x) =2X2 + 1 and g(x)
Example:
=x
- 4 find
f(g(x)).
f(g(x))
= f(x
=
- 4)
2(x - 4i + 1
= 2(x2 - 8x + 16) + 1
=2x2-16x+32+1
f(g(x))
Let f(x)
= 2x + 1 and g(x) = 2X2 -l. Find each.
12. f(2)
=
15. f[g(-2)J=
Let
f(x)
=
13. g(-3)=
_
sinx
__
-
16. g[f(m+2)J=
= 2X2 -16x + 33
_
14. f(t+l)=
17.
_
Find each exactly.
19. f(23ll")=
_
_
f(x + h) - f(x)
h
Let f(x)
=
X2, g(x)
=
2x
+ 5, and hex)
= X2 -1.
Find each.
21. f[g(X-1)]=
20. h[f(-2)]=
Find f(x + h) - f(x)
h
__
for the given function f.
23. f(x)=9x+3
24. f(x)=5-2x
Intercepts and Points of Intersection
To find the x-intercepts,
To find the y-intercepts,
= 0 in your
let x = 0 in your
let y
= X2 - 2x - 3
x - int. (Let y = 0)
Example:
equation and solve.
equation and solve.
y
0= X2 -2x-
0= (x - 3)(x + 1)
x=-lorx=3
x - intercepts
y- int. (Let x
3
(-1,0) and (3,0)
y
=
0
2
-
=
0)
2(0) - 3
y=-3
y - intercept (0, - 3)
Find the x and y intercepts for each.
25.
y= 2x-5
27.
26.
Y =x2 +x-2
28.
i =x
3
-4x
Use substitution or elimination method to solve the system of equations.
Example:
+ 39 = 0
x 2 -y 2 - 9 = 0
X2 + Y -16x
Elimination Method
Substitution Method
2x2-16x+30=0
Solve one equation for one variable.
X2 - 8x+ 15 = 0
(x - 3)(x - 5)
x
y2 =_X2 +16x-39
=0
X2 _(_x2 + 16x- 39)- 9
= 3 and x = 5
Plug x = 3 and x
32 - y2 - 9
=0
=5
i -9 = 0
2X2 -16x + 30
=0
-i = 0
16 = y2
X2 - 8x + 15 = 0
y=O
y=±4
(x - 3)(x - 5)
Points of Intersection (5,4), (5,-4) and (3,0)
= 0 Plug what
l
is equal
to into second equation.
into one original
52 -
(1 st equation solved for y)
x
= 3 or x - 5
=0
(The rest is the same as
previous example)
Find the point(s) of intersection of the graphs for the given equations.
29.
x +y
=8
30.
4x - y = 7
X2
+Y = 6
x +y
=4
31.
X2 -
4l
- 20x - 64Y - 172 = 0
16x2 + 4l-
320x + 64y + 1600 = 0
Interval Notation
32. Complete the table with the appropriate notation or graph.
Solution
Interval Notation
Graph
-2 < x ~ 4
[-1,7)
J-.
~
8
Solve each equation. State your answer in BOTH interval notation and graphically.
33.
2x -120
34.
-4 ~ 2x-
3< 4
35.
x
x
2
3
--->5
Domain and Range
Find the domain and range of each function.
36.
f(x)
= X2
-
Write your answer in INTERVAL notation.
37. f(x) = -Jx + 3
5
38. f(x) = 3sinx
39. f(x) = _2_
x-I
Inverses
To find the inverse of a function, simply switch the x and the y and solve for the new "y" value.
Example:
= ~x + 1
f(x)
y
= ~x
x
=
+1
~y+ 1
Rewrite f(x) as y
Switch x and y
Solve for your new y
(x y = (~y+ 1) Cube both sides
x3
y
=y+1
=x
3
f-I(X)
-
1
= x3 -1
Simplify
Solve for y
Rewrite in inverse notation
Find the inverse for each function.
2
40.
f(x)=2x+l
41. f(x)=~
3
Also, recall that to PROVE one function is an inverse of another function, you need to show
that:
f(g(x))
= g(.f(x)) = x
Example:
x-9
If: f(x) = --
f(g(x))
and g(x)
4
= 4(X-9)
-4-
= 4x + 9 show f(x) and g(x) are inverses of each other.
g(f(x))=
+9
(4X+:)-9
4x+ 9- 9
=x-9+9
4
4x
=x
4
=x
f(g(x))
= g(.f(x)) = x therefore they are inverses
of each other.
Prove f and g are inverses of each other.
3
42. f(x)
=x
2
g(x)
= ifb
43. f(x)=9-x2,x?,O
g(x)=.J9-x
Equation of a line
Slope intercept form:
Point-slope form:
y = mx + b
y-y, = m(x-x,)
44. Use slope-intercept
Vertical line: x
=c
Horizontal line: y
(slope is undefined)
=c
(slope is 0)
form to find the equation of the line having a slope of 3 and a y-intercept of 5.
45. Determine the equation of a line passing through the point (5, -3) with an undefined slope.
46. Determine the equation of a line passing through the point (-4,2) with a slope of 0.
47. Use point-slope form to find the equation of the line passing through the point (0, 5) with a slope
of 2/3.
