AP Precalculus Glossary

Half the vertical distance between the maximum and minimum values of a periodic function. For $y = a \cdot \sin(x)$, the amplitude is $|a|$.

A line that a function's graph approaches but never reaches (or reaches only in the limit). Can be vertical, horizontal, or oblique.

The operation of applying one function to the output of another, written $(f \circ g)(x) = f(g(x))$. Evaluated from inside out.

Describes the curvature direction of a graph. Concave up (like a cup) means the rate of change is increasing; concave down (like a cap) means it is decreasing.

The set of all valid input values (x-values) for a function. Restricted by division by zero, square roots of negatives, and logarithms of non-positives.

The trend of a function's output as x approaches positive or negative infinity. Determined by the leading term for polynomials and by asymptotes for rational and exponential functions.

A function satisfying $f(-x) = f(x)$ for all $x$ in its domain. Its graph is symmetric about the y-axis. Examples include $x^2$, $\cos(x)$, and $|x|$.

A function of the form $f(x) = a \cdot b^x$ where $b > 0$ and $b \neq 1$. Characterized by a constant multiplicative rate of change over equal intervals.

A relation that assigns exactly one output to each input in its domain. Passes the vertical line test when graphed.

A visual test to determine if a function is one-to-one. If every horizontal line intersects the graph at most once, the function has an inverse. If any horizontal line intersects more than once, restrict the domain before finding an inverse.

A function $f^{-1}$ that reverses $f$, satisfying $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$. Exists only for one-to-one functions.

The inverse of an exponential function. $\log_b(x) = y$ means $b^y = x$. Domain is $x > 0$; range is all real numbers.

The number of times a factor $(x - r)$ appears in a polynomial's factored form. Determines graph behavior at the corresponding zero.

A function satisfying $f(-x) = -f(x)$ for all $x$ in its domain. Its graph is symmetric about the origin. Examples include $x^3$, $\sin(x)$, and $\tan(x)$.

A function where each output corresponds to exactly one input. Equivalently, it passes the horizontal line test. Only one-to-one functions have inverses.

The simplest form of a function family, used as the starting point for transformations. Examples: $y = x^2$, $y = \sqrt{x}$, $y = |x|$, $y = e^x$.

The smallest positive value $p$ such that $f(x + p) = f(x)$ for all $x$. For $y = \sin(bx)$, the period is $2\pi/|b|$.

A function defined by different expressions on different intervals of its domain. Each piece applies over a specified range of x-values, and the function may or may not be continuous at the boundary points.

A unit of angle measure where one full rotation equals $2\pi$ radians. One radian is the angle subtended by an arc equal in length to the radius. $\pi$ radians = 180°.

The set of all output values (y-values) a function actually produces over its domain.

The acute angle formed between the terminal side of an angle and the x-axis. Used to evaluate trig functions in any quadrant by applying the appropriate sign based on the ASTC rule (All Students Take Calculus).

A line passing through two points on a function's graph. Its slope equals the average rate of change of the function over that interval.

An operation that modifies a parent function's graph through shifts, stretches, compressions, or reflections without changing its fundamental shape.

A circle of radius 1 centered at the origin used to define trigonometric functions. The coordinates of a point at angle $\theta$ are $(\cos\theta, \sin\theta)$.

An x-value where $f(x) = 0$. Also called a root or x-intercept. The multiplicity of a zero affects whether the graph crosses or bounces at that point.