AP Precalculus Formula Sheet

Average Rate of Change

Measures how much the function value changes over an interval relative to the change in x.

Increasing Function

A function where the values rise as x increases, indicating a positive slope/rate of change.

Decreasing Function

A function where the values fall as x increases, indicating a negative slope/rate of change.

Constant Function

A function that neither increases nor decreases over an interval.

Concave Up

The graph of the function looks like a smile, indicating that the slope is increasing.

Concave Down

The graph of the function looks like a frown, indicating that the slope is decreasing.

Linear Function

A function with a constant rate of change, represented by the equation y = mx + b.

Quadratic Function

A function where rates of change form a linear pattern, represented by the equation f(x) = ax^2 + bx + c.

Polynomial Function

A function of degree n, where nth order rates of change are constant when x values are evenly spaced.

Local Maximum

A point where a function transitions from increasing to decreasing.

Local Minimum

A point where a function transitions from decreasing to increasing.

Point of Inflection

The point at which concavity changes; the graph switches from curving upwards to curving downwards.

Complex Roots

If a complex number z = a + bi is a root of a polynomial function with real coefficients, its complex conjugate z = a - bi must also be a root.

Polynomial End Behavior

Determined by the degree and the sign of the leading coefficient.

Rational Functions Zeroes

Zeroes of R are zeroes of P that are not also zeroes of Q.

Hole in the Graph

Occurs when both P and Q equal zero at x = c; indicates that the function is undefined.

Vertical Asymptote

Occurs at x-values where Q(x) = 0 while P(x) is non-zero.

End Behavior of Rational Functions

Indicates whether the function approaches 0, infinity, or negative infinity as x approaches positive/negative infinity.

Binomial Expansion

Can be found using box multiplication or Pascal’s Triangle.

Base Logarithmic Function

A logarithmic function assumed to be base 10 when no subscript is provided.

Natural Logarithms

Equal to regular logarithms with e as the subscript.

Standard Position in Trigonometry

The vertex is at the origin with one ray of the angle on the positive x-axis.

Arc Length Formula

Calculation method for the length of an arc on a circle.

Unit Circle

A circle with a radius of one used in trigonometry to define sine, cosine, and tangent.

Trigonometric Functions Equation

y = Asin(B(x+c)) + D where A is amplitude, period is 2π/B, phase shift is C, and the midline is y = D.

Modeling with Semi-log Plot

When the vertical axis is logarithmically scaled and the graph appears linear, it indicates an exponential function.

Residual Plot Interpretation

A pattern in a residual plot means the model is inappropriate, while no pattern indicates appropriateness.