Appendix A

Continuos Graph

A continuous graph is when the points on a graph are connected, and it is clear that they are supposed to be connected.

Discrete Graph

A discrete graph is a sequence of separate points.

Function

A function is a relationship in which for each input value there is one and only one output value. One example of a function is: f(x)=x+4

Geometric Sequence

A geometric sequence is a sequence that is generated by a multiplier. This means that each term of a geometric sequence can be found by multiplying the previous term by a constant.

Linear Function

A linear function is a polynomial function of degree one or zero, with general equation f(x)=a(x-h)+k. The graph of a linear function is a line.

y-intercept

A point where a graph intersects the y-axis. In two dimensions the coordinates of the y-intercept are (0, y). In three dimensions, the coordinates are (0, y, 0).

Recursive Sequence

A recursive sequence is a sequence in which each term depends on the term(s) before it. The equation of a recursive sequence requires at least one term to be specified.

Sequence

A sequence is an enumerated set of objects or numbers that follow one another in a specific pattern.

Term

A single number, variable, or product of numbers and variables. A component of a sequence.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers where the difference between the sequential terms is constant. Each term in the sequence can be generated by adding the common difference to the previous term.

First Term

An equation of a sequence written in first term form uses the first term of the sequence and its common difference or ratio. The first term form of an arithmetic sequence is an=a1+d(n-1), where a1 is the first term and d is the common difference. The first term form of a geometric sequence is an=a1r^n-1, where a1 is the first term and r is the common ratio.

Explicit Equation (for a sequence)

An explicit equation is an equation for a term in a sequence that determines the value of any term t(n) directly from n, without necessarily knowing any other terms in the sequence.

Exponential Function

An exponential function in this course has an equation of the form, y=ab^x+c, where a is the initial value, b is positive and is the multiplier, and y=c is the equation of the horizontal asymptote.

Common Ratio

Common ratio is another name for the multiplier or generator of a geometric sequence. It is the number to multiply one term by to get the next one. In the sequence: 96, 48, 24,... the common ratio is 1/2.

Domain

Domain is the set of all input values for a relation or function. For variables, the domain is the set of numbers the variable may represent.

Multiplier

In a geometric sequence the number multiplied times each term to get the next term is called the multiplier or the common ratio or the generator. The multiplier is also also the number you can multiply by in order to increase or decrease an amount by a given percentage in one step. For example, to increase a number by 4% the multiplier is 1.04. The multiplier for decreasing by 4% is 0.96.

Term Number

In a sequence, a number that gives the position of a term in the sequence. A replacement value for the independent variable in a function that determines the sequence.

Practice Problem #1 Determine if the following sequences are arithmetic, geometric, or neither: a. -7, -3, 1, 5, 9,... b. -64, -16, -4, -1,... c. 1, 0, 1, 4, 9 d. 0, 2, 4,...

Practice Problem #1 Answer: a. arithmetic b. geometric c. neither, d. arithmetic

Practice Problem #2 Create an explicit equation for each recursively-defined sequence below. a. a1 = 17, an + 1 = an - 7 b. t(1) = 3, t(n+1) = 5 * t(n)

Practice Problem #2 Answer: a .t(n) = 24 - 7n b. t(n) = 3/5(5)^n

Common Difference

The difference between consecutive terms of arithmetic sequence or the generator of the sequence. The common difference is positive when the sequence increases, and negative when the sequence decreases. For example in the sequence: 2, 4, 6,... The common difference is 2.

Generator

The generator of a sequence tells what you do to each of term to get the next term. The generator only tells you how to find the following term, when you already know at least one term.

Initial Value

The initial value of a sequence is the first term of the sequence.

Range

The range of a function is the set of possible outputs for a function. It consists of all the values of the dependent variable, that is every number y can represent for the function f(x)=y.

Function Notation

When a rule expressing a function is written using function notation, the function is given a name, most commonly "f," "g," or "h." The notation f(x) represents the output of a function, named f, when x is the input.

t(0)

t(0) represents the 0th term of a sequence. This comes before the first term of the sequence. To find the 0th term, you need to do the reverse of what the multiplier or common difference is. For example the sequence: 3, 6, 9, 12,... the common difference is +3; therefore, you subtract 3 to find t(0) which equals 0 in this example.