Glossary

Absolute value. The distance of a real number from zero on the number line.

Average rate of change. The ratio of net output change to input change over an interval.

Completing the square. Rewriting a quadratic by adding and subtracting the value needed to form a perfect-square trinomial.

Composite function. The function \((f\circ g)(x)=f(g(x))\), formed by using the output of \(g\) as the input of \(f\).

Data matrix. A rectangular arrangement in which rows represent cases and columns represent variables.

Domain. The set of inputs for which a function is defined.

End behaviour. The behaviour of a function as \(x\to\infty\) and as \(x\to-\infty\).

Equation. A statement that two expressions have equal values.

Equivalent formula. A rearranged equation that expresses the same relationship while isolating a different variable.

Exponent. A number that indicates the power to which a base is raised.

Exponential function. A function \(f(x)=ab^x\), where \(a\ne0\), \(b>0\), and \(b\ne1\).

Factor. An expression multiplied by another expression to produce a given product.

Fraction. A quotient \(a/b\) in which \(a\) is the numerator, \(b\) is the denominator, and \(b\ne0\).

Fractional expression. A quotient of two algebraic expressions, with all values that make the denominator zero excluded.

Function. A rule that assigns exactly one output to every input in its domain.

Hole. A removable discontinuity created when a common factor cancels but its zero remains excluded from the original domain.

Horizontal asymptote. A line \(y=L\) approached by a function as \(x\to\infty\) or \(x\to-\infty\).

Increasing function. A function whose outputs rise as inputs increase on a specified interval.

Inequality. A comparison between quantities using \(<\), \(\le\), \(>\), or \(\ge\).

Interquartile range. The spread of the middle half of the data, \(IQR=Q_3-Q_1\).

Interval. A continuous set of real numbers between specified endpoints.

Inverse function. A function \(f^{-1}\) that reverses a one-to-one function, so \(f^{-1}(f(x))=x\) on the domain of \(f\).

Inverse sine. The function \(\arcsin:[-1,1]\to[-\pi/2,\pi/2]\) satisfying \(\sin(\arcsin x)=x\).

Local extremum. A local maximum or minimum value compared with nearby function values.

Logarithm. For \(b>0\) and \(b\ne1\), \(\log_b(y)=x\) exactly when \(b^x=y\).

Logistic model. A growth model that approaches a limiting carrying capacity, commonly written \(P(t)=d/(1+ke^{-ct})\) for positive constants \(c\), \(d\), and \(k\).

Log-log graph. A graph with logarithmic scales on both axes.

Multiplicity. The number of times a factor corresponding to a zero occurs in a polynomial.

Observational study. A study that records naturally occurring variables without assigning treatments.

One-to-one function. A function in which distinct inputs always produce distinct outputs.

Onto function. A function whose range equals its stated codomain.

Parent function. A basic function whose graph provides a starting shape for a family of related functions.

Polynomial. A finite sum of terms whose variable exponents are nonnegative integers.

Population. The complete collection of cases about which a study seeks information.

Quadratic function. A polynomial function \(f(x)=ax^2+bx+c\) with \(a\ne0\).

Radian. The central angle that intercepts an arc equal in length to the circle’s radius.

Range. The set of outputs a function actually produces.

Rational expression. A fractional expression whose numerator and denominator are polynomials.

Rational number. A number that can be written as \(a/b\), where \(a\) and \(b\) are integers and \(b\ne0\).

Real number. A number represented by a point on the real number line. The set of all real numbers is denoted by \(\mathbb{R}\).

Root. A value that produces a given number when raised to a specified power.

Sample. The subset of the population from which data are collected.

Sample mean. The arithmetic average \(\bar x=(\sum_{i=1}^n x_i)/n\).

Sample standard deviation. The square root of sample variance, \(s=\sqrt{\sum_{i=1}^n(x_i-\bar x)^2/(n-1)}\).

Sampling bias. A systematic tendency for a sampling method to overrepresent or underrepresent parts of the population.

Semi-log graph. A graph with one logarithmic axis and one linear axis.

Set. A collection of distinct objects called elements.

Skewed distribution. A distribution with a longer tail on one side than the other.

Slope. The vertical change divided by the horizontal change between two points on a nonvertical line.

Variable. A recorded characteristic that can differ from one case to another.

Vertical asymptote. A line \(x=a\) approached by a function whose magnitude grows without bound as \(x\) approaches \(a\) from at least one side.

Zero of a polynomial. A value \(c\) for which \(P(c)=0\).