Prerequisite Skills
Curtis, Chris, Camil
Properties of Exponents



Product rule

anam=an+m

Ex. 5253=55
Quotient rule

an/am=an-m

Ex. 55/52=53
Power rule

(an)m=anm

Ex. (93)2=96
 Negative exponents
 a-n=1/an
 Ex. 4-3=1/43
 Rational exponents
 an/m=m√an
 Ex. 52/3=3√52
Properties of Logarithms
 Product Rule
 logan + logam = loganm
 Ex. Log28 + log232 = log2256

 Quotient Rule
 logan – logam = loga(n/m)
 Ex. Log2256 – log232 = log28

 Power Rule
 nlogam = logamn
 Ex. 3log28 = log2512
Power of a log

alogam(n) = m

Ex. 9log9(10) = 10
Base Law

logaam = m

log9910 = 10
Converting

The exponential function an=y can be expressed in logarithmic form as logay=n

Ex. 43=64 (exponential)

Ex. log12144=2 (logarithmic)
log464=3 (logarithmic)
122=144 (exponential)
The Exponential Function

y=bx

The base b is positive and b cannot equal 1

The y-intercept is y=1

Horizontal asymptote at the x-axis

The domain is any real value of x

The range is all positive values

The function is increasing when b > 1

The function is decreasing when 0 < 𝑏 < 1
y=2x
Ex. The value of a section of land costs $30000 and it’s value is expected
to increase by 15% every 2 years.
The logarithmic Function

The inverse of y=bx is

x=by
y=2x
Or

logbx=y (logarithmic function)
y=log2x
Trigonometric Ratios
Special Triangles:
y=sinx
y=cosx
y=tanx
Radian Measure

A radian is an arc of a circle that is equal to the radius

𝜋r=180°
Converting degrees to radians:
Ex. 60° to radians

𝜋
60°( )
180
60𝜋
180
=
𝜋
=
3
Converting radians to degrees:
4𝜋
 Ex. 3 radians to degrees
180
4𝜋
)x(
)
𝜋
3
720𝜋
= 3𝜋
(
=240°
SYR CXR TYX & SOH CAH TOA
When solving for the value of a trigonometric ratio these following rules are needed:
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
sinΘ= ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
cosΘ=ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
tanΘ=𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
When solving for a trig ratio within a circle:
𝑦
sinΘ=𝑟𝑎𝑑𝑖𝑢𝑠
𝑥
cosΘ= 𝑟𝑎𝑑𝑖𝑢𝑠
tanΘ=
𝑦
𝑥
C.A.S.T Rule
π/2
S
A
π
2π or 0
C
T
3π
2
All Ratios (+)
Cos (+)
Tan (+)
4th Qtr
Examples of finding exact values
Find the exact values between 0 and 2π
3sinx = sinx+1
2sinx = 1
1
sinx = 2
π
x = 6 or
5π
6
π
-tanx = 6
1
x= - √3
tanx =
5π
6
Transformations of graphs

Base sine graph: y=acos(bx+c)+q / y=asin(bx+c)+q
Where A= 1 B= 1 C= 0
The A value controls the vertical stretch or compression. If the A value is greater than
one, then the base graph is stretched by a factor of A. If the value is less than one,
then it is compressed by a factor of A. The A value is known as the amplitude.
The B value controls the horizontal stretch. If the value is less than one, then you
stretch by a factor of the denominator. If It is greater than one, you compress by a
factor of the value.
The C value is responsible for the phase shift left/right on the horizontal plane. If the
value is negative, you move the graph to the right, and if it is positive, you move to
the left.
The Q value is responsible for the vertical shift on the graph. Move up or down by the
corresponding value.
The value B is the number of cycles it completes in an interval of 0 to
B affects the period. The period of sine and cosine is
2𝜋
𝐵
.
360
.The
2𝜋
value
Problem solving

Identify values and what they do:
y = 2cosx
y = cos(x+1)
y = -sinx
1
y = 3sin(2x+6)-3
Ex. The price of snowboards fluctuates between a maximum of $150 and a minimum
of $100 over a year. The peak selling time is in January (t=0) and the slowest time is
in July (t=6). Sketch the graph.
Trig Identities

Reciprocal Identities

1
csc∅ = 𝑠𝑖𝑛∅
1
 sec∅ =
𝑐𝑜𝑠∅
1
 cot∅ =
𝑡𝑎𝑛∅
 Reflection Identities
 sin(-∅) = −𝑠𝑖𝑛∅
 cos(-∅) = −𝑐𝑜𝑠∅
 Pythagorean Identities
 sin2∅ + cos2∅ = 1
 tan2∅ + 1 = sec2∅
 1 + cot2∅ = csc2∅
 Cofunction Identities
𝜋
 cos( 2 − 𝑥) = sin𝑥
𝜋
 sin( 2 − 𝑥) = cos𝑥
 Quotient Identity
𝑠𝑖𝑛∅
 tan∅ = 𝑐𝑜𝑠∅
 cot∅ =
𝑐𝑜𝑠∅
𝑠𝑖𝑛∅