Source: SYMKC Series, Calculus Prep (Texas A&M University)
Tags: algebra 2, arithmetic series, geometric series, probability, permutations, combinations, transformations, fractions, conjugate, horizontal asymptote, vertical asymptote, imaginary numbers, complex numbers, parent functions, fraction operations
Algebra 2 builds on everything from Algebra 1 and introduces sequences and series, deeper function transformations, complex numbers, asymptote behaviour, and probability/combinatorics. This sheet is your quick-reference for the formulas and rules that come up again and again in Pre-Cal and Calculus.
Arithmetic series
A sequence where each term differs from the previous by a constant amount (the common difference, d).
Geometric series
A sequence where each term is found by multiplying the previous by a constant ratio (the common ratio, r).
Permutation (nPr)
The number of ways to arrange r items chosen from n distinct items, where order matters.
Combination (nCr)
The number of ways to choose r items from n distinct items, where order does not matter.
Conjugate
For a binomial a + b, the conjugate is a – b. Multiplying a binomial by its conjugate eliminates the middle term.
Horizontal asymptote
A horizontal line that the graph of a function approaches as x → ±∞.
Vertical asymptote
A vertical line where the function is undefined and the output grows without bound.
Imaginary unit (i)
Defined as i = √(–1). The foundation of complex numbers.
nth term: tₙ = a₁ + (n – 1)d
Sum of n terms: Sₙ = n/2 · (a₁ + aₙ), or equivalently Sₙ = n/2 · (2a₁ + (n – 1)d)
nth term: tₙ = a₁ · r⁽ⁿ⁻¹⁾
Sum of n terms: Sₙ = a₁(1 – rⁿ) / (1 – r), where r ≠ 1
Sum of an infinite geometric series (|r| < 1): S = a₁ / (1 – r)
Permutations: nPr = n! / (n – r)!
Combinations: nCr = C(n, r) = n! / (r!(n – r)!)
Basic counting principle: if there are m ways to do one thing, n ways to do another, and l ways to do a third, there are m × n × l total outcomes
y = f(x) – k: shift down k units
y = f(x – h): shift right h units
y = –f(x): reflection over the x-axis
y = f(–x): reflection over the y-axis
y = f⁻¹(x): the inverse function (reflection over y = x)
y = a·f(x): vertical stretch if a > 1, vertical compression if 0 < a < 1
y = f(bx): horizontal compression if b > 1, horizontal stretch if 0 < b < 1
These come up so often that hesitation costs real time:
0 / # = 0 (zero divided by anything nonzero is zero)
0 / 0 = undefined (indeterminate form)
ab + ac = a(b + c) — factor out the common term
a/b + c/d = (ad + bc) / bd — common denominator addition
(a + b) / c = a/c + b/c — splitting a fraction
(a/b) / (c/d) = (a/b) · (d/c) = ad / bc — dividing fractions means multiply by the reciprocal
(a/b) / c = a / (bc)
a / (b/c) = ac / b
a(b/c) = ab / c
(a – b) / (c – d) = (b – a) / (d – c) — flipping signs in numerator and denominator
(ab + ac) / a = b + c — cancel the common factor
The conjugate of a + b is a – b
(a – b)(a + b) = a² – b² (difference of squares)
This is particularly useful for rationalising denominators containing radicals.
For y = (axᵐ + ...) / (bxⁿ + ...):
If m < n (degree of numerator < degree of denominator): y = 0
If m = n (degrees are equal): y = a/b (ratio of leading coefficients)
If m > n (degree of numerator > degree of denominator): no horizontal asymptote (there may be an oblique/slant asymptote)
Set the denominator equal to zero and solve. Each solution that does not also zero the numerator gives a vertical asymptote. If both numerator and denominator share a zero, there is a hole (removable discontinuity) rather than an asymptote.
√(–1) = i
i² = –1
i³ = –i
i⁴ = 1
The powers of i cycle with period 4: i, –1, –i, 1, i, –1, ...
Know the shape and key features of the parent function y = x (linear). Every transformation rule above modifies the parent. In Algebra 2 you also work with y = x², y = x³, y = |x|, y = √x, y = 1/x, and y = 2ˣ as parents.
Arithmetic series nth term: tₙ = a₁ + (n – 1)d
Geometric series nth term: tₙ = a₁ · r⁽ⁿ⁻¹⁾
Infinite geometric sum: S = a₁ / (1 – r), valid only when |r| < 1
Permutations: nPr = n! / (n – r)!
Combinations: nCr = n! / (r!(n – r)!)
Difference of squares: (a – b)(a + b) = a² – b²
⚠️ The infinite geometric series formula only works when |r| < 1. If |r| ≥ 1, the series diverges.
⚠️ Horizontal asymptote rules depend on comparing the degrees of numerator and denominator. Students often confuse which case gives y = 0 vs y = a/b.
⚠️ The powers of i repeat every four: i¹ = i, i² = –1, i³ = –i, i⁴ = 1. To evaluate i raised to a large power, divide the exponent by 4 and use the remainder.
⚠️ When dividing fractions, a common error is forgetting to flip the second fraction. "Dividing by a fraction" means "multiplying by its reciprocal."
⚠️ Transformations: y = f(x – h) shifts right (not left). The sign is counterintuitive.
Q: Find the 10th term of an arithmetic sequence where a₁ = 3 and d = 5.
A: t₁₀ = 3 + (10 – 1)(5) = 3 + 45 = 48.
Q: What is the sum of the infinite geometric series 8 + 4 + 2 + 1 + ...?
A: r = 1/2, so S = 8 / (1 – 1/2) = 8 / (1/2) = 16.
Q: What is i⁴⁷?
A: 47 ÷ 4 = 11 remainder 3. So i⁴⁷ = i³ = –i.
Q: Determine the horizontal asymptote of y = (3x² + 1) / (5x² – 2).
A: Degrees are equal, so HA is y = 3/5.
Q: How many ways can you arrange 3 books chosen from a shelf of 8?
A: 8P3 = 8! / 5! = 8 × 7 × 6 = 336.
algebra 2 review, SYMKC, arithmetic sequence, geometric sequence, common difference, common ratio, series sum formula, infinite series, permutation, combination, counting principle, transformations, vertical shift, horizontal shift, reflection, stretch, compression, inverse function, fraction operations, conjugate, difference of squares, horizontal asymptote, vertical asymptote, removable discontinuity, hole, imaginary numbers, complex numbers, powers of i, parent functions