Algebra 2 Units 3-5 Flashcards

Questions and Answers

What is the value of the mathematical constant e?

2.71828182845... (correct)

3.14159265359...

1.61803398875...

0.57721566490...

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?

The highest exponent occurring on x when the polynomial is written out as a sum of non-zero constants times powers of x. Signup and view all the answers

What is meant by the end behavior of a function?

How the outputs of a function change as the input values move further from zero. Signup and view all the answers

An even function satisfies the condition f(x) = f(-x) for all inputs of x.

True Signup and view all the answers

Define a geometric sequence.

A sequence in which each term is found by multiplying the previous term by the same number. Signup and view all the answers

What is a horizontal asymptote?

The line y=c is a horizontal asymptote if the outputs of the function approach c as inputs increase or decrease indefinitely. Signup and view all the answers

An identity equation is one that is true for all values of the variables in it.

True Signup and view all the answers

What is an imaginary number?

A number on the imaginary line, written as bi, where b is a real number and i^2 = -1. Signup and view all the answers

What is a logarithm?

The exponent to which the base 10 must be raised to produce x, written as log10(x). Signup and view all the answers

Define a logarithmic function.

A function given by f(x) = k loga(x), where k is any number and a is a positive number. Signup and view all the answers

What is multiplicity in polynomials?

The power to which a factor occurs in the factored form of a polynomial. Signup and view all the answers

What is a natural logarithm?

The logarithm to the base e of x, written as ln(x), such that e^y = x. Signup and view all the answers

An odd function satisfies f(x) = -f(-x) for all inputs of x.

False Signup and view all the answers

What is a polynomial function?

A function given by the sum of terms where each term is a constant times a whole number power of x. Signup and view all the answers

Define a rational function.

A function defined by a fraction with polynomials in the numerator and denominator. Signup and view all the answers

What is a real number?

A number on the number line. Signup and view all the answers

What is a relative maximum?

A point on the graph of a function that is higher than any points around it. Signup and view all the answers

What is a relative minimum?

A point on the graph of a function that is lower than any of the points around it. Signup and view all the answers

Define a sequence.

A list of numbers possibly going on forever. Signup and view all the answers

What is a term of a sequence?

One of the numbers in a sequence. Signup and view all the answers

What is a vertical asymptote?

The line x=a is a vertical asymptote if f is undefined at x=a and outputs grow large when x approaches a. Signup and view all the answers

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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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.