AP Precalculus Ultimate Guide

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.

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

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

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

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

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

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

A rate of change is increasing in a function.

A rate of change is decreasing in a function.

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

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)).

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

When one quantity increases, the other quantity decreases.

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

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

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.

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

Zeros of a polynomial that are real numbers.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

shifting or scaling an exponential function.

the standard equation for an exponential function.

adding a constant to an exponential function.

the property of negative exponents in exponential functions.

the property of multiplying exponential functions with the same base.

the property of raising an exponential function to a power.

the inverse relationship between two functions.

a function that returns the same value as its input.

the inverse relationship between exponential and logarithmic functions.

functions that model proportional growth or repeated multiplication.

the property of multiplying logarithmic functions.

the property of dividing logarithmic functions.

the property of exponentiating logarithmic functions.

the property of logarithmic functions with base e.

solving equations and inequalities involving exponential and logarithmic functions.

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

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

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

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

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

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

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

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

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.

The ray on the x-axis.

An angle's other ray in standard position.

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

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

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.

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

An interval where the polar function is increasing.

An interval where the polar function is decreasing.