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Questions and Answers
What is the primary focus of algebra?
What type of equation has the highest power of the variable(s) as 1?
What property of equations allows you to add or subtract the same value to both sides?
What is the set of input values for which a function is defined?
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What is the point at which a graph crosses the x-axis?
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What is the equation of a line in the form y = mx + b?
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What is the process of combining two or more functions to create a new function?
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What is the coefficient of a term in an algebraic expression?
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Study Notes
Algebra
Definition
- Algebra is a branch of mathematics that deals with variables and their relationships, often expressed through the use of symbols, equations, and functions.
Key Concepts
- Variables: Letters or symbols that represent unknown values or quantities.
- Constants: Numbers that do not change value.
- Algebraic Expressions: Combinations of variables, constants, and mathematical operations.
- Equations: Statements that express the equality of two algebraic expressions.
Types of Equations
- Linear Equations: Equations in which the highest power of the variable(s) is 1.
- Quadratic Equations: Equations in which the highest power of the variable(s) is 2.
- Polynomial Equations: Equations involving variables and coefficients of whole-number exponents.
Solving Equations
- Addition and Subtraction Properties: Adding or subtracting the same value to both sides of an equation does not change its solution.
- Multiplication and Division Properties: Multiplying or dividing both sides of an equation by a non-zero value does not change its solution.
- Inverse Operations: Reversing the operation to isolate the variable (e.g., adding the opposite to both sides to eliminate a term).
Functions
- Domain: The set of input values for which a function is defined.
- Range: The set of output values of a function.
- Composition of Functions: Combining two or more functions to create a new function.
Graphing
- Coordinate Plane: A two-dimensional plane with x and y axes used to graph algebraic equations.
- X-Intercept: The point at which a graph crosses the x-axis.
- Y-Intercept: The point at which a graph crosses the y-axis.
- Slope-Intercept Form: The equation of a line in the form y = mx + b, where m is the slope and b is the y-intercept.
Algebra
Definition
- Deals with variables and their relationships using symbols, equations, and functions.
Key Concepts
Variables and Constants
- Variables: represent unknown values or quantities, denoted by letters or symbols.
- Constants: numbers with fixed values, do not change.
Algebraic Expressions and Equations
- Algebraic Expressions: combinations of variables, constants, and mathematical operations.
- Equations: statements expressing equality of two algebraic expressions.
Types of Equations
Linear, Quadratic, and Polynomial Equations
- Linear Equations: highest power of variables is 1, e.g., 2x + 3 = 5.
- Quadratic Equations: highest power of variables is 2, e.g., x^2 + 4x + 4 = 0.
- Polynomial Equations: involve variables and coefficients of whole-number exponents.
Solving Equations
Properties and Inverse Operations
- Addition and Subtraction Properties: adding or subtracting same value to both sides doesn't change solution.
- Multiplication and Division Properties: multiplying or dividing both sides by non-zero value doesn't change solution.
- Inverse Operations: reversing operations to isolate variables, e.g., adding opposites to eliminate terms.
Functions
Domain, Range, and Composition
- Domain: set of input values for which a function is defined.
- Range: set of output values of a function.
- Composition of Functions: combining two or more functions to create a new function.
Graphing
Coordinate Plane and Intercepts
- Coordinate Plane: two-dimensional plane with x and y axes for graphing algebraic equations.
- X-Intercept: point where graph crosses x-axis.
- Y-Intercept: point where graph crosses y-axis.
- Slope-Intercept Form: equation of a line in form y = mx + b, where m is slope and b is y-intercept.
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Description
Test your understanding of algebra fundamentals, including variables, constants, algebraic expressions, and equations.