48. Find the equation of a line passing through the point (2, 8) and parallel to the line y
49. Find the equation of a line perpendicular
= ~ x-I.
6
to the y- axis passing through the point (4, 7).
50. Find the equation of a line passing through the points (-3, 6) and (1,2).
51. Find the equation of a line with an x-intercept (2, 0) and a y-intercept (0, 3).
Radian and Degree Measure
U se
180°
t 0 get nid 0 f ra dlians an d
n radians
Use
convert to degrees.
52. Convert to degrees:
n radians
180°
.
to get rid of degrees and
convert to radians.
a.
5Jr
b.
6
4Jr
c. 2.63
5
radians
53. Convert to radians:
a. 45°
b.
-lr
c.237"
Angles in Standard Position
54. Sketch the angle in standard position.
a.
l l zr
6
c.
SJr
d. 1.8 radians
3
Reference Triangles
55. Sketch the angle in standard position. Draw the reference triangle and label the sides, if
possible.
2
a. -Jr
3
b. 225
0
c.
d. 30'
4
Unit Circle
You can determine the sine or cosine of a quadrantal angle by using the unit circle. The x-coordinate
of the circle is the cosine and the y-coordinate
is the sine of the
;:mnll=
~
(0,1)
Example:
56.
a.) sin180°
sin90°
n
cos2
=1
=0
-1,0)
'\
V
!"
(1,0)
./
(0,-1)
b.) cos270°
(0,1)
/'
-1,0)
d.) sin zr
-,
"'\
(1,0)
/
(0,-1)
e.) cos360o
f.)
cos(-n)
Graphing Trig Functions
f(x) :'-sin (x)-- -
--
f(x) =-COS (x)
2
y = sin x and y = cos x have a period of 2 Jr and an amplitude of 1. Use the parent graphs above to
help you sketch a graph of the functions below.
For f(x)
=
A sin(Bx + C) + K , A
= amplitude,
period,
C
B
= phase
shift (positive CIS shift left, negative CIS shift right) and K
Graph two complete periods of the function.
57. f(x) = 5sinx
59. f(x) =
-cos(
58. f(x) = sin2x
x - :)
60. f(x)
= cosx
- 3
= vertical
shift.
2Jr
B
=
Trigonometric
Equations:
Solve each of the equations for 0::; x < 2:rr. Isolate the variable, sketch a reference triangle, find all
the solutions within the given domain, 0::; x < 2:rr. Remember to double the domain when solving for
a double angle. Use trig identities, if needed, to rewrite the trig functions.
(See formula sheet at the
end of the packet.)
.
1
61. smx=-2
63. cos 2x
62. 2 cosx
1
=
J2
65. sin2x = _
13
2
67. 4 cos ' x - 3 = 0
64.
.
7
Sllf"
= 13
1
x =2
66. 2cos2x-l-cosx=O
68. sin" x + cos2x - cosx
=0
Inverse Trigonometric
Recall:
Functions:
Inverse Trig Functions can be written in one of ways:
arcsin
(x)
sin "
(x)
Inverse trig functions are defined only in the quadrants as indicated below due to their restricted
domains.
costx < 0
sin-1x > 0
cos'
X
>0
tan' x > 0
sini x c O
tarr ' x< 0
Example:
Express the value of "y" in radians.
-1
y = arctan
Draw a reference triangle.
J3
~-1
This means the reference angle is 30° or
Jr.
6
-Jr
-<y<2
Jr
2
So, y
=-
Jr
6
so that it falls in the interval from
Answer: y
= - -st
6
For each of the following, express the value for "y" in radians.
69. y
-13
-
= arcsin-
2
Example:
70. y
=
( )
arccos \-1
71. y
= arctan(-1)
Find the value without a calculator.
cos( arctan~)
Draw the reference triangle in the correct quadrant first.
Find the missing side using Pythagorean
Thm.
Find the ratio of the cosine of the reference triangle.
cosB= --
6
J6l
6
For each of the following give the value without a calculator.
(
21
72. tanl arccos})
. (
121
74. sml arctan 5)
i .
121
13)
I
73. secl sm-
Vertical Asymptotes
Determine the vertical asymptotes for the function.
the x-value for which the function is undefined.
76. f(x)
=
Set the denominator
equal to zero to find
That will be the vertical asymptote.
2
1
77.
-2
f(x)
= --;-
78.
f(x)=
x- -4
X
,2+x
x-(l- x)
Horizontal Asymptotes
Determine the horizontal asymptotes
using the three cases below.
Case I. Degree of the numerator is less than the degree of the denominator.
y
The asymptote is
= o.
Case II. Degree of the numerator is the same as the degree of the denominator.
The asymptote is
the ratio of the lead coefficients.
Case III. Degree of the numerator is greater than the degree of the denominator.
horizontal asymptote.
The function increases without bound.
exactly 1 more than the degree of the denominator,
determined by long division.)
There is no
(If the degree of the numerator is
then there exists a slant asymptote, which is
Determine all Horizontal Asymptotes.
80. f(x) =
----------
5x3
4x-
-
2X2
,
+8
3x> + 5
4x5
81. f(x)=-,x--7
----------
...