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Things to Remember
Linear Functions
Exponent Rules
𝑥0 = 1
1
(𝑥 𝑎 )𝑏 = 𝑥 𝑎𝑏
𝑥 −𝑎 = 𝑎
𝑥
𝑎
𝑏
𝑎+𝑏
(𝑥 ∙ 𝑦)𝑎 = 𝑥 𝑎 𝑦 𝑎
𝑥 ∙𝑥 =𝑥
𝑥𝑎
𝑥𝑏
𝑎
𝑥𝑏
= 𝑥 𝑎−𝑏
𝑏
𝑥 𝑎
𝑥𝑎
𝑦
𝑦𝑎
( ) =
𝑎
𝑏
= ( √𝑥 ) = √𝑥 𝑎
Simplify Polynomials
‘like’ terms: same variable & same exponent
( ) + ( )  drop the parentheses then combine like terms
( ) – ( )  change to add the opposite
( ) ( )  distribute
( ) ÷ ( )  long division (may have a remainder)
* order of operations (PEMDAS) when given just numbers
Factor Polynomials
1. GCF (negative included)
2. Number of terms
2 terms  DOTS
3 terms  rainbow
4 terms  grouping
Graph a Line y = mx + b
Δ𝑦
m  slope =
Δ𝑥
b  y-intercept @ (0, b)
(x,y)  a point on the line
Absolute Value y = |x|
Solve a System
2 Linear Equations
algebraic (elimination/substitution)
graphic (identify slope & y-intercept)
number of solutions: 0, 1, 
Linear-Quadratic Equations
algebraic (substitution)
graphic (table with turning point)
number of solutions: 0, 1, 2
Quadratic Equations
algebraic (substitution)
graphic
number of solutions: 0, 1, 2, 
Linear-Circle Equations
algebraic (substitution)
graphic (center & radius)
number of solutions: 0, 1, 2
Quadratic-Circle Equat
algebraic(substitution)
graphic
# of sol: 0, 1, 2, 3, 4
Functions
domain: all possible x values (input)
range: all possible y values (output)
relation: need 2 variables
function: passes the vertical line test
one-to-one: passes the horizontal line test
onto: all members of range are used
evaluate: plug it in!
composite: (𝑔 ∘ 𝑓)(𝑥) = 𝑔(𝑓(𝑥))
inverse: flip the graph over y=x
transform: y = A(Bx – C) + D
A: vertical stretch
B: horizontal compression
C: horizontal shift
D: vertical shift
Sequences
Arithmetic: add a constant to get to next term
Ex: 5, 7, 9, …
Recursion: pattern
Explicit: Specific term
an+1 = an + d
an = a1 + (n – 1)d
a1 = 5; an+1 = an + 2
a15 = 5+(15-1)(2) = 33
Geometric: multiply to get to next term
Ex: 5, 10, 20, …
Recursive:
Explicit:
an+1 = r(an)
a1 = 5; an+1 = 2(an)
Summation: The sum of the first n terms
n-1
an=a1(r)
a15=5(2)
15-1
Sn =
= 81,920
𝑎1 − 𝑎1 𝑟 𝑛
S15 =
1−𝑟
5 – 5(2)15
1−2
= 65,531
Rational Equations
Undefined: A fraction is undefined when
the denominator is 0.
Simplify
Add/Subtract: common denominator
Multiply: factor to reduce
Divide: change to multiply the reciprocal
Complex Fractions:
old school: one fraction on top, one fraction on bottom;
keep-change-flip
master blaster: use Least Common Denominator
2
4
2
4
−
−
𝑥 2 𝑥 = [𝑥 2 𝑥 ] ∙ 𝑥 2 = 2 − 4𝑥 = 2(1 − 2𝑥) = −1
4
2
4
2
4𝑥 − 2
2(2𝑥 − 1)
−
−
𝑥 𝑥2
𝑥 𝑥2
Solve:
old school: (simplify both sides then cross-multiply)
master blaster: blast each piece with LCD
1
1
5
+ = 2
𝑚2 −𝑚
𝑚
𝑚 −𝑚
1
1
5
[ 2
+
= 2
] ∙ 𝑚(𝑚 − 1)
𝑚 −𝑚 𝑚
𝑚 −𝑚
1 + (m – 1) = 5
m = 5 Check your answer for extraneous roots!
