Questions and Answers
What is the slope-intercept form of a line?
What is Standard Form?
Ax+By=C
What is Point-Slope Form?
y - y1 = m(x - x1)
What are real numbers?
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The order of operations is defined as Parentheses, Exponents, Addition, Subtraction, Multiplication, and Division.
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What does a 'No Solution' in a linear equation indicate?
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What does 'All Real Numbers' mean in solving a linear equation?
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Define a relation.
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What is the domain in a function?
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What is the range in a function?
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A function is a relation where no input has exactly one output.
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What indicates if a relation is a function using the Vertical Line Test?
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What does f(x) mean?
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What is the formula for slope?
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What does m represent in slope?
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What does the variable b represent in the slope-intercept form?
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How do you find an equation of a line that passes through two points?
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What does the Discriminant tell you?
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If b² - 4ac > 0, the equation has two imaginary solutions.
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If b² - 4ac = 0, the equation has two real solutions.
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If b² - 4ac < 0, the equation has two real solutions.
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What is the Quadratic Formula?
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Study Notes
Algebra Concepts and Definitions
- Slope-intercept Form: Expressed as y = mx + b, where m is the slope and b is the y-intercept.
- Standard Form: Written as Ax + By = C, encompassing all linear equations.
- Point-Slope Form: Given as y - y1 = m(x - x1) for a line passing through point (x1, y1) with slope m.
- Real Numbers: Consist of both rational and irrational numbers, encompassing all possible values along the number line.
- Order of Operations (PEMDAS): A mnemonic for solving expressions in the order of Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).
Linear Equations
- No Solution: Occurs when contradictory equations, e.g., 10 = -3, arise when solving.
- All Real Numbers: Happens when the equation simplifies to a true statement, such as 0 = 0.
- Relation: A pairing of input and output values, typically represented as coordinate points (x, y).
- Domain: The set of all possible input values (x values) for a relation.
- Range: The collection of all output values (y values) corresponding to the domain.
Functions
- Function: A relation where each input has exactly one output; x values cannot repeat.
- Vertical Line Test: A graphical method to determine if a relation is a function; a vertical line should intersect the graph at most once.
- Function Notation: Represented as f(x), indicating the value of the function for a given x.
Slope and Line Characteristics
- Slope (m): Defined as rise/run, or (y2 - y1) / (x2 - x1).
- Positive Slope: Indicates an upward line.
- Negative Slope: Represents a downward line.
- Zero Slope: Characterizes a horizontal line.
- Undefined Slope: Pertains to a vertical line, denoted as m = number/0.
- Parallel Lines: Lines with identical slopes that never intersect.
- Perpendicular Lines: Lines intersecting at right angles; their slopes are opposite reciprocals, e.g., m1 * m2 = -1.
Quadratic Functions and Equations
- Quadratic Formula: x = -b ± √(b² - 4ac) / 2a, used to find roots of quadratic equations.
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Discriminant (b² - 4ac): Indicates the nature of the roots of a quadratic equation;
- Greater than zero indicates two real solutions.
- Equal to zero indicates one real solution.
- Less than zero indicates two imaginary solutions.
- Completing the Square: A method for solving quadratic equations, rearranging them into a perfect square trinomial.
Graphing Quadratics
- Vertex: The highest or lowest point of a parabola formed by a quadratic function.
- Axis of Symmetry: The vertical line that divides the parabola into two symmetrical halves, passing through the vertex.
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Graphing Quadratic Inequalities:
- Parabola drawn based on the inequality type; dashed for > or <, solid for ≥ or ≤.
- Test points inside the parabola to determine shaded regions.
Polynomial Theorems
- Rational Zero Theorem: Provides potential rational roots based on the factors of the constant term and leading coefficient.
- Factor Theorem: States that (x - k) is a factor of polynomial f(x) if f(k) = 0.
- Remainder Theorem: The remainder of f(x) divided by (x - k) is f(k).
- Fundamental Theorem of Algebra: Asserts that any polynomial of degree n has at least one complex root.
Complex Numbers
- Complex Conjugates: Two complex numbers of form a + bi and a - bi; their product yields a real number.
- Absolute Value of Complex Numbers: Calculated as |z| = √(a² + b²), where a and b are real and imaginary components respectively.
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Test your knowledge of key Algebra 2 concepts with these flashcards. Each card provides a definition of important terms, including different forms of equations and the order of operations. Perfect for students looking to reinforce their understanding of algebraic principles.