Questions and Answers
What is the value of the mathematical constant e?
Define an arithmetic sequence.
A sequence in which each term is found by adding the same number to the previous term.
What is a complex number?
A number of the form a + bi where a and b are real numbers and i is the square root of -1.
What does the degree of a polynomial refer to?
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What is meant by the end behavior of a function?
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An even function satisfies the condition f(x) = f(-x) for all inputs of x.
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Define a geometric sequence.
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What is a horizontal asymptote?
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An identity equation is one that is true for all values of the variables in it.
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What is an imaginary number?
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What is a logarithm?
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Define a logarithmic function.
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What is multiplicity in polynomials?
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What is a natural logarithm?
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An odd function satisfies f(x) = -f(-x) for all inputs of x.
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What is a polynomial function?
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Define a rational function.
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What is a real number?
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What is a relative maximum?
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What is a relative minimum?
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Define a sequence.
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What is a term of a sequence?
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What is a vertical asymptote?
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Study Notes
Mathematical Constants and Sequences
- e (Mathematical Constant): An irrational number approximately equal to 2.71828182845, important in finance and science as the base for exponential functions.
- Arithmetic Sequence: A sequence where each term is obtained by adding a constant value to the previous term.
- Geometric Sequence: A sequence where each term is found by multiplying the previous term by a constant factor.
Complex Numbers and Functions
- Complex Number: A number represented as a + bi, with a and b being real numbers, and i representing the square root of -1.
- Even Function: A function f where f(x) = f(-x) for all x; characterized by symmetry around the y-axis.
- Odd Function: A function f where f(x) = -f(-x) for all x; exhibits symmetry through a 180-degree rotation about the origin.
Polynomial and Rational Functions
- Degree of a Polynomial: The highest exponent of the variable x in a polynomial expression when arranged as a sum of terms.
- Polynomial Function: A function defined by a sum of terms where each term is a constant multiplied by a whole number power of x.
- Rational Function: A function defined as a fraction of polynomials, encompassing polynomials themselves as they can be written with a denominator of 1.
Properties of Functions
- End Behavior of a Function: Describes how function outputs behave as input values become increasingly large or small, approaching specific behavior at infinity.
- Horizontal Asymptote: The line y = c indicates that functions approach the value c as inputs grow large or small, affecting the graph's behavior at extremes.
- Vertical Asymptote: The line x = a indicates points where the function is undefined and where outputs tend towards infinity as they approach a from either side.
Logarithms and Logarithmic Functions
- Logarithm: The exponent needed for a given base (e.g., base 10) to yield a particular number; the natural logarithm is denoted as ln(x).
- Logarithmic Function: Defined as f(x) = k loga(x), grows slowly and does not approach a horizontal asymptote.
Miscellaneous Definitions
- Identity Equation: An equation that holds true for all possible values of the variables.
- Imaginary Number: A number expressed as bi, where b is real, and i is defined such that i^2 = -1.
- Multiplicity: Refers to the number of times a specific factor appears in the factored form of a polynomial.
Graphical Points and Sequences
- Relative Maximum: A point on a graph that is higher than all nearby points, showcasing a peak in the function's values.
- Relative Minimum: A point on a graph that is lower than surrounding points, indicating a trough in the function's values.
- Sequence: An ordered list of numbers that can theoretically extend indefinitely, such as the series of odd positive integers.
- Term of a Sequence: Any individual number within a sequence.
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Description
Test your understanding of key concepts in Algebra 2 with these flashcards covering Units 3 to 5. Each card provides definitions and explanations of important terms like the mathematical constant e, arithmetic sequences, and complex numbers. Perfect for quick revision and concept reinforcement.