This Algebra 1 reference sheet provides a concise overview of key concepts and formulas, serving as a valuable resource for students throughout their learning journey. We've organized the information for easy access and understanding, focusing on clarity and practicality. Remember to consult your textbook and teacher for further clarification and detailed examples.
I. Real Numbers and Operations
- Real Numbers: The set of all rational (fractions, decimals, integers) and irrational (non-repeating, non-terminating decimals like π) numbers.
- Properties of Real Numbers:
- Commutative Property: a + b = b + a; a * b = b * a
- Associative Property: (a + b) + c = a + (b + c); (a * b) * c = a * (b * c)
- Distributive Property: a(b + c) = ab + ac
- Identity Property: a + 0 = a; a * 1 = a
- Inverse Property: a + (-a) = 0; a * (1/a) = 1 (where a ≠ 0)
- Order of Operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
II. Variables and Expressions
- Variable: A symbol (usually a letter) representing an unknown value.
- Constant: A fixed value.
- Algebraic Expression: A combination of variables, constants, and operations.
- Terms: Parts of an expression separated by + or - signs.
- Coefficients: The numerical factor of a term.
- Like Terms: Terms with the same variables raised to the same powers. These can be combined.
III. Equations and Inequalities
- Equation: A statement showing two expressions are equal (=).
- Solution: The value(s) that make an equation true.
- Solving Equations: Use inverse operations to isolate the variable.
- Inequalities: Statements showing a relationship between two expressions using < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to).
- Solving Inequalities: Similar to solving equations, but remember to reverse the inequality sign when multiplying or dividing by a negative number.
IV. Linear Equations and Graphs
- Linear Equation: An equation whose graph is a straight line. Often written in the form y = mx + b (slope-intercept form).
- Slope (m): The steepness of a line; calculated as (y₂ - y₁) / (x₂ - x₁).
- y-intercept (b): The point where the line crosses the y-axis.
- x-intercept: The point where the line crosses the x-axis (set y = 0 and solve for x).
- Slope-Intercept Form: y = mx + b
- Point-Slope Form: y - y₁ = m(x - x₁)
- Standard Form: Ax + By = C
V. Systems of Linear Equations
- System of Equations: A set of two or more equations with the same variables.
- Solution to a System: The point(s) where the graphs of the equations intersect.
- Methods for Solving Systems:
- Graphing: Graph both equations and find the intersection point.
- Substitution: Solve one equation for one variable and substitute into the other equation.
- Elimination: Multiply equations by constants to eliminate a variable when adding the equations.
VI. Polynomials
- Polynomial: An expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
- Degree of a Polynomial: The highest power of the variable.
- Monomial: A polynomial with one term.
- Binomial: A polynomial with two terms.
- Trinomial: A polynomial with three terms.
- Adding and Subtracting Polynomials: Combine like terms.
- Multiplying Polynomials: Use the distributive property (FOIL method for binomials).
VII. Factoring
- Factoring: Rewriting an expression as a product of simpler expressions.
- Common Factoring: Factoring out the greatest common factor (GCF).
- Factoring Trinomials: Finding two binomials whose product equals the trinomial (often using the "X" method or trial and error).
- Difference of Squares: a² - b² = (a + b)(a - b)
VIII. Quadratic Equations
- Quadratic Equation: An equation of the form ax² + bx + c = 0 (where a ≠ 0).
- Solving Quadratic Equations:
- Factoring: Factor the quadratic expression and set each factor equal to zero.
- Quadratic Formula: x = [-b ± √(b² - 4ac)] / 2a
- Completing the Square: A method to rewrite the equation into a perfect square trinomial.
This reference sheet offers a starting point for your Algebra 1 studies. Remember to practice regularly and seek help when needed. Consistent effort and understanding of the underlying concepts are key to success.
