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Algebra 2 formula sheet

49 formulas ยท what each one means ยท a free lesson for each

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steplyclass.com/formula-sheets/algebra-2/ ยท free, step-by-step lessons for every formula

Functions

Function notation
f(x) = 3x + 2 โ†’ f(4) = 3 ร— 4 + 2
Put the number in for x.Lesson: Functions โ†’
Combine functions
(f + g)(x) = f(x) + g(x) (fg)(x) = f(x) ร— g(x)
Lesson: Combining functions โ†’
Composition
(f โˆ˜ g)(x) = f(g(x))
Work out g first, then put the answer into f.Lesson: Composition and inverse functions โ†’
Inverse
f(fโˆ’1(x)) = x
Swap x and y, then solve for y. The graph is mirrored in y = x.Lesson: Composition and inverse functions โ†’
Average rate of change
f(b) โˆ’ f(a)b โˆ’ a
The slope between two points of the graph.Lesson: Average rate of change โ†’
Domain
no 0 at the bottom, nothing negative under an even root
The x values the function can use.Lesson: Domain and range โ†’

Moving graphs

Shift
y = f(x โˆ’ h) + k
h moves it right, k moves it up.Lesson: Graph transformations โ†’
Reflect
y = โˆ’f(x) y = f(โˆ’x)
โˆ’f(x) flips over the x-axis; f(โˆ’x) flips over the y-axis.Lesson: Graph reflections โ†’
Stretch
y = a f(x)
a > 1 stretches it up; 0 < a < 1 squashes it.Lesson: Graph stretches โ†’

Polynomials

Remainder theorem
f(x) รท (x โˆ’ c) โ†’ remainder f(c)
Put c in to get the remainder.Lesson: Remainder and factor theorem โ†’
Factor theorem
f(c) = 0 โ‡” (x โˆ’ c) is a factor
Lesson: Remainder and factor theorem โ†’
Rational root theorem
ยฑpq
p is a factor of the last number, q of the first.Lesson: Finding all zeros โ†’
Polynomial from its zeros
f(x) = a(x โˆ’ r1)(x โˆ’ r2)โ€ฆ
Each zero r gives a bracket (x โˆ’ r).Lesson: Writing polynomials from zeros โ†’
End behavior
led by axn
Even n: both ends the same way; odd n: opposite ways. a < 0 flips them.Lesson: Polynomial graphs โ†’
Multiplicity
(x โˆ’ r)k
k odd: the graph crosses at r; k even: it touches and turns back.Lesson: Polynomial graphs โ†’

Rational functions

Vertical asymptote
bottom = 0
Where the bottom is 0 (and the factor does not cancel). A factor that cancels makes a hole.Lesson: Rational functions โ†’
Horizontal asymptote
degrees: top < bottom: y = 0 ยท equal: y = ab
a and b are the leading numbers. Top bigger by 1: slant asymptote.Lesson: Rational functions โ†’
Add rational expressions
ab + cd = ad + bcbd
Same as fractions: a common bottom first.Lesson: Adding and subtracting rational expressions โ†’

Sequences and series

Arithmetic sequence
an = a1 + (n โˆ’ 1)d
Add the same d each time.Lesson: Sequences โ†’
Arithmetic series
Sn = n(a1 + an)2
Number of terms ร— the mean of the first and last.Lesson: Series โ†’
Geometric sequence
an = a1 rnโˆ’1
Multiply by the same r each time.Lesson: Sequences โ†’
Geometric series
Sn = a1(1 โˆ’ rn)1 โˆ’ r
Lesson: Series โ†’
Infinite geometric series
S = a11 โˆ’ r, |r| < 1
Only when r is between โˆ’1 and 1.Lesson: Series โ†’
Sigma notation
ฮฃni=1 ai = a1 + a2 + โ€ฆ + an
Add the terms from i = 1 to n.Lesson: Sigma notation โ†’
Binomial theorem
(a + b)n = ฮฃnk=0 (nk) anโˆ’k bk
The numbers in front come from Pascal's triangle.Lesson: The binomial theorem โ†’
Binomial coefficient
(nk) = n!k!(n โˆ’ k)!
Lesson: The binomial theorem โ†’

Log rules

Definition
logb x = y โ‡” by = x
A log is a power: the power of b that gives x.Lesson: Exponential equations and logarithms โ†’
Product rule
logb(MN) = logb M + logb N
Times inside โ†’ plus outside.Lesson: Log properties with letters โ†’
Quotient rule
logb MN = logb M โˆ’ logb N
Divide inside โ†’ minus outside.Lesson: Log properties with letters โ†’
Power rule
logb(Mp) = p logb M
The power comes down in front.Lesson: Log properties with letters โ†’
Change of base
logb x = log xlog b = ln xln b
How a calculator finds any log.Lesson: Exponential equations and logarithms โ†’
Special values
logb b = 1 logb 1 = 0 blogb x = x
Lesson: Exponential equations and logarithms โ†’
Natural log
ln x = loge x, e โ‰ˆ 2.718
Lesson: Natural logarithms โ†’
Log equations
check every answer: no log of 0 or a negative
Some answers are false and must be thrown away.Lesson: Log equations with false answers โ†’

Interest and growth

Compound interest
A = P(1 + rn)nt
n = times a year it is added (monthly: 12), t = years.Lesson: Compound and continuous interest โ†’
Continuous interest
A = Pert
Lesson: Compound and continuous interest โ†’
Half-life
A = A0 (12)t/h
h is the half-life: the time for half to go.Lesson: Half-life and time with logs โ†’
Savings plan (future value)
FV = PMT ร— (1 + i)N โˆ’ 1i
i = rate per payment, N = number of payments.Lesson: Savings plans โ†’
Loan payment
PMT = P ร— i1 โˆ’ (1 + i)โˆ’N
Rounded up to the next cent.Lesson: Loan payments โ†’

Complex numbers

The number i
i2 = โˆ’1 โˆšโˆ’a = iโˆša
Lesson: Complex numbers โ†’
Powers of i
i, โˆ’1, โˆ’i, 1, โ€ฆ
They repeat every 4: divide the power by 4 and use the remainder.Lesson: Complex numbers โ†’
Multiply
(a + bi)(c + di) = (ac โˆ’ bd) + (ad + bc)i
Open the brackets, then iยฒ = โˆ’1.Lesson: Complex numbers โ†’
Conjugate
(a + bi)(a โˆ’ bi) = a2 + b2
Used to divide: multiply top and bottom by the conjugate.Lesson: Dividing complex numbers โ†’

Matrices

Add and times a number
A + B, kA
Entry by entry.Lesson: Matrices โ†’
Multiply
row of A ร— column of B
AB is not the same as BA.Lesson: Multiplying matrices โ†’
2 ร— 2 determinant
abcd = ad โˆ’ bc
Lesson: Determinants โ†’
2 ร— 2 inverse
Aโˆ’1 = 1ad โˆ’ bcdโˆ’bโˆ’ca
Swap a and d, change the signs of b and c, divide by the determinant.Lesson: Inverse of a 2 ร— 2 matrix โ†’
System with the inverse
AX = B โ†’ X = Aโˆ’1B
Lesson: Systems with the inverse matrix โ†’
Cramer's rule
x = DxD y = DyD
D_x: the answer column in place of the x column.Lesson: Cramer's rule โ†’

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