Algebra 1 – Complete Reference Guide

Real Number System

Rational: Can be written as $a/b$. Includes integers, terminating & repeating decimals.

Irrational: Non-terminating, non-repeating decimals ($\pi$).

Properties of Algebra

Real Number Properties

Commutative: $a+b=b+a$

Associative: $(a+b)+c=a+(b+c)$

Distributive: $a(b+c)=ab+ac$

Identity: Add: 0, Mult: 1

Inverse: Add: $-a$, Mult: $1/a$

Properties of Equality

Addition/Sub: If $a=b$, then $a+c=b+c$.

Mult/Div: If $a=b$, then $ac=bc$.

Linear Equations & Functions

Key Forms

Slope-Intercept: $y = mx + b$

Standard: $Ax + By = C$

Point-Slope: $y - y_1 = m(x - x_1)$

Slope Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Types of Slope

Parallel & Perpendicular

Parallel lines ($||$): Slopes are equal ($m_1 = m_2$).

Perpendicular lines ($\perp$): Slopes are negative reciprocals ($m_1 \cdot m_2 = -1$).

Graphing Strategies

Slope-intercept: Plot $b$ (y-int), then use $m$ (Rise/Run).

Intercepts: Set $x=0$ to find y-int; set $y=0$ to find x-int.

Solving Inequalities

Solve like equations, but FLIP symbol when mult/div by negative number.

Compound Inequalities

AND ($x > -2$ AND $x < 5$): Intersection (overlap).

OR ($x < -2$ OR $x> 5$): Union (moving apart).

Systems of Equations

Methods

Graphing: Find intersection point.

Substitution: Solve for one var, plug into other eq.

Elimination: Add eqs to cancel a variable.

Solutions

One Sol: Intersecting

No Sol: Parallel Lines ($m_1=m_2, b_1 \neq b_2$)

Inf Sols: Same Line ($m_1=m_2, b_1=b_2$)

Exponents & Polynomials

Exponent Rules

Product: $x^a \cdot x^b = x^{a+b}$

Quotient: $\frac{x^a}{x^b} = x^{a-b}$

Power: $(x^a)^b = x^{ab}$

Negative: $x^{-n} = \frac{1}{x^n}$

Zero: $x^0 = 1$ ($x \neq 0$)

Polynomials

Standard Form: Terms in descending order of degree.
$Ax^2 + Bx + C$

Adding: Combine like terms.

Multiplying (Binomials): FOIL (First, Outer, Inner, Last).
$(x+3)(x-2) = x^2 - 2x + 3x - 6 = x^2 + x - 6$

Factoring

GCF: Pull out largest common factor first!
$4x^2 + 8x = 4x(x+2)$

Diff of Squares: $a^2 - b^2 = (a-b)(a+b)$
$x^2 - 9 = (x-3)(x+3)$

Trinomials ($x^2+bx+c$): Find factors of $c$ that add to $b$.
$x^2 + 5x + 6 = (x+2)(x+3)$

Slip & Slide ($ax^2+bx+c$): Mult $a \cdot c$, factor, then divide.

Quadratic Functions

$y = ax^2 + bx + c$ (Parabola)

Key Features

Vertex: $x = -b/(2a)$, plug in to find $y$.

Axis of Sym: $x = -b/(2a)$

Solutions (Roots): Quadratic Formula:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Discriminant ($b^2-4ac$):
$>0$: 2 real solutions
$=0$: 1 real solution
$<0$: 0 real solutions

ALGEBRA 1 Complete Reference