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: $y = \frac{\text{Lead Coef of Numerator}}{\text{Lead Coef of Denominator}}$
- 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 — Support Skill

Not a separate Algebra 2 unit — collected here because radicals turn up whenever you simplify quadratic-formula roots, conic distances, and exact trig values.

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

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!

Systems & Matrices

Write the system as the augmented matrix $[A \mid \mathbf{b}]$, then row-reduce:

$$\begin{cases} 2x + 3y = 7 \\ 4x - y = 5 \end{cases} \;\longrightarrow\; \left[\begin{array}{cc|c} 2 & 3 & 7 \\ 4 & -1 & 5 \end{array}\right]$$

Elementary Row Operations

Swap: $R_i \leftrightarrow R_j$

Scale: $kR_i \to R_i$ with $k \neq 0$

Add a multiple: $R_i + kR_j \to R_i$

Each one leaves the solution set unchanged.

Reading the Row Echelon Form

One solution: a leading $1$ in every variable column.

Infinitely many: a zero row and no contradiction — a free variable.

No solution: a row reading $[\,0\ \ 0\ \ 0 \mid k\,]$ with $k \neq 0$.

Matrix Arithmetic

Add/subtract only matrices of identical dimensions, element by element.

$c_{ij} = \sum_{k} a_{ik} b_{kj}$ — row $i$ of $A$ dotted with column $j$ of $B$.

$A_{m\times n} \cdot B_{n\times p} = C_{m\times p}$: the inner dimensions must match.

Determinants

$\det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc$

$3 \times 3$: cofactor expansion along a row, signs alternating $+,-,+$.

$\det(AB) = \det(A)\det(B)$; swapping two rows negates $\det$.

$A$ is invertible $\iff \det(A) \neq 0$.

Inverse of a 2×2

$\begin{bmatrix} a & b \\ c & d \end{bmatrix}^{-1} = \frac{1}{ad-bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$

$AA^{-1} = A^{-1}A = I$; solve $AX = B$ with $X = A^{-1}B$.

Cramer's Rule

$x = \dfrac{D_x}{D}$, $y = \dfrac{D_y}{D}$, $z = \dfrac{D_z}{D}$, valid when $D \neq 0$.

$D$ is the coefficient determinant; $D_x$ replaces the $x$-column with the constants.

Above: $D = -14$, $D_x = -22$, $D_y = -18$, so $x = \tfrac{11}{7}$, $y = \tfrac{9}{7}$.

$D = 0$ means Cramer's Rule cannot decide — row-reduce instead.

Conic Sections

Every conic comes from $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$. Complete the square in $x$ and in $y$ to reach standard form.

Circle ($e = 0$)

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

Center $(h,k)$, radius $r$.

Ellipse ($0 \lt e \lt 1$)

$\dfrac{(x-h)^2}{a^2} + \dfrac{(y-k)^2}{b^2} = 1$, $a \gt b \gt 0$

Vertices $(h \pm a,\, k)$; co-vertices $(h,\, k \pm b)$.

Foci $(h \pm c,\, k)$ with $c^2 = a^2 - b^2$; $e = \dfrac{c}{a}$.

Swap $a^2$ and $b^2$ for a vertical major axis.

Parabola ($e = 1$)

$(x-h)^2 = 4p(y-k)$ or $(y-k)^2 = 4p(x-h)$

Vertex $(h,k)$; focus $p$ units from it, directrix $p$ units the other way.

Hyperbola ($e \gt 1$)

$\dfrac{(x-h)^2}{a^2} - \dfrac{(y-k)^2}{b^2} = 1$ (horizontal transverse axis)

Vertices $(h \pm a,\, k)$; foci $(h \pm c,\, k)$ with $c^2 = a^2 + b^2$.

Asymptotes $y - k = \pm\dfrac{b}{a}(x-h)$.

Vertical version $\dfrac{(y-k)^2}{a^2} - \dfrac{(x-h)^2}{b^2} = 1$ has asymptotes $y-k = \pm\dfrac{a}{b}(x-h)$.

Identifying a Conic

Discriminant $\Delta = B^2 - 4AC$: $\Delta \lt 0$ circle or ellipse · $\Delta = 0$ parabola · $\Delta \gt 0$ hyperbola.

No $xy$-term: $A = C$ circle · same signs but $A \neq C$ ellipse · $A = 0$ or $C = 0$ parabola · opposite signs hyperbola.

Focus rule: the ellipse subtracts ($c^2 = a^2 - b^2$), the hyperbola adds ($c^2 = a^2 + b^2$).

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 $|r| \lt 1$)

Trigonometry

On the unit circle $\cos\theta = x$, $\sin\theta = y$, and $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$ (undefined where $\cos\theta = 0$). Convert with $\text{rad} = \text{deg} \times \dfrac{\pi}{180}$; a full circle is $360^\circ = 2\pi$.

Unit Circle — Special Angles

At $\theta = 0,\ \tfrac{\pi}{6},\ \tfrac{\pi}{4},\ \tfrac{\pi}{3},\ \tfrac{\pi}{2}$: $\sin\theta = 0,\ \tfrac{1}{2},\ \tfrac{\sqrt2}{2},\ \tfrac{\sqrt3}{2},\ 1$

$\cos\theta$ runs the same list backwards: $1,\ \tfrac{\sqrt3}{2},\ \tfrac{\sqrt2}{2},\ \tfrac{1}{2},\ 0$

$\tan\tfrac{\pi}{6} = \tfrac{\sqrt3}{3}$, $\tan\tfrac{\pi}{4} = 1$, $\tan\tfrac{\pi}{3} = \sqrt3$, $\tan\tfrac{\pi}{2}$ undefined.

Pythagorean & Reciprocal

$\sin^2\theta + \cos^2\theta = 1$

$1 + \tan^2\theta = \sec^2\theta$ · $1 + \cot^2\theta = \csc^2\theta$

$\csc\theta = \dfrac{1}{\sin\theta}$, $\sec\theta = \dfrac{1}{\cos\theta}$, $\cot\theta = \dfrac{1}{\tan\theta}$

Even / Odd & Co-function

$\cos(-\theta) = \cos\theta$; $\sin(-\theta) = -\sin\theta$; $\tan(-\theta) = -\tan\theta$

$\sin\!\left(\dfrac{\pi}{2} - \theta\right) = \cos\theta$ and $\cos\!\left(\dfrac{\pi}{2} - \theta\right) = \sin\theta$

Sum & Difference

$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$

$\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$

$\tan(A \pm B) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$

Double & Half Angle

$\sin 2\theta = 2\sin\theta\cos\theta$

$\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$

$\tan 2\theta = \dfrac{2\tan\theta}{1 - \tan^2\theta}$

$\sin\dfrac{\theta}{2} = \pm\sqrt{\dfrac{1-\cos\theta}{2}}$, $\cos\dfrac{\theta}{2} = \pm\sqrt{\dfrac{1+\cos\theta}{2}}$

Graphing $y = A\sin(Bx - C) + D$

Amplitude $|A|$ · Period $\dfrac{2\pi}{|B|}$ · Phase shift $\dfrac{C}{B}$ · Midline $y = D$

$\sin x$, $\cos x$: period $2\pi$, range $[-1,1]$. $\tan x$: period $\pi$, range all reals.

ALGEBRA 2 Complete Reference