Cheatsheets - Algebra 1 Properties, and Formulas

1. Basic Operations

• Addition: \( a + b \)
• Subtraction: \( a - b \)
• Multiplication: \( a \times b \) or \( ab \)
• Division: \( a \div b \) or \( \frac{a}{b} \)

2. Properties of Operations

• Commutative Property:
• Addition: \( a + b = b + a \)
• Multiplication: \( ab = ba \)
• Associative Property:
• Addition: \( (a + b) + c = a + (b + c) \)
• Multiplication: \( (ab)c = a(bc) \)
• Distributive Property:
• \( a(b + c) = ab + ac \)

3. Solving Equations

• Linear Equation:
  • General Form: \( ax + b = c \)
  • To solve: \( x = \frac{c - b}{a} \)
• Quadratic Equation:
  • Standard Form: \( ax^2 + bx + c = 0 \)
  • Solutions (using Quadratic Formula):
  • \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

4. Inequalities

• Linear Inequality:
  • \( ax + b < c \) or \( ax + b > c \)
  • Solution: Similar to solving equations, but flip the inequality sign when multiplying/dividing by a negative number.

5. Functions

• Function Notation:
  • \( f(x) = mx + b \) (linear function)
• Slope-Intercept Form:
  • \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept.
• Point-Slope Form:
  • \( y - y_1 = m(x - x_1) \)

6. Systems of Equations

• Graphical Method:
  • Solve by finding the intersection point of two lines.
• Substitution Method:
  • Solve one equation for a variable and substitute it into the other.
• Elimination Method:
  • Add or subtract equations to eliminate a variable.

7. Factoring

• Factoring Quadratics:
• \( ax^2 + bx + c = (px + q)(rx + s) \)
• Common Factoring Techniques:
  • Factor out the greatest common factor (GCF).
  • Difference of squares: \( a^2 - b^2 = (a - b)(a + b) \)
  • Perfect square trinomial: \( a^2 \pm 2ab + b^2 = (a \pm b)^2 \)

8. Exponents and Radicals

• Exponent Rules:
  • \( a^m \times a^n = a^{m+n} \)
  • \( \frac{a^m}{a^n} = a^{m-n} \)
  • \( (a^m)^n = a^{mn} \)
• Square Root:
  • \( \sqrt{a^2} = a \)

9. Polynomials

• Addition/Subtraction: Combine like terms.
• Multiplication:
  • \( (a + b)(c + d) = ac + ad + bc + bd \) (FOIL for binomials)
• Degree of a Polynomial: Highest exponent of a term.

10. Statistics

• Mean (Average):
  • \( \text{Mean} = \frac{\text{Sum of Values}}{\text{Number of Values}} \)
• Median: Middle value in a sorted list.
• Mode: Most frequently occurring value.

11. Coordinate Geometry

• Distance Formula:
  • \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
• Midpoint Formula:
  • \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)