Things to Remember
Radicals
Imaginary Numbers
Simplify a radical: the index tells how many of a kind are needed to exit the
radical
√81𝑎2 𝑏 3 = √9 ∙ 9 ∙ 𝑎 ∙ 𝑎 ∙ 𝑏 ∙ 𝑏 ∙ 𝑏 = 9𝑎𝑏√𝑏
3
3
3
√81𝑎2 𝑏 3 = √9 ∙ 9 ∙ 𝑎 ∙ 𝑎 ∙ 𝑏 ∙ 𝑏 ∙ 𝑏 = 𝑏 √81𝑎2
Simplify an Expression
Add/Subtract: combine like radicals
Ex: 3√2 + 4√2 − 8√3 = 7√2 − 8√3
Multiply/Divide: coefficients; radicands; simplify
Ex: 5√2(3√6) = 15√12 = 15(2√3) = 30√3
Ex: 20√40 ÷ 5√5 = 4√8 = 4(2√2) = 8√2
Rationalize (the denominator):
√−1 = 𝑖
cycle: i0 = 1 i1 = I
i2 = -1 i3 = -i
complex numbers: a + bi  ‘a’ is real
part and ‘bi’ is imaginary part
Ex:
Ex:
5
√2
=
5
3 + √6
5
√2
=
⋅
√2
√2
5
=
3 + √6
2𝑖
∙
3 − √6
=
=
=
=
=
5(3 – √6)
9−3√6+3√6−6
=
5
49
4
5±7
16
𝑥=
4
, x={3,
2
}
2𝑖
2(−1)
2
3+𝑖
5
∙
3−𝑖
3+𝑖
3−𝑖
15−5𝑖
=
5(3−𝑖)
9−3𝑖+3𝑖−𝑖 2
9−(−1)
15−5𝑖
3
2
10
−
1
2
𝑖
3
5±7
4
12 −2
𝑥={ , }
𝑥=
−1
𝑖
5(3 – √6)
Solve:
1. Get one radical by itself, then get rid of it
2. Get the other radical by itself, then get rid of it
3. Get x by itself
4. Check your answer(s)!
Quadratic
Solve 2x2 – 5x = 3
Factoring
Complete the Square
Quadratic Formula
2𝑥 2 5𝑥
3
2x2-5x-3=0
−𝑏 ± √𝑏 2 − 4𝑎𝑐
−
=
𝑥=
(2x + 1)(x – 3) = 0
2
2
2
2𝑎
5
25
3
0 = 2x+1
x-3=0
−(−5) ± √(−5)2 − 4(2)(−3)
2
𝑥 − 𝑥+
=
𝑥=
−1
2(2)
2
16
2
=𝑥
x=3
2
25
+
5 ± √25 + 24
16
𝑥=
2
4
5
49
5 ± √49
(𝑥 − ) =
𝑥=
4
16
4
𝑥 = ±√
2𝑖
5
Ex:
5√2
2
3 −√6
Simplify an Expression
Add/Subtract: like parts
Ex: 3 + 5i – 7 = -4 + 5i Multiply/Divide:
Ex: 8 + 6𝑖 ÷ 2 = 8 + 3i
Rationalize (the denominator):
3
3
𝑖
3𝑖
3𝑖
3
Ex: =
∙ = 2=
= − 𝑖
4
x = {3,
4
−1
2
}
Quadratic Roots at y = 0
Nature of Roots
b2 - 4ac < 0  2 imaginary roots
b2 - 4ac = 0  1 real root, rational
b2 - 4ac > 0  2 real roots
rational if perfect square
irrational if not perfect square
−𝑏
Sum of Roots =
𝑎
Product of Roots =
𝑐
𝑎
Circles
Center-Radius Form: (x – h)2 + (y – k)2 = r2
center at (h, k)
radius at r
Complete the Square (double)
x2 – 6x + y2 + 8y = 0
x2 – 6x + 9 + y2 + 8y + 16 = 0 + 9 + 16
(x – 3)2 + (y + 4)2 = 25
center at (3, -4)
radius of 5
Things to Remember
Exponential Functions
Logarithmic Functions
Simplify
Expressions
Ex: 6𝑥 ∙ 6𝑦 = 6𝑥+𝑦
Ex:
6𝑥
6𝑦
= 6𝑥−𝑦
Solve an Equation
Ex: 2𝑥 = 25
x=5
Ex: 5x=7
log(5)x=log(7)
x log(5) = log(7)
x=
log(7)
log(5)
Ex: 82 = 42𝑥+1
(23 )2 = (22 )2𝑥+1
26 = 24x+2
6 = 4x + 2
4 = 4x
1=x
Convert
exp  log
xa = b  logx(b) = a
Base 10
log(𝑥) = 𝑙𝑜𝑔10 (𝑥)
Simplify Expression
log(x) + log(y)=log(xy)
𝑥
log(x) - log(y)=log( )
𝑦
log(x)n=n∙log(x)
Solve an Equation
Ex: log(x) + log(8) =
log(24)
log(8x) = log(24)
8x = 24  x = 3
Ex: log(x2) = 2 + log(3)
log(x2) – log(3) = 2
𝑥2
Change of base formula
log(𝑎)
𝑙𝑜𝑔𝑏 (𝑎) =
log( ) = 2
3
log(𝑏)
2
10 =
100 =
Right Triangles
Pythagorean’s Theorem
SohCahToa
Specials
Functions
sin 𝜃 = 𝑦
cos 𝜃 = 𝑥
sin 𝜃 𝑦
tan 𝜃 =
=
cos 𝜃 𝑥
1
1
csc 𝜃 =
=
sin 𝜃 𝑦
1
1
sec 𝜃 =
=
cos 𝜃 𝑥
1
𝑥
cot 𝜃 =
=
tan 𝜃 𝑦
Reference Angles
Sinusoids
Co-Functions are complementary!
sin(20°)=cos(70°)
tan(5°)=cot(85°)
sec(90°)=csc(0°)
y = A sin B(x + C) + D
amplitude = |A|
2𝜋
frequency = B period =
𝐵
horizontal shift (right/left)= C
vertical shift (up/down)= D
The Unit Circle
Trigonometric Graphs
Quadrant I
Radians
120° =
120 𝑑𝑒𝑔𝑟𝑒𝑒𝑠
1
∙
𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
180 𝑑𝑒𝑔𝑟𝑒𝑒𝑠
=
2
3
Inverse Functions
𝜋
√3
sin-1( ) = 60°
2
𝜋
𝜋 𝑟𝑎𝑑 180 𝑑𝑒𝑔
𝑟𝑎𝑑 =
∙
= 30°
6
6
𝜋 𝑟𝑎𝑑
arccos (
√2
) = 45°
2
𝑥2
3
𝑥2
3
 x = √300
Things to Remember
Probability
(organize) Sample Space
- Tree Diagram
- Two-Way Frequency Table
- Venn Diagram
Independence
P(A) = P(A|B)
Rules
P(A and B) = P(A∩B)
= P(A)∙P(B) for independent events
= P(A)∙P(B|A) for dependent events
P(not A) = P(AC) = 1-P(A)
P(A or B) = P(A∪B) = P(A) + P(B) - P(A∩B)
𝑃(𝐴∩𝐵)
P(A given B) = P(A|B) =
𝑃(𝐵)
Statistics
calculator
enter list: stat-edit
examine: stat-calc-1 var stats
sort list: stat-sortA
skewed
(positive) right
symmetric
(negative) left
Normal Distribution
standardized --> 𝑥̅ = 0
z-score =
normalcdf(lower boundary, upper boundary)
central tendency: median
variability: IQR = Q3-Q1
observational studies
- unbiased sample randomly selected
- population: the entire set of observations that can be made
- sample: a piece of the population (≈ 10% of population)
- survey asks questions
central tendency: mean(𝑥̅ )
normalcdf(lower boundary, upper boundary, 𝑥̅ , Sx)
variability: standard deviation (Sx)
experimental studies
- impose treatment (cause) to find a response (effect)
- subjects are randomly assigned a treatment