Precalculus vocabulary & definitions

An absolute maximum or minimum is the greatest or least output over the entire domain being considered. It may not exist.

The change in output over an interval divided by the change in input: $\frac{f(b)-f(a)}{b-a}$, where $a\ne b$. It is the slope of the secant line through the endpoints.

The set of points at a fixed positive distance $r$ from a center $(h,k)$. Its equation is $(x-h)^2+(y-k)^2=r^2$.

A plane with perpendicular horizontal x- and vertical y-axes used to locate points.

A function is decreasing on an interval when its outputs get smaller as inputs increase through that interval.

The expression $\frac{f(x+h)-f(x)}{h}$, where $h\ne0$ and the function values exist. It gives an average rate of change between $x$ and $x+h$.

The distance between $(x_1,y_1)$ and $(x_2,y_2)$ is $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$.

The set of all allowed input values of a function or relation.

A function satisfying $f(-x)=f(x)$ for every $x$ in its domain, with $-x$ also in the domain. Its graph is symmetric about the y-axis.

Numbers, variables, and operations combined without an equality or inequality statement.

A number or expression multiplied by another to form a product. Factoring rewrites an expression as a product.

A relationship assigning exactly one output to each input in its domain. Different inputs may have the same output.

Notation such as $f(x)$ for the output of function $f$ at input $x$. It means a function value, not $f$ multiplied by $x$.

The visible ranges of x- and y-values on a graphing calculator. A poor window can hide intercepts, turning points, or important behavior.

The largest common numerical factor together with all variable factors shared by every term, using each variable’s smallest exponent.

A function is increasing on an interval when its outputs get larger as inputs increase through that interval.

A way to describe sets of real numbers with endpoints. Parentheses exclude an endpoint; brackets include it. Infinity always uses a parenthesis.

An equation whose graph is a straight line, such as $Ax+By=C$, where $A$ and $B$ are not both zero.

A straight-line relationship used to represent a situation with a constant rate of change and a starting value.

The point halfway between two points: $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$.

A function satisfying $f(-x)=-f(x)$ for every $x$ in its domain, with $-x$ also in the domain. Its graph is symmetric about the origin.

The agreed order for evaluating expressions: grouping, exponents, multiplication and division from left to right, then addition and subtraction from left to right.

Two numbers $(x,y)$ identifying a point: the first gives horizontal position and the second gives vertical position.

The point $(0,0)$, where the coordinate axes intersect.

Distinct lines in a plane that never intersect. Nonvertical parallel lines have equal slopes; all vertical lines are parallel to one another.

Lines meeting at a right angle. Their slopes have product −1 when both slopes are defined; a horizontal and a vertical line are also perpendicular.

The equation $y-y_1=m(x-x_1)$ for a line with slope $m$ through $(x_1,y_1)$.

A sum of terms with real coefficients and variables raised only to nonnegative integer powers.

One of the four regions formed by the axes, numbered I, II, III, IV counterclockwise from the upper right. Points on an axis are not in any quadrant.

The set of all output values actually produced by a function or relation.

The change in one quantity divided by the corresponding change in another, with units such as dollars per hour. A line has a constant rate of change equal to its slope.

A local maximum or minimum: an output at least as large as, or at most as large as, nearby outputs in the domain.

The change in y divided by the corresponding nonzero change in x: $m=\frac{y_2-y_1}{x_2-x_1}$. Vertical lines have undefined slope; horizontal lines have slope zero.

The equation $y=mx+b$, where $m$ is the slope and $(0,b)$ is the y-intercept. Vertical lines cannot be written in this form.

Replacing a variable or expression with an equal value or expression.

A graph has symmetry when a reflection or rotation maps it onto itself. Common tests use the x-axis, y-axis, or origin.

A graph represents $y$ as a function of $x$ exactly when no vertical line intersects the graph more than once.

A point where a graph meets the x-axis, so $y=0$. Its x-coordinate is also a zero of the function when the graph represents a function.

A point where a graph meets the y-axis, so $x=0$.

An input $x$ for which $f(x)=0$. On the graph, it is the x-coordinate of an x-intercept.

If a product equals zero, at least one factor equals zero. For example, $ab=0$ implies $a=0$ or $b=0$.