Algebra Engine Reference Sheet
ALGEBRA II QUICK REFERENCE
Essential Rules, Core Formulas & Visual Models
High School / College Prep
Formulas • Graphs • Properties
Parent Functions & Graph Signatures
Key behavior, asymptotes, and vertices
Linear \(f(x)=x\)
Domain & Range: \((-\infty, \infty)\)
Quadratic \(f(x)=x^2\)
Vertex: \((0,0)\) • Range: \([0, \infty)\)
Abs Value \(f(x)=|x|\)
Symmetric across \(y\)-axis
Square Root \(f(x)=\sqrt{x}\)
Domain & Range: \([0, \infty)\)
Cubic \(f(x)=x^3\)
Point of Inflection: \((0,0)\)
Rational \(f(x)=\frac{1}{x}\)
Asymptotes: \(x=0, y=0\)
Exponential \(f(x)=b^x\)
\(y\)-int: \((0,1)\) • HA at \(y=0\)
Logarithm \(f(x)=\log_b x\)
\(x\)-int: \((1,0)\) • VA at \(x=0\)
Function Transformations
\(y = a \cdot f\big(b(x - h)\big) + k\)
\(a\)
Vertical Stretch / Shrink: Stretch if \(|a| > 1\); Compress if \(0 < |a| < 1\). If \(a < 0\), reflect over \(x\)-axis.
\(h\)
Horizontal Shift: \((x - h)\) moves RIGHT \(\rightarrow\); \((x + h)\) moves \(\leftarrow\) LEFT. (Opposite sign!)
\(k\)
Vertical Shift: \(+k\) shifts UP \(\uparrow\); \(-k\) shifts \(\downarrow\) DOWN.
\(b\)
Horizontal Scaling: Factor \(\frac{1}{|b|}\). If \(b < 0\), reflect over \(y\)-axis.
Inside parentheses affects \(x\) (opposite operation). Outside affects \(y\) directly.
Quadratics & Complex Numbers
Quadratic Formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
Vertex Formula \(h = -\frac{b}{2a}, \; k = f(h)\)
| Discriminant \(b^2 - 4ac\) | Roots | Graph Intercepts |
|---|
| \(\Delta > 0\) | 2 Real Solutions | Two \(x\)-intercepts |
| \(\Delta = 0\) | 1 Real Double Root | Vertex on \(x\)-axis |
| \(\Delta < 0\) | 2 Complex Conjugates | No \(x\)-intercepts |
Imaginary Unit \(i\):
\(i^1 = i\) \(i^2 = -1\) \(i^3 = -i\) \(i^4 = 1\)
Exponents & Logarithms
\(\log_b(x) = y \iff b^y = x\) Natural log: \(\ln(x) = \log_e(x)\) • Common log: \(\log(x) = \log_{10}(x)\)
Product Rule \(\log_b(uv) = \log_b u + \log_b v\)
Quotient Rule \(\log_b\left(\frac{u}{v}\right) = \log_b u - \log_b v\)
Power Rule \(\log_b(u^p) = p \cdot \log_b u\)
Change of Base \(\log_b a = \frac{\log a}{\log b} = \frac{\ln a}{\ln b}\)
\(x^{-n} = \frac{1}{x^n}\) \(x^{m/n} = \sqrt[n]{x^m}\) \((x^m)^n = x^{mn}\) \(x^0 = 1\)
Polynomial Factoring & Theorems
Diff of Squares: \(a^2 - b^2 = (a - b)(a + b)\)
Perfect Trinomial: \(a^2 \pm 2ab + b^2 = (a \pm b)^2\)
Sum/Diff of Cubes: SOAP
\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)
\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
Same sign, Opposite sign, Always Positive
Remainder Theorem: Evaluating \(P(c)\) gives the remainder when dividing \(P(x)\) by \((x - c)\).
Fund. Thm of Algebra: Degree \(n\) polynomial has exactly \(n\) complex roots.
Sequences, Series & Rational Functions
Arithmetic (\(+d\))
\(n\)-th term: \(a_n = a_1 + (n-1)d\)
Sum: \(S_n = \frac{n(a_1 + a_n)}{2}\)
Geometric (\(\times r\))
\(n\)-th term: \(a_n = a_1 r^{n-1}\)
Sum: \(S_n = \frac{a_1(1-r^n)}{1-r}\)
Infinite: \(S_\infty = \frac{a_1}{1-r}\) (\(|r| < 1\))
Rational Asymptotes: \(\frac{P(x)}{Q(x)}\)
• Vertical (VA): Where denominator \(Q(x)=0\) (after canceling).
• Hole: Canceled factor zero \((x-c)\).
• Horizontal (HA): Compare degrees \(m = \deg P, n = \deg Q\):
\(m < n \Rightarrow y=0\) • \(m = n \Rightarrow y = \frac{a}{b}\) • \(m > n \Rightarrow\) none
