Questions and Answers
What is a complex number?
What is the absolute value?
The distance between a number and zero.
Define an expression in mathematics.
A representation of a quantity that may contain numbers, variables, or operation symbols.
What is a variable?
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What is a term in mathematics?
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Define a polynomial.
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What does the degree of a polynomial refer to?
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What is the leading coefficient?
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What are the factors of a monomial?
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What is a prime polynomial?
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If $ab = 0$, then $a = 0$ or $b = 0$.
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What are zeros in a function?
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What are x-intercepts?
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What is a vertical line represented by?
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What is a horizontal line represented by?
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Define an inequality.
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What is the domain of a function?
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What is the range of a function?
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What is function notation?
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What represents direct variation?
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Define inverse variation.
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What does joint variation represent?
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Define an arithmetic sequence.
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What is a geometric sequence?
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Define a permutation.
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What is a combination?
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Define a z-score.
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What is a literal equation?
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What is the standard form of a quadratic equation?
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What is the vertex of a quadratic function?
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What does the quadratic formula do?
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What are increasing/decreasing intervals?
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What are extrema in a function?
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Define end behavior in terms of functions.
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What is the linear parent function?
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What is the quadratic parent function?
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Define the absolute value parent function.
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What is the square root parent function?
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What is the cubic parent function?
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Define the cube root function.
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What is the rational parent function?
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What does the exponential growth parent function represent?
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What is the logarithmic parent function?
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What does translation of a parent function involve?
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What does dilation of a parent function refer to?
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Define reflection of a parent function.
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What is the inverse of a function?
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Define continuity in functions.
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What is an asymptote?
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Define a hole in graphing functions.
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What is the fundamental counting principle?
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What is the permutation formula?
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What is the combination formula?
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What does ∑ (capital sigma) represent?
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Define the empirical rule in statistics.
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Study Notes
Complex Numbers and Absolute Value
- A complex number is represented as a ± bi, combining real (a) and imaginary (bi) parts.
- Absolute value reflects the distance of a number from zero on the number line.
Algebraic Concepts
- An expression constitutes numbers, variables, and operation symbols.
- Variables, like x or y, represent quantities that can adjust in mathematical scenarios.
- Terms can be a single number/variable or a product of both, separated by addition or subtraction.
Polynomials and Their Characteristics
- Polynomials encompass algebraic expressions such as monomials, binomials, and trinomials.
- The degree of a polynomial is defined by the largest exponent or sum of exponents in its terms.
- The leading coefficient is the coefficient of the first term when written in decreasing order of exponents.
Functions and Graphs
- Zeros of a function indicate where the function equals zero (f(x) = 0).
- X-intercepts occur where the graph crosses the x-axis.
- Vertical lines have an equation x = a with undefined slopes; horizontal lines are y = c with a zero slope.
Relationships and Variations
- Inequalities compare two quantities in algebraic sentences.
- Domain refers to all possible values that an independent variable can take.
- Range represents all possible values of a dependent variable.
Variations Defined
- Direct variation follows the formula y = kx, indicating a constant ratio between x and y.
- Inverse variation is expressed as k = xy, indicating a constant product relationship.
- Joint variation presents y = kxz, where one variable depends on two or more others.
Sequences and Combinatorics
- An arithmetic sequence has a consistent difference between consecutive terms.
- A geometric sequence involves multiplying each term by a constant ratio.
- Permutations are ordered arrangements; combinations involve selection without regard to order.
Statistical Measures
- Z-score indicates how many standard deviations an element is from the mean.
- Literal equations predominantly consist of variables, often used to express relationships.
Quadratic Functions
- Standard form for quadratic equations is ax² + bx + c = 0.
- Vertex of a quadratic function can be found using special formulas based on its standard and vertex forms.
- The quadratic formula x = -b ± √(b² - 4ac)/2a solves quadratic equations.
Function Behavior
- Increasing/decreasing intervals characterize the behavior of functions over specified ranges.
- Extrema denote the maximum and minimum values of functions, either locally or globally.
- End behavior describes the function's value as x approaches positive or negative infinity.
Parent Functions
- Parent functions represent the simplest forms:
- Linear: f(x) = x
- Quadratic: f(x) = x²
- Absolute Value: f(x) = |x|
- Square Root: f(x) = √x
- Cubic: f(x) = x³
- Cube Root: f(x) = ³√x
- Rational: f(x) = 1/x
- Exponential Growth: f(n) = bⁿ (b > 1)
- Logarithmic: f(x) = logₙ(x) (n > 1)
Transformations and Inverses
- Translations of parent functions shift graphs horizontally/vertically.
- Dilations stretch or compress graphs either vertically or horizontally.
- Reflections flip graphs over axes, altering their orientations.
Continuity and Discontinuities
- Continuity indicates a function is unbroken at all points in its domain.
- Asymptotes are lines that approach a curve but do not intersect it, indicating discontinuity.
- Holes in functions occur where both the numerator and denominator are zero, causing discontinuity.
Counting Principles
- Fundamental counting principle: If there are m ways for one event and n ways for another, there are m ∙ n ways for both.
- Permutations and combinations have specific formulas: nPr for permutations and nCr for combinations.
Statistical Distribution
- The empirical rule (68-95-99.7 rule) describes the distribution of elements in a normal distribution curve.
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