Questions and Answers
What does it mean to evaluate an expression?
To figure out or compute the value of an expression.
What does it mean to simplify an expression?
To rewrite an expression as simply as possible using arithmetic and algebra rules.
What does it mean to solve in mathematics?
To find all solutions to an equation, inequality, or system of equations and/or inequalities.
What is a variable?
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What is an expression in mathematics?
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What is the Identity Property of Addition?
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What is the Inverse Property of Addition?
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What is the Commutative Property of Addition?
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What is the Associative Property of Addition?
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What is the Identity Property of Multiplication?
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What does the Distributive Property state?
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What does a trinomial consist of?
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What does the term 'inequality' refer to?
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What is the definition of a consistent system in mathematics?
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What is a polynomial?
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What is a quadratic function?
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Study Notes
Evaluating and Simplifying Expressions
- Evaluate means to compute a mathematical expression, such as determining that 12 + 5 simplifies to 17.
- Simplify refers to rewriting an expression in its simplest form using arithmetic and algebra rules.
Solving Variables and Expressions
- Solve involves finding all solutions to equations, inequalities, or systems thereof.
- A variable is a symbol representing a quantity that can change, often depicted as letters like x or y.
- An expression is a mathematical combination of numbers and variables using operations, excluding equality and inequality signs.
Properties of Operations
- Identity Property of Addition: a + 0 = a, meaning adding zero does not change the value of a.
- Inverse Property of Addition: a + (-a) = 0, indicating each number has an opposite that results in zero when added.
- Commutative Property of Addition: a + b = b + a, demonstrating that the order of addition does not affect the sum.
- Associative Property of Addition: (a + b) + c = a + (b + c), showing that grouping does not alter the sum.
Properties of Multiplication
- Identity Property of Multiplication: a * 1 = a, meaning multiplying by one retains the original value.
- Inverse Property of Multiplication: a * (1/a) = 1, signifying that each number has a reciprocal that results in one.
- Commutative Property of Multiplication: ab = ba, depicting that the order of multiplication doesn't matter.
- Associative Property of Multiplication: (ab)c = a(bc), indicating that how factors are grouped does not change the product.
Algebraic Properties and Relations
- Distributive Property: a(b + c) = ab + ac; demonstrates distributing a term across parentheses.
- Substitution Property: If a = b, then b can replace a in equations or inequalities.
- Reflexive Property of Equality: a = a, indicating any quantity equals itself.
- Symmetric Property of Equality: If a = b, then b = a, showing equality is interchangeable.
- Transitive Property of Equality: If a = b and b = c, then a = c, establishing a chain of equality.
Inequalities and Their Properties
- Inequality is an algebraic statement comparing two quantities.
- Graph of an Inequality shows solutions visually on a coordinate plane.
- Transitive Property for Inequality: If a < b and b < c, then a < c (and similarly for greater than).
- Addition/Subtraction Property: Adding or subtracting the same value on both sides of an inequality doesn't change it.
Equations and Systems
- An equation is a statement that equates two mathematical expressions.
- A trinomial is a polynomial with three distinct terms.
- A consistent system of equations has at least one solution, while a dependent system must have infinitely many.
- An inconsistent system of equations has no solutions.
Functions and Their Characteristics
- A quadratic function can be represented in the form ax² + bx + c, where a ≠ 0.
- The vertex is the point of a parabola where it turns; the axis of symmetry is the vertical line through the vertex.
- The discriminant provides information on the roots of a quadratic equation within the quadratic formula: x = (-b ± √D) / (2a), where D is the discriminant.
- Roots of a function are solutions where f(x) = 0, which may be real or complex.
Graphing Concepts
- The end behavior of a graph reflects how it behaves as x approaches infinity or negative infinity.
- x-intercept and y-intercept are points where the graph crosses the respective axes.
- The average rate of change measures how much a function's output changes with respect to its input over a given interval.
Advanced Algebra Topics
- Linear programming utilizes systems of inequalities to optimize functions under constraints.
- Completing the Square is a method for rewriting quadratic equations as perfect square trinomials.
- The vertical shift and horizontal shift describe movements of graphs up, down, or laterally.
- Extrema refer to maximum or minimum values of a function, which can be classified using relative maximum and minimum.
Representation of Functions
- Various forms to represent quadratic functions include graphical, symbolic, and tabular forms.
- Complex numbers, such as 3 - 2i, combine real and imaginary numbers and contribute to the solutions of certain equations.
- Polynomial functions are made up of the sum or difference of terms with variables raised to nonnegative integer powers.
General Definitions
- Factor: An integer or algebraic expression that divides another expression evenly.
- Degree of a Monomial: The exponent of the variable; for multiple variables, the degree is the sum of exponents.
- Product is defined as the result of multiplying quantities.
These notes summarize essential mathematical concepts and terms crucial for Algebra 2, providing a solid foundation for final exam preparation.
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Prepare for your Algebra 2 final exam with these flashcards. Each card covers essential terms and definitions that will help reinforce your understanding of key concepts like evaluation, simplification, and solving equations. Great for review and building confidence ahead of the exam!
