Algebra II Final Cheat Sheet

A Education › Mathematics cheat sheet in English by Melinluvsu

For the use of Algebra II Final

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Periodic Functions

Periodic Function: repeats a pattern of y-values (outputs) at regular intervals

Cycle: may begin at any point in a graph

Period: is the horizontal length of one cycle.

Special Right Angles

45-45-90
h = sqrt 2 times l

30-60-90
h = 2 times s
l = sqrt 3 times s

s = short leg
l = long leg

Properties Of Sine Functions

y = a sin b theta

period = 2pi/b

|a| = amplitude
b = number of cycles (0 to 2pi)

Quadratic Functions

Standard Form

f(x) = ax² + bx + c

ax²
Quadratic term

bx
Linear term

c
constant term

Exponential Growth & Exponential Decay

b= 1 + r
b>1 = epon. growth

When b<1, b is a decay factor
x-axis = asymptote
0<b<1
b= 1+(-r)
y=ab^x

b= growth factor
r= increase in rate

e & Its Importance

A = Pe^rt

A= amount in account
P=principal (what you start with)
r = rate in interest (annually)
t= time (in years)

Unit Circle

radian 2pi, tangent 0
radian pi/6, tangent sqrt 3/3
radian pi/4, tangent 1
radian pi/3, tangent sqrt 3
radian pi/2, tangent undefined
radian 2pi/3, tangent -sqrt 3
radian 3pi/4, tangent -1
radian 5pi/6, tangent -sqrt3/3
radian pi, tangent 0
radian 7pi/6, tangent sqrt3/3
radian 5pi/4, tangent 1
radian 4pi/3, tangent sqrt3
radian 3pi/2, tangent undefined
radian 5pi/3, tangent -sqrt3
radian 7pi/4, tangent -1
radian 11pi/6, tangent sqrt3/3

Sine, Cosine, Tangent

Sine = opp./adj.

Cosine = Adj./Hypo.

Tangent = Opp./Adj.

Mazimun & Minimum

y = ax²+bx+c
AOS: = x = -b/2a

1. vertex
2. c
3. another point

Area= length times width

Trigonometric Identities

Reciprocal Identities
csc theta = 1/sin theta

Sec theta = 1/ cos theta

Cot theta = 1/ tan theta

Tangent & Cotangent Identities
Tan theta = sin theta/ cos theta

Cot theta = cos theta/ sin theta

Pythagorean Identities
Cos² theta + Sin² theta = 1

1+ Tan ² theta = Sec² theta

1+ Cot² theta = Csc² theta

Angle Identities

Angle Difference Identities
sin (A-B) = sinA cosB-cosA sinB

cos (A-B) = cosA cosB + sinA sinB

tan (A-B) = tanA - tan B/1+ tanA tanB

Angle Sum Identities
sin (A+B) = sinA cosB + cosA SinB

cos (A+B) = cosA cosB - sinA sinB

tan (A+B) = tanA + tan B/1-tanA tanB

Identities

Double-Angle Identities
cos2 x = cos² x- sin² x

cos2 x = 2cos² x-1

cos2 x = 1- 2sin²x

sin2 x = 2sin x cos x

tan2 x = 2tan x/1-tan²x

Half Angle Identities
sin A/2 = +/- sqrt 1-cosA/2

cos A/2 = +/- sqrt 1+cosA/2

tan A/2 = +/- sqrt 1-cosA/1+cosA

Logarithms

- to base b of a positive number y is defined as...
If y=ab^x, then logb y= x

Log In Life
pH= -log[H⁺]

b is not equal to 1
b must be positive
log of 0 or negative number = undefined

log= log base 10

Log Are Inverses Of Exponentials

1. Graph exponenetial function
2. Graph y = x
3. Reflect exponential function over y = x (reverse coodinates)

Properties Of Log

logb MN = logb M+ logb N <----product property

logb M/N= logb M - logb N <---- Quotient property

logb M^x = x logb M <----Power property

WATCH OUT FOR ERRORS
logb a/logb c does not equal logb a/c

logb a times c does not equal logb a times logb c

Expanding Log

log2 7b = log2 7 + log 2 b

left to right = expand

right to left = simplify

Natural Log

Write 3ln6 - ln8 as a single natural log

ln 6³/8 ---> ln 216/8 ---> ln 27

Solving Log Equations

Pt 1

solve log(3x+1) = 5
3x+1 = 10⁵
3x+1 = 100000
3x = 99,999
x = 33,333

Pt 2
Solve 2log x- log 3 = 2
log(x²/3)=2
x²/3 = 10^2

x2^= 2(100)
x=10sqrt3 or 17.32

Pairs Of Relations are Inverse Of Each Other

y = x - 7/2
y = 2x+7

y = 3x - 1
y = x +1/3

y = -x + 4
y = -x + 4/-1

y = x + 4/5
y = 5x - 4

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