Algebra 2 – Complete Reference Guide

Parent Functions & Transformations

$y = a \cdot f(b(x - h)) + k$

Vertical Changes (Outside)

$|a| > 1$: Vertical Stretch

$0 < |a| < 1$: Vertical Compression

$a < 0$: Reflect over x-axis

$k$: Shift Up (+)/Down (-)

Horizontal Changes (Inside)

$|b| > 1$: Horizontal Compression

$0 < |b| < 1$: Horizontal Stretch

$b < 0$: Reflect over y-axis

$h$: Shift Right (-)/Left (+) (Opposite!)

Complex Numbers & Quadratics

Imaginary Unit

$i = \sqrt{-1}$ and $i^2 = -1$

Powers of $i$: $i^1=i, i^2=-1, i^3=-i, i^4=1$

Complex Numbers

$a + bi$ form.

Add/Sub: Combine real & imag parts.

Mult: FOIL and use $i^2 = -1$.

Div: Multiply by conjugate $a-bi$.

Vertex Form

$y = a(x-h)^2 + k$

Vertex: $(h, k)$

Polynomial Functions

End Behavior

Zeros & Multiplicity

Odd Multiplicity: Crosses x-axis.

Even Multiplicity: Touches (bounces) and turns.

Theorems

Remainder: $f(c) = R$ when dividing by $x-c$.

Rat Root: Possibles = $\pm$ (Factors of Constant / Factors of Lead)

Rational Functions

$f(x) = \frac{p(x)}{q(x)}$ where $p, q$ are polynomials.

Asymptotes

Vertical (VA): Set denominator $q(x) = 0$.

Horizontal (HA): Compare degrees.
- Deg Top < Bottom: $y=0$
- Deg Top = Bottom: $y = \frac{\text{Lead Coef}}{\text{Lead Coef}}$
- Deg Top > Bottom: No HA (Slant)

Holes: When factor cancels out top & bottom.

Exponential & Logarithmic

$y = b^x \iff x = \log_b y$

Properties of Logs

$\log(xy) = \log x + \log y$

$\log(x/y) = \log x - \log y$

$\log(x^n) = n \log x$

$\log_b b = 1$, $\log_b 1 = 0$

Natural Log (ln)

Base $e \approx 2.718$. $\ln x = \log_e x$.

Compound Interest

Finite: $A = P(1 + r/n)^{nt}$

Continuous: $A = Pe^{rt}$

Radicals & Rational Exponents

$x^{m/n} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m$

Simplifying Radicals

Find largest perfect $n$-th power factor.

$\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$

Solving Radical Equations

1. Isolate radical.

2. Raise both sides to power $n$.

3. Solve.

Always check for extraneous solutions!

Sequences & Series

Arithmetic (+d)

$a_n = a_1 + (n-1)d$

$S_n = \frac{n}{2}(a_1 + a_n)$

Geometric ($\times r$)

$a_n = a_1 \cdot r^{n-1}$

$S_n = a_1 \frac{1-r^n}{1-r}$

$S_\infty = \frac{a_1}{1-r}$ (when $|r| \lt 1$)

Stats & Probability

Normal Distribution

Empirical Rule: 68% - 95% - 99.7%

Combinatorics

Permutation (Order matters): $_nP_r = \frac{n!}{(n-r)!}$

Combination (Order doesn't): $_nC_r = \frac{n!}{r!(n-r)!}$

ALGEBRA 2 Complete Reference