Algebra Reference: Parent Functions and Core Properties

Linear and Basic Line Forms

• Linear parent: $y = x)$
• Domain & Range: $(-\infty, \infty)$ for both
• Odd function: $f(-x) = -f(x)$
• Point-slope form: $y - y<em>1 = m(x - x</em>1)$
• Slope-intercept form: $y = mx + b$

Constant and Basic Linearity

• Constant function: $y = c$
• Domain: $(-\infty, \infty)$; Range: {$c$}

Quadratic and Polynomial Functions

• Standard form: $y = ax^2 + bx + c$
• Vertex form: $y = a(x - h)^2 + k$
• Domain: $(-\infty, \infty)$; Range depends on orientation (up if a>0, down if a<0)

Exponential Functions

• Standard form: $y = ab^x$
• Domain: $(-\infty, \infty)$
• Range: $(0, \infty)$ if a>0
• Horizontal asymptote: $y = 0$ (as x → -∞ when base > 0)

Logarithmic Functions

• Standard form: $y = \log_b x$ (including natural log $\ln x$ for base $e$)
• Domain: $(0, \infty)$; Range: $(-\infty, \infty)$
• Vertical asymptote: $x = 0$
• Base restrictions: b>0, b\neq 1

Radical Functions

• Even roots (e.g., $\sqrt{x}$): domain $[0, \infty)$
• Odd roots (e.g., $\sqrt[3]{x}$): domain $(-\infty, \infty)$
• Basic idea: inverse of appropriate power functions

Rational Functions

• Form: $\dfrac{P(x)}{Q(x)}$
• Domain excludes zeros of the denominator $Q(x) = 0$
• Behavior governed by degrees of numerator and denominator; horizontal/oblique asymptotes accordingly

Exponent Rules

• $a^m a^n = a^{m+n}$
• $(a^m)^n = a^{mn}$
• $\dfrac{a^m}{a^n} = a^{m-n}$
• $(ab)^n = a^n b^n$
• $a^0 = 1$
• $a^{-n} = \dfrac{1}{a^n}$

Logarithm Rules

• $\log<em>b (xy) = \log</em>b x + \log_b y$
• $\log<em>b \left(\dfrac{x}{y}\right) = \log</em>b x - \log_b y$
• $\log<em>b (x^k) = k \log</em>b x$
• $\log<em>b (a^n) = n \log</em>b a$
• $e^{\ln x} = x\;\text{and}\; \ln (e^x) = x$

Radical Rules

• $\sqrt[m]{a}\;\sqrt[m]{b} = \sqrt[m]{ab}$ compatible with index rules
• $\big(\sqrt[n]{a}\big)^m = a^{m/n}$ if defined
• $\sqrt{a^2} = |a|$

Factoring (Techniques and Patterns)

• Greatest common factor (GCF) factoring
• Difference of squares: $a^2 - b^2 = (a - b)(a + b)$
• Difference of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$
• Sum of cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
• Factoring trinomials via grouping or simple product-sum methods

Quick Example Patterns (from common practice)

• Example factoring: $2x^2 - 5x - 3 = (2x + 1)(x - 3)$
• Example factoring by difference of squares: $x^2 - 9 = (x - 3)(x + 3)$
• Example factoring by difference of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$
• Example sum of cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

Quick References for Last-Minute Review

• Always check domain restrictions first for rational, radical, and logarithmic forms
• For exponents, simplify to common base when possible
• For factoring, start with GCF, then move to special products (diff. of squares, cubes) before trying to factor quadratics
• Use point-slope form to quickly write equations from a point and slope
• Recognize standard forms to identify parent functions quickly: linear, quadratic, exponential, logarithmic, radical, rational