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: ${}<em>nP</em>r = \frac{n!}{(n-r)!}$ (order matters)

• Combinations: ${}<em>nC</em>r = \frac{n!}{(n-r)!\, r!}$ (order does not matter)

• Examples:

  • Choosing 2 from 20 (identical jobs): ${}<em>{20}C</em>2 = \frac{20!}{18!\,2!} = 190$

  • 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: ${}<em>{10}P</em>3 = 10\cdot9\cdot8 = 720$

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): $\sum<em>{i=1}^{6} (3i+2)$ corresponds to first six terms with term form $a</em>i = 3i+2$.

Geometric & Arithmetic Series

• Finite geometric sum: $S<em>n = \frac{a</em>1(1 - r^{n})}{1 - r}, \quad r \neq 1$

• Infinite geometric sum (|r|<1): $S<em>\infty = \frac{a</em>1}{1 - r}$

• 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: $a<em>n = a</em>1 + d\,(n-1)$; 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: $a<em>n = a</em>1 r^{(n-1)}$; 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., $\sum_{i=1}^{6}(3i+2)$.