Prerequisite SkillsCurtis, 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=man Ex. 52/3=52
Properties of Logarithms
Power of a log alogam(n) = m Ex. 9log9(10) = 10
Base Law logaam = m log9910 = 10
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
Converting
The exponential function an=y can be expressed in logarithmic form as logay=n Ex. 43=64 (exponential) log464=3 (logarithmic) Ex. log12144=2 (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 The function is decreasing when 0
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
Or logbx=y (logarithmic function)
y=2x
y=log2x
Trigonometric RatiosSpecial Triangles:
y=sinx y=cosx y=tanx
Radian Measure
A radian is an arc of a circle that is equal to the radius r=180°Converting degrees to radians:
Ex. 60° to radians60°==
Converting radians to degrees: Ex. radians to degrees()x()==240°
SYR CXR TYX & SOH CAH TOA
When solving for the value of a trigonometric ratio these following rules are needed: sinΘ= cosΘ= tanΘ=When solving for a trig ratio within a circle:sinΘ= cosΘ= tanΘ=
C.A.S.T Rule
All Ratios (+)Cos (+)Tan (+)4th Qtr
AS
T C
π/2
π
3π2
2π or 0
Examples of finding exact values
Find the exact values between 0 and 2π 3sinx = sinx+1 2sinx = 1 sinx = x = or tanx = -tanx = x= -
Transformations of graphs Base sine graph: y=acos(bx+c)+q / y=asin(bx+c)+qWhere 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 .The value B affects the period. The period of sine and cosine is .
Problem solving Identify values and what they do: y = 2cosxy = cos(x+1)y = -sinxy = sin(2x+6)-3Ex. 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 csc= sec= cot =
Pythagorean Identities sin22 1 tan22 1 + cot2 = csc2
Quotient Identity tan cot
Reflection Identities sin(- cos(-
Cofunction Identities cos( sin(