AP Precalculus Ultimate Guide

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.

Input Values

Also known as the domain or independent variable (x), these are the values that are used as input in a function.

Output Values

Also known as the range or dependent variable (y), these are the values that are produced as output by a function.

Function Rule

The rule that determines how the input values are transformed into output values in a function.

Increasing Function

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

Decreasing Function

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

Graph

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

Concave Up

A rate of change is increasing in a function.

Concave Down

A rate of change is decreasing in a function.

x-intercepts

The zeros of the function, which are the values of x for which the function equals zero.

Average Rate of Change

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

Positive Rate of Change

When one quantity increases, the other quantity increases as well.

Negative Rate of Change

When one quantity increases, the other quantity decreases.

Local/Relative Maximum/Minimum

Points where a polynomial changes between increasing and decreasing or includes an endpoint with restricted domain.

Global/Absolute Maximum/Minimum

The greatest local maximum or least local minimum in a polynomial.

Points of Inflection

Points where the rate of change of a function changes from increasing to decreasing or from decreasing to increasing, resulting in a change in concavity.

Complex Numbers

Numbers that include both real numbers and non-real numbers.

Real Zeros

Zeros of a polynomial that are real numbers.

Even Function

A function that is symmetric over the line x = 0 and satisfies the property f(-x) = f(x).

Odd Function

A function that is symmetric over the point (0, 0) and satisfies the property f(-x) = -f(x).

End Behavior

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

Rational Function

The ratio of two polynomials where the polynomial in the denominator is not equal to zero.

Vertical Asymptote

Zeros of the polynomial in the denominator that are not zeros of the numerator.

Hole

A point where a zero appears more times in the numerator than the denominator in a rational function.

Equivalent Representations

Different forms of expressing polynomial and rational expressions, such as standard form and factored form.

Polynomial Long Division

A method used to find the equations of slant asymptotes of graphs of rational functions.

Binomial Theorem

A theorem used to expand terms in the form (a + b)^n and polynomials functions in the form of (x + c)^n.

Transformations of Functions

Changes made to a parent function, such as vertical or horizontal translations, dilations, or reflections.

Function Model Selection

Choosing the appropriate type of function model based on the characteristics of the data or scenario.

Assumptions and Restrictions

The assumptions made and restrictions applied when constructing a function model.

Function Model Construction

The process of creating a function model based on restrictions, transformations, technology, or piece-wise functions.

Application of Function Models

Using function models to draw conclusions about data sets or scenarios and making appropriate use of units of measure.

Sequence

an ordered list of numbers, with each listed number being a term. It could be finite or infinite.

Arithmetic Sequence

when each successive term in a sequence has a common difference (constant rate of change).

Geometric Sequence

when each successive term in a sequence has a common ratio (consistent proportional change).

Linear Function

if the output values of a function change at a constant rate.

Exponential Function

if the output values of a function change at a proportional rate.

Nth Term (Arithmetic)

the formula to find the nth term in an arithmetic sequence.

Nth Term (Geometric)

the formula to find the nth term in a geometric sequence.

Exponential Functions

always increasing or always decreasing, with no extrema or inflection points.

Horizontal Translation/Vertical Dilation

shifting or scaling an exponential function.

General Form of an Exponential Function

the standard equation for an exponential function.

Additive Transformation of an Exponential Function

adding a constant to an exponential function.

Negative Exponent Property

the property of negative exponents in exponential functions.

Product Property

the property of multiplying exponential functions with the same base.

Power Property

the property of raising an exponential function to a power.

Inverse Functions

the inverse relationship between two functions.

Identity Function

a function that returns the same value as its input.

Inverse of Exponential Functions

the inverse relationship between exponential and logarithmic functions.

Logarithmic Functions

functions that model proportional growth or repeated multiplication.

Log Product Property

the property of multiplying logarithmic functions.

Log Quotient Property

the property of dividing logarithmic functions.

Log Exponential Property

the property of exponentiating logarithmic functions.

Natural Log Property

the property of logarithmic functions with base e.

Exponential and Logarithmic Equations and Inequalities

solving equations and inequalities involving exponential and logarithmic functions.

Semi-Log Plots

plots with logarithmic scaling on the y-axis and linear scaling on the x-axis.

Linear Model for Semi-Log Plot

the equation for a linear model on a semi-log plot.

Periodic Phenomena

Occurrences or relationships that display a repetitive pattern over time or space.

Periodic Function

A function that replicates a sequence of y-values at fixed intervals.

Period

The gap between repetitions of a periodic function, representing the length of one complete cycle.

Intervals of Increase and Decrease

Sets of x-values from the lower to the maximum point and from the upper to the minimum point, respectively.

Concavity

Determines whether the function is concave up or down, influencing its behavior.

Average Rate of Change

Calculated as the change in output divided by the change in input.

Standard position

If an angle's initial side is parallel to the positive x-axis and its vertex is at the origin, it is said to be in the standard position.

Initial side

The ray on the x-axis.

Terminal side

An angle's other ray in standard position.

Positive Angle

If you rotate something counterclockwise, it's considered a positive angle.

Negative Angle

If you rotate something clockwise, it's considered a negative angle.

Radian angle measures

The measure of an angle in radians using the formula θ=s/r, where s is the arc length and r is the circle's radius.

Coterminal Angles

Angles that end up in the same position but have different measures due to multiple rotations around a circle.

Special Triangles

Triangles on the unit circle with specific side length ratios used to evaluate trigonometric functions with exact ratios.

Quadrant Positivity

Determining in which quadrants sine and cosine are positive and identifying positive ratios in each quadrant.

Phase Shift

The horizontal shift in a sine or cosine function upon adding an angle.

Sinusoidal Functions

Sine and cosine functions that share a sinusoidal nature and exhibit the same shape and traits.

Sign Determination

Relating point coordinates on the unit circle to trigonometric functions and understanding how quadrant positions affect the signs of cosine and sine.

Graph of Sine Function

Using unit circle angles for x-axis representation and a coordinate range of [-1,1] for the y-axis.

Graph of Cosine Function

Using unit circle angles for x-axis representation and a coordinate range of [-1,1] for the x-axis.

Transformations of Sine and Cosine Functions

Modifying the critical features of sine and cosine functions through amplitude, vertical shift, period, and phase shift.

Sinusoidal Function

A function in the form y=asin[b(x-c)]+d that represents a sine wave pattern and can be transformed through amplitude, frequency, and midline changes.

Interpreting, Verifying, and Reporting with Models

Selecting and verifying suitable models for periodic phenomena problems and reporting findings with relevant information.

Tangent Function

Constructing representations of the tangent function using the unit circle and describing its key characteristics.

Additive and Multiplicative Transformations

Transformations involving the tangent function that involve adding or multiplying angles.

Tangent Function

The tangent function, f(θ)=tanθ, is a periodic function with a domain and range based on the unit circle.

Period of Tangent Function

The tangent function has a period of π and is undefined where cosθ=0, leading to a discontinuous function.

Transformations of Tangent Function

The tangent function, f(θ)=atan[b(θ−c)]+d, can be transformed by adjusting frequency and midline.

Key Features of Tangent Function

The key features of the tangent function include domain, range, x-intercepts, y-intercept, period, amplitude, and midline.

Inverse Trigonometric Functions

The inverse trigonometric functions, denoted as arcsin, arccos, and arctan, introduce a nuanced understanding of function inverses.

Analytical and Graphical Representations of Inverse Trigonometric Functions

Inverse trigonometric functions can be represented analytically and graphically, such as sin(arcsinx)=x and arcsin(sinx)=x.

Solving Trigonometric Equations

Trigonometric equations involve one or more of the six trigonometric functions and can be solved using inverse functions and algebraic manipulations.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions, including cosecant (csc), secant (sec), and cotangent (cot), relate to the fundamental trigonometric functions and can be used to solve equations.

Characteristics of Reciprocal Trigonometric Functions

The cosecant, secant, and cotangent functions have specific characteristics such as domain, range, x-intercepts, y-intercept, period, amplitude, and midline.

Equivalent Representations of Trigonometric Functions

Trigonometric expressions can be rewritten using the Pythagorean identity and sine and cosine sum identities, which can also be used to solve equations.

Trigonometry and Polar Coordinates

Polar coordinates involve measurements of distance (r) from the origin and angle (θ) from the positive horizontal axis, and can be converted from rectangular coordinates.

Graphs of Polar Functions

Polar functions can represent circles, roses, and limacons, each with their own characteristics based on parameters such as radius and petal count.

Rates of Change in Polar Functions

Polar functions have characteristics such as rate of change, intervals of increase and decrease, positive and negative intervals, and extrema.

Average rate of change

The ratio of the change in radius values to the change in θ, indicating how the radius changes per radian.

Δr

The change in radius values between two points on a polar function.

Δθ

The change in θ (angle) between two points on a polar function.

Increasing interval

An interval where the polar function is increasing.

Decreasing interval

An interval where the polar function is decreasing.