Glossary

amplitude

The maximum distance of the values of a periodic function above or below the midline.


arithmetic sequence

A sequence in which each term is the previous term plus a constant.


complex number

A number in the complex plane. It can be written as \(a + bi\), where \(a\) and \(b\) are real numbers and \(i^2 = \text-1\).

 


degree

The degree of a polynomial in \(x\) is the highest exponent occuring on \(x\) when you write the polynomial out as a sum of non-zero constants times powers of \(x\) (with like terms collected).


$e$ (mathematical constant)

The number \(e\) is an irrational number with an infinite decimal expansion that starts \(2.71828182845\ .\ .\ .\), which is used in finance and science as the base for an exponential function.


end behavior

How the outputs of a function change as we look at input values further and further from 0.

This function shows different end behavior in the positive and negative directions. In the positive direction the values get larger and larger. In the negative direction the values get closer and closer to -3.


even function

A function \(f\) that satisfies the condition \(f(x) = f(\text-x)\) for all inputs \(x\). You can tell an even function from its graph: Its graph is symmetric about the \(y\)-axis.


experimental study

An experimental study collects data by directly influencing something to determine how another thing is changed.


geometric sequence

A sequence in which each term is a constant times the previous term.


horizontal asymptote

The line \(y =c\) is a horizontal asymptote of a function if the outputs of the function get closer and closer to \(c\) as the inputs get larger and larger in either the positive or negative direction. This means the graph gets closer and closer to the line as you move to the right or left along the \(x\)-axis.


identity

An equation which is true for all values of the variables in it.


imaginary number

A number on the imaginary number line. It can be written as \(bi\), where \(b\) is a real number and \(i^2 = \text-1\).


logarithm

The logarithm to base 10 of a number \(x\), written \(\log_{10}(x)\), is the exponent you raise 10 to get \(x\), so it is the number \(y\) that makes the equation \(10^y = x\) true. Logarithms to other bases are defined the same way with 10 replaced by the base, e.g. \(\log_2(x)\) is the number \(y\) that makes the equation \(2^y = x\) true. The logarithm to the base \(e\) is called the natural logarithm, and is written \(\ln(x)\).


logarithmic function

A logarithmic function is a constant multiple of a logarithm to some base, so it is a function given by \(f(x) = k \log_{a}(x)\) where \(k\) is any number and \(a\) is a positive number (10, 2, or \(e\) in this course). The graph of a typical logarithmic function is shown. Although the function grows very slowly, the graph does not have a horizontal asymptote.


margin of error

The maximum expected difference between an estimate for a population characteristic and the actual value of the population characteristic. 


midline

The value halfway between the maximum and minimum values of a period function. Also the horizontal line whose \(y\)-coordinate is that value.


multiplicity

The power to which a factor occurs in the factored form of a polynomial. For example, in the polynomial \((x-1)^2(x+3)\), the factor \(x-1\) has multiplicity 2 and the factor \(x+3\) has multiplicity 1.


natural logarithm

The natural logarithm of \(x\), written \(\ln(x)\), is the log to the base \(e\) of \(x\). So it is the number \(y\) that makes the equation \(e^y = x\) true.


normal distribution

mean = 10. standard deviation = 1

A bell-shaped distribution.

mean = 10. standard deviation = 2

A bell-shaped distribution.

mean = 8. standard deviation = 2

A bell-shaped distribution.

A specific distribution in statistics whose graph is symmetric and bell-shaped, has an area of 1 between the \(x\)-axis and the graph, and has the \(x\)-axis as a horizontal asymptote. 


observational study

An observational study collects data without influencing the subjects directly.


odd function

A function \(f\) that satisfies \(f(x) = \text-f(\text-x)\) for all inputs \(x\). You can tell an odd function from its graph: Its graph is taken to itself when you reflect it across both the \(x\)- and \(y\)-axes. This can also be seen as a 180\(^\circ\) rotation about the origin.


period

The length of an interval at which a periodic function repeats. A function \(f\) has a period, \(p\), if \(f(x+p) = f(x)\) for all inputs \(x\).


periodic function

A function whose values repeat at regular intervals. If \(f\) is a periodic function then there is a number \(p\), called the period, so that \(f(x + p) = f(x)\) for all inputs \(x\).


polynomial

A polynomial function of \(x\)  is a function given by a sum of terms, each of which is a constant times a whole number power of \(x\). The word polynomial is used to refer both to the function and to the expression defining it.


Pythagorean identity

The identity \(\sin^2(x) + \cos^2(x) = 1\) relating the sine and cosine of a number. It is called the Pythagorean identity because it follows from the Pythagorean theorem.


random selection

A selection process by where each item in a set has an equal probability of being selected.


rational function

A rational function is a function defined by a fraction with polynomials in the numerator and denominator. Rational functions include polynomials because a polynomial can be written as a fraction with denominator 1.


real number

A number on the number line.


relative frequency histogram
Histogram. 

A histogram where the height of each bar is the fraction of the entire data set that falls into the corresponding interval (that is, it is the relative frequency with which the data values fall into that interval).


relative maximum

A point on the graph of a function that is higher than any of the points around it.


relative minimum

A point on the graph of a function that is lower than any of the points around it.


sample

A sample is a subset of a population.


sequence

A list of numbers, possibly going on forever, such as all the odd positive integers arranged in order: 1, 3, 5, 7, . . . .


survey

A survey is a set of questions given to people to seek their responses.


term (of a sequence)

One of the numbers in a sequence.


treatment

In an experiment where you are comparing two groups, one of which is being given a treatment and the other of which is the control group without any treatment, the treatment is the value of the variable that is changed for the treatment group.


unit circle

The circle in the coordinate plane with radius 1 and center the origin.


vertical asymptote

The line \(x=a\) is a vertical asymptote for a function \(f\) if \(f\) is undefined at \(x=a\) and its outputs get larger and larger in the negative or positive direction when \(x\) gets closer and closer to \(a\) on each side of the line. This means the graph goes off in the vertical direction on either side of the line.