Equations: Definition, Types, and Solving

Equations

• Defining Equations: An equation is a statement that says two expressions are equal.
• Types of Equations:
  • Linear Equations: ax + by = c, where a, b, and c are constants.
  • Quadratic Equations: ax^2 + bx + c = 0, where a, b, and c are constants.
  • Exponential Equations: a^x = b, where a is a constant and x is the variable.
• Solving Equations:
  • Addition and Subtraction Properties: Add or subtract the same value to both sides of the equation.
  • Multiplication and Division Properties: Multiply or divide both sides of the equation by the same non-zero value.

Functions

• Defining Functions: A function is a relation between a set of inputs (domain) and a set of possible outputs (range).
• Types of Functions:
  • Linear Functions: f(x) = mx + b, where m is the slope and b is the y-intercept.
  • Quadratic Functions: f(x) = ax^2 + bx + c, where a, b, and c are constants.
  • Exponential Functions: f(x) = a^x, where a is a constant.
• Function Operations:
  • Composition: (f ∘ g)(x) = f(g(x))
  • Inverse: f^(-1)(x) is the function that reverses f(x)

Systems of Equations

• Defining Systems of Equations: A system of equations is a set of two or more equations with variables.
• Methods for Solving Systems:
  • Substitution Method: Solve one equation for one variable and substitute into the other equation.
  • Elimination Method: Add or subtract the equations to eliminate one variable.
  • Graphing Method: Graph the equations on the same coordinate plane and find the point of intersection.

Graphing

• Coordinate Plane: A graph with an x-axis and a y-axis that intersect at the origin (0,0).
• Graphing Linear Equations:
  • Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept.
  • Standard Form: Ax + By = C, where A, B, and C are constants.
• Graphing Quadratic Equations:
  • Vertex Form: f(x) = a(x - h)^2 + k, where (h,k) is the vertex.
  • Factored Form: f(x) = a(x - r)(x - s), where r and s are the x-intercepts.

Quadratic Equations

• Defining Quadratic Equations: A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants.
• Methods for Solving Quadratic Equations:
  • Factoring: Factor the equation into the product of two binomials.
  • Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / 2a
  • Graphing: Graph the equation and find the x-intercepts.

All Algebra 1 Topics for EOC Review

• Algebraic Properties:
  • Commutative Property: a + b = b + a
  • Associative Property: (a + b) + c = a + (b + c)
  • Distributive Property: a(b + c) = ab + ac
• Inequalities:
  • Linear Inequalities: ax + by >, <, ≥, or ≤ c
  • Quadratic Inequalities: ax^2 + bx + c >, <, ≥, or ≤ 0
• Functions and Relations:
  • Domain and Range: The set of inputs and possible outputs of a function.
  • Function Operations: Composition, Inverse, and Identity.
• Systems of Equations and Inequalities:
  • Substitution and Elimination Methods for solving systems of equations.
  • Graphing systems of inequalities on a coordinate plane.

Equations

• An equation is a statement that says two expressions are equal.
• Linear Equations: ax + by = c, where a, b, and c are constants.
• Quadratic Equations: ax^2 + bx + c = 0, where a, b, and c are constants.
• Exponential Equations: a^x = b, where a is a constant and x is the variable.

Solving Equations

• Addition and Subtraction Properties: Add or subtract the same value to both sides of the equation.
• Multiplication and Division Properties: Multiply or divide both sides of the equation by the same non-zero value.

Functions

• A function is a relation between a set of inputs (domain) and a set of possible outputs (range).
• Linear Functions: f(x) = mx + b, where m is the slope and b is the y-intercept.
• Quadratic Functions: f(x) = ax^2 + bx + c, where a, b, and c are constants.
• Exponential Functions: f(x) = a^x, where a is a constant.
• Function Operations:
  • Composition: (f ∘ g)(x) = f(g(x))
  • Inverse: f^(-1)(x) is the function that reverses f(x)

Systems of Equations

• A system of equations is a set of two or more equations with variables.
• Methods for Solving Systems:
  • Substitution Method: Solve one equation for one variable and substitute into the other equation.
  • Elimination Method: Add or subtract the equations to eliminate one variable.
  • Graphing Method: Graph the equations on the same coordinate plane and find the point of intersection.

Graphing

• Coordinate Plane: A graph with an x-axis and a y-axis that intersect at the origin (0,0).
• Graphing Linear Equations:
  • Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept.
  • Standard Form: Ax + By = C, where A, B, and C are constants.
• Graphing Quadratic Equations:
  • Vertex Form: f(x) = a(x - h)^2 + k, where (h,k) is the vertex.
  • Factored Form: f(x) = a(x - r)(x - s), where r and s are the x-intercepts.

Quadratic Equations

• A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants.
• Methods for Solving Quadratic Equations:
  • Factoring: Factor the equation into the product of two binomials.
  • Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / 2a
  • Graphing: Graph the equation and find the x-intercepts.

Algebraic Properties

• Commutative Property: a + b = b + a
• Associative Property: (a + b) + c = a + (b + c)
• Distributive Property: a(b + c) = ab + ac

Inequalities

• Linear Inequalities: ax + by >, <, ≥, or ≤ cx + dy