AP Precalculus Ultimate Guide

A comprehensive set of flashcards covering key concepts in AP Precalculus, including polynomial and rational functions, exponential and logarithmic functions, and trigonometric functions.

Function

A mathematical relationship that maps a set of input values to a set of output values such that each input value is mapped to exactly one output value.

Domain

The set of input values for a function, also known as the independent variable (x).

Range

The set of output values for a function, also known as the dependent variable (y).

Increasing Function

A function is increasing over an interval if as the input values increase, the output values always increase.

Decreasing Function

A function is decreasing over an interval if as the input values increase, the output values always decrease.

Graph

A visual display of input-output pairs that shows how values vary.

Concave Up

A graph is concave up if the rate of change is increasing.

Concave Down

A graph is concave down if the rate of change is decreasing.

Average Rate of Change

The slope of the secant line over a closed interval [a, b] from (a, f(a)) to (b, f(b)).

Positive Rate of Change

When one quantity increases, the other quantity also increases.

Negative Rate of Change

When one quantity increases, the other quantity decreases.

Local Maximum

A point where a function changes from increasing to decreasing.

Local Minimum

A point where a function changes from decreasing to increasing.

Global Maximum

The greatest local maximum of a function.

Global Minimum

The least local minimum of a function.

Points of Inflection

Points where the rate of change of a function changes from increasing to decreasing or vice versa.

Polynomial Function

A function represented as a polynomial with a degree n, where n is a positive integer.

Leading Term

The term in a polynomial of the highest degree.

Leading Coefficient

The coefficient of the leading term in a polynomial.

Even Degree Polynomial

A polynomial with an even degree that has a global maximum or minimum.

Complex Numbers

Numbers that include real numbers and non-real numbers.

Zero/root of function

A value a such that p(a) = 0; it contributes to the polynomial's x-intercepts.

End Points

The behavior of a polynomial function as input values increase or decrease without bound.

Rational Function

A function defined as the ratio of two polynomials where the denominator is not zero.

Vertical Asymptote

The zeros of the polynomial in the denominator that lead to undefined function values.

Horizontal Asymptote

The behavior of a function as the input values grow larger or smaller in magnitude.

Hole in Function

Occurs when a zero appears more times in the numerator than in the denominator.

Standard Form

A representation of a polynomial or rational function used to find end behavior.

Factored Form

A representation of polynomial or rational functions used to find x-intercepts.

Binomial Theorem

A method to expand expressions in the form (a + b)^n.

Transformations of Functions

Changes made to a function using vertical/horizontal shifts or dilations.

Linear Function

A function with a constant rate of change, represented as a straight line.

Quadratic Function

A function that models roughly linear relationships and has a unique maximum/minimum.

Piecewise Function

A function composed of multiple sub-functions defined over nonoverlapping domain intervals.

Rational Expression

An expression that can be written as the ratio of two polynomials.

Transverse Function

A function that represents a change from one variable to another.

Inverse Function

A function that reverses the mapping of another function.

Exponential Function

A function of the form f(x) = a*b^x, where b > 0.

Logarithm Function

The inverse of an exponential function, given as f(x) = log_b(x) where b > 0.

Coterminal Angles

Angles that share the same terminal side after more than one rotation.

Radian

A unit of angle measure based on the radius of a circle.

Unit Circle

A circle with a radius of one centered at the origin of a coordinate plane.

Trigonometric Function

Functions defined as ratios of sides of a right triangle.

Sinusoidal Function

Functions such as sine and cosine that produce a wave-like pattern.

Amplitude

The height of the wave from the midline in a sinusoidal function.

Period of a Function

The length of one complete cycle of a periodic function.

Frequency

The number of complete cycles of a periodic function that occur in a given interval.

Phase Shift

The horizontal shift of a periodic function along the x-axis.

Midline

The horizontal line that represents the average value of a sinusoidal function.

Frequency Control

Adjustment of sine or cosine functions impacting their graphs.

Inverse Trigonometric Functions

Functions that reverse the effect of the standard trigonometric functions.

Pythagorean Identity

The identity that relates sine and cosine: sin²θ + cos²θ = 1.

Polar Coordinates

A system identifying points by a distance from the origin and an angle.

Limacon

Polar equations expressing shapes determined by parameters a and b.

Rose Curve

Defined by equations r = acos(kθ) or r = asin(kθ) with petal structure determined by k.

Average Rate of Change

The change in output value divided by the change in input value over an interval.

Function Model Validation

Confirming that a function model accurately represents a given data set.

For a Full Rotation in Degrees

360 degrees represents a complete turn around a circle.

For a Full Rotation in Radians

2π radians represent a complete turn around a circle.

Arcsin Function

The function that returns the angle whose sine is a given number.

Arccos Function

The function that returns the angle whose cosine is a given number.

Arctan Function

The function that returns the angle whose tangent is a given number.

Cosecant Function

Reciprocal of sine, expressed as cscθ = 1/sinθ.

Secant Function

Reciprocal of cosine, expressed as secθ = 1/cosθ.

Cotangent Function

Reciprocal of tangent, expressed as cotθ = cosθ/sinθ.

Transformation of a Function

Change made to a function's graph or equation.

Reciprocal Trigonometric Functions

Functions derived from standard trigonometric functions.

Total Rotation of an Angle

Can exceed 2π radians, measuring full turns.

Mathematical Argument

A reasoning process used to conclude based on premises.

Sum of Angles Identity

Formulas for the sine and cosine of a sum of angles.

Factor of Function

A number or expression that divides another function evenly.

Radian Measure of Angle

Formed by the arc length of a circle divided by the radius.

Radius of Polar Function

The distance from the origin in polar coordinates.

Theta (θ)

An angle measurement in a polar coordinate system.

Initial Side of Angle

The ray representing the starting point of an angle measured in standard position.

Terminal Side of Angle

The ray that completes the angle measure after rotation.

Periodic Phenomena

Occurrences that repeat regularly over time or space.

Function Model Construction

Designing a mathematical representation based on various criteria.

Transformation Component

Part of a function that alters its position or shape.

Inverse of an Angle Function

The process to derive the angle from its trigonometric ratio.

Graph of Rational Function

The representation of a rational expression in a coordinate plane.

Behavior at Infinity

How a function behaves as its input values grow infinitely large or small.

Trigonometric Ratio

The ratios derived from the sides of a right triangle.

Function Type

Categories based on characteristics and behavior of mathematical expressions.

Growth Model

A representation of patterns in increasing values over intervals.

Characteristics of Functions

Distinctive properties that define various types of mathematical functions.

Model Validation Process

Making sure a mathematical model correctly represents reality.

Force of Function Representation

The strength of a function in describing real-world situations.

Value Definitions in Functions

Specific terms used in mathematical functions to avoid ambiguity.

Output Behavior of Functions

How a function's output changes with varying inputs.

Root of Polynomial

A value that makes a polynomial function equal to zero.

Graphical Interpretation

Understanding mathematical concepts through visual representations.

Parameter of Function

A constant that defines a family of functions.

Unit Circle Radius

The unit length of radius considered in trigonometric functions.

Angle Measurement Techniques

Methods employed to quantify angles, both in degrees and radians.

Statistical Modeling

Creating functions based on predicted behaviors in data sets.

Phase Synchronization

Condition in which functions align at specific intervals.

Forms of Exponential and Logarithmic Functions

Different representations of exponential and logarithmic functions.

Symmetry in Polynomial Functions

Reflection properties that serve to find overall characteristics.

Visual Function Representation

Modeling functions graphically to comprehend behavior.