Algebra II Final Exam – Quick Reference Notes

Inverse & Joint/Combined Variations

Inverse variation: y = \frac{a}{x} (a is the constant of variation, also written as K). Given y=3 when x=-8, then 3 = \frac{a}{-8} \Rightarrow a = -24. Equation: y = \frac{-24}{x}. If x=4, then y = \frac{-24}{4} = -6.
Joint variation: z = a\,x\,y. Given z=-72 when x=9, y=-4: -72 = a(9)(-4) \Rightarrow a = 2. Equation: z = 2xy . For x=6, y=4, z = 2\cdot 6 \cdot 4 = 48.
Combined variation (direct with one variable and inverse with another):
- If d varies directly with m and inversely with the cube of p: d = \frac{a\,m}{p^3}.
- If r varies jointly with y and the square root of x: r = a\, y\, \sqrt{x}.

Graphing Inverse Variation Form y = \dfrac{a}{x-h} + k

General form: shifted inverse variation with center (h, k). Vertex/asymptotes:
- Vertical asymptote: x = h
- Horizontal asymptote: y = k
Example: y = -\dfrac{2}{x-1} + 3 has vertical asymptote x=1 and horizontal asymptote y=3.
Concept: take the parent function y = \dfrac{a}{x}, then shift by h units right/left and by k units up/down; the factor with x in the denominator determines horizontal shift, the added value sets vertical shift.

Factoring & Excluded Values

Difference of squares: x^2 - 16 = (x+4)(x-4).
Denominators (quadratics): factor and identify zeros to determine excluded values. For example, 3x^2+11x-4 = (3x-1)(x+4), so zeros are x = -4, \tfrac{1}{3}; before cancellation these are the excluded values.
After factoring, cancel common factors only after identifying excluded values. Final simplified form may still require noting excluded values (from the original denominator).
Example 1: \frac{(x^2-16)}{(x^2-7x+12)} = \frac{(x+4)(x-4)}{(x-3)(x-4)} \rightarrow \frac{x+4}{x-3}, \text{ with excluded } x=3,4.
Example 2: \frac{5x+20}{3x^2+11x-4} = \frac{5(x+4)}{(3x-1)(x+4)} = \frac{5}{3x-1}, \text{ with excluded } x=-4, \tfrac{1}{3}.
Division by a fraction (keep-change-flip):
- \frac{A}{B} ÷ \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C}.
When solving rational equations, cancel common factors top/bottom after forming a common denominator; be mindful of extraneous solutions where any canceled factor would have been zero.

Fractions: Common Denominator & Numerator Addition/Subtraction

To subtract fractions with polynomial denominators, build a common denominator: for \frac{P}{x-5} - \frac{Q}{x+4} use common denom \,(x-5)(x+4).
Multiply each numerator by the factor it’s missing to form the common denominator, then combine numerators.
After clearing denominators you may factor the new numerator to see cancellations; cancel only after ensuring domain exclusions.

Proportions, Equations & Extraneous Solutions

Cross-multiplication: if \frac{A}{B} = \frac{C}{D}, then AD = BC.
Clearing denominators by multiplying both sides by the least common multiple simplifies solving fractions.
Always check potential extraneous solutions that arise from multiplying both sides by expressions that could be zero (denominator zeros).

Permutations, Combinations & Counting Principles

Permutations: {}nPr = \frac{n!}{(n-r)!} (order matters)
Combinations: {}nCr = \frac{n!}{(n-r)!\, r!} (order does not matter)
Examples:
- Choosing 2 from 20 (identical jobs): {}{20}C2 = \frac{20!}{18!\,2!} = 190
- Distinguishable permutations of "algebra": \frac{7!}{2!}=2520 (two A’s)
- Mississippi word: \frac{11!}{4!\,4!\,2!}=\text{(value)}
- Finish order in a race: {}{10}P3 = 10\cdot9\cdot8 = 720

Binomial Theorem, Coefficients & Terms

Binomial term: \binom{n}{k} a^{n-k} b^{k}
Coefficient example: coefficient of x^4 in (2x-3)^7 is
\binom{7}{3} 2^{4} (-3)^{3} = -15120.
Third term rule (general): in expansion of (x+y)^n, the k-th term is \binom{n}{k-1} x^{n-(k-1)} y^{k-1}; use Pascal’s triangle pattern to locate terms quickly.
Summation notation (Sigma): \sum{i=1}^{6} (3i+2) corresponds to first six terms with term form ai = 3i+2.

Geometric & Arithmetic Series

Finite geometric sum: Sn = \frac{a1(1 - r^{n})}{1 - r}, \quad r \neq 1
Infinite geometric sum (|r|<1): S\infty = \frac{a1}{1 - r}
Identify first term a_1 and common ratio r to apply formulas.

Probability, Normal Distribution & Margin of Error

Probability: P(B) = \frac{\text{# blue}}{\text{total}} (example: 7/15)
Summary statistics: mean, median, mode, range, standard deviation, outliers.
Margin of error (sample proportion context, per transcript): \text{MOE} = \pm \sqrt{\dfrac{1}{n}} (n = sample size).
Z-score: z = \dfrac{x - \mu}{\sigma}; interpret as number of standard deviations from the mean.
For normal distribution: use standard normal table (or z-table) to estimate probabilities.

Sequences: Arithmetic & Geometric

Arithmetic sequence: an = a1 + d\,(n-1); given a7 and d, solve for a1. Example: if a7 = 12 and d = -3, then a1 = 30 and a_n = 30 - 3(n-1) = 33 - 3n.
Geometric sequence: an = a1 r^{(n-1)}; given a1 and r, find general term. Example: if a3 = 8, r = \tfrac12, then a1 = 32, and a_n = 32\left(\tfrac12\right)^{n-1}.

Trigonometry: Trig Functions, Unit Circle, Inverses

Special value recall (unit circle): know common values for sine, cosine, tangent at key angles; use reference angles and quadrant signs.
Inverse trig ranges: \sin^{-1}, \tan^{-1} \in [-\tfrac{\pi}{2}, \tfrac{\pi}{2}]; \cos^{-1} \in [0, \pi].
Pythagorean identities: \sin^2 x + \cos^2 x = 1; reciprocals: \csc x = \frac{1}{\sin x}, \sec x = \frac{1}{\cos x}, \cot x = \frac{\cos x}{\sin x}.

Trigonometric Graphs: Amplitude, Period, Phase Shift & Vertical Shift

For y = a\sin(Bx - H) + K:
- Amplitude: |a|
- Period: \dfrac{2\pi}{|B|}
- Phase shift: \dfrac{H}{B} (shift left if negative in the inner, depending on sign conventions)
- Vertical shift: K
Example: sine with amplitude 2, period 4π (i.e., B = \tfrac{\pi}{2}) etc. (graphical intuition)

Graphing: Tangent

Tangent standard features: vertical asymptotes where cosine = 0; period = \dfrac{\pi}{|B|} after transformation y = A\tan(Bx).
Graph scaling affects height (amplitude not defined for tan) and period.

Law of Sines & Law of Cosines (Solving Triangles)

Law of Sines: \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}
Law of Cosines: c^2 = a^2 + b^2 - 2ab\cos C (and cyclic forms)
Use when triangles are non-right or when given enough info to form a ratio.

Angles, Coterminals, & Conversions

Coterminal angles: add or subtract multiples of 2\pi (or 360°) to obtain equivalent angles.
Conversions: 180^{\circ} = \pi\text{ rad}; to convert \theta\text{ rad} \to {}^{\circ}: \theta\cdot\dfrac{180^{\circ}}{\pi}; to convert degrees to radians: \theta\times\dfrac{\pi}{180^{\circ}}.
Angles in radians for exact values: e.g., \pi/5, 2\pi/3, 7\pi/6,\ldots

Arc Length & Sector Area (Radians)

Arc length: s = r\,\theta (θ must be in radians)
Sector area: A = \tfrac{1}{2} r^2\theta (θ in radians)
Proportional approach: part/whole arguments also work via similar triangles in circles.

3D & Area of Triangles (SAS) & Heron

SAS area: \text{Area} = \tfrac{1}{2} a b \sin C (or two sides with included angle)
Heron's formula: for sides a,b,c, semiperimeter s = \dfrac{a+b+c}{2}, area = \sqrt{s(s-a)(s-b)(s-c)}.

Angles of Depression & Elevation

Angle of elevation: angle above the horizontal; angle of depression is measured from the horizontal downward.
If a pair of lines are parallel, alternate interior angles yield equal angles used to compute missing angles.

Area/Volume & Miscellaneous

Solve triangle problems with unit circle, law of sines/cosines as applicable.
Know area formulas, arc-length formulas, and sector area formulas for quick reference.

Quick Reference: Key Notation & Tips

Excluded values: always check original denominators before canceling factors.
When clearing denominators, verify no division by zero for any potential solution.
Use the most efficient term-by-term approach (binomial term selection, Pascal’s triangle shortcut, etc.).
For binomial coefficients, remember: \binom{n}{k} = \dfrac{n!}{k!(n-k)!}.
For summations: write the index, the range, and the term properly, e.g., \sum_{i=1}^{6}(3i+2).