AP Precalculus Unit 1 Reference Sheet Flashcards

Vocabulary practice flashcards covering AP Precalculus Unit 1 topics including graphical behaviors, rates of change, polynomial properties, and rational functions.

Average Rate of Change (AROC)

The slope between two points of a function $f$ over the interval $[a, b]$ ($a \le x \le b$), given by the formula $\text{AROC} = \frac{f(b) - f(a)}{b - a}$.

Rate of Change (at a single point)

The slope of the graph of a function $f$ at a single point, conceptually thought of as the steepness of a roller coaster car at that exact position.

Increasing Function (Graphical Behavior)

A function behavior where y-values of $f$ decrease as x-values increase from left to right, which is equivalent to saying the rate of change of $f$ is positive.

Decreasing Function (Graphical Behavior)

A function behavior where the y-values of $f$ decrease as x-values increase from left to right, which is equivalent to saying the rate of change of $f$ is negative.

Concave Up (Graphical Behavior)

A function behavior where the graph bends upward, which is equivalent to saying the rate of change of the function is increasing.

Concave Down (Graphical Behavior)

A function behavior where the graph bends downward, which is equivalent to saying the rate of change of the function is decreasing.

Increasing at an Increasing Rate

The graphical behavior where the graph of $f$ is increasing (rate of change is positive) and concave up (rate of change is increasing).

Increasing at a Decreasing Rate

The graphical behavior where the graph of $g$ is increasing (rate of change is positive) and concave down (rate of change is decreasing).

Decreasing at an Increasing Rate

The graphical behavior where the graph of $h$ is decreasing (rate of change is negative) and concave up (rate of change is increasing).

Decreasing at a Decreasing Rate

The graphical behavior where the graph of $k$ is decreasing (rate of change is negative) and concave down (rate of change is decreasing).

Point of Inflection

A point where the graph of $f$ changes concavity from concave up to concave down, or vice versa.

Output Differences and Concavity (Tables)

Over equal-length input-value intervals, if the differences in outputs ($y$-values) are increasing, the function is concave up; if the differences in outputs are decreasing, the function is concave down.

Constant Finite Differences Property

Over equal-length input-value intervals, a polynomial of degree $n$ has constant $n\text{th}$ differences in outputs (for example, a quadratic function has constant second differences).

Complex Zeros Property

Non-real zeros of a polynomial always come in conjugate pairs; for example, if $x = -2 + 4i$ is a zero, then $x = -2 - 4i$ is also a zero.

Multiplicity of a Zero

The number of times $n$ a factor is repeated in a polynomial; if a zero has an even multiplicity, the graph will bounce off the $x$-axis at that zero.

Polynomial End Behavior

The behavior of $f(x)$ as $x$ increases or decreases without bound, determined by the leading term: a negative leading coefficient makes the right side go down, while an odd degree causes the left side to exhibit opposite behavior to the right side.

Even Function

A function whose graph has symmetry over the $y$-axis and satisfies $f(-x) = f(x)$; plugging in a number and its opposite yields the same answer.

Odd Function

A function whose graph has symmetry over the origin and satisfies $f(-x) = -f(x)$; plugging in a number and its opposite yields opposite answers.

Sign Chart for Polynomial Inequalities

A number line marked with all real zeros used to determine intervals of positive and negative values, where signs alternate across successive intervals unless a zero has an even multiplicity (which causes a bounce).

Top-Heavy Rational Function ($N > D$)

A rational function where the degree of the numerator is greater than the degree of the denominator, resulting in no horizontal asymptotes.

Same-Degree Rational Function ($N = D$)

A rational function where the degree of the numerator equals the degree of the denominator, resulting in a horizontal asymptote determined by the ratio of the leading coefficients ($y = \frac{a}{b}$).

Bottom-Heavy Rational Function ($N < D$)

A rational function where the degree of the numerator is less than the degree of the denominator, resulting in a horizontal asymptote at $y = 0$.

Slant Asymptote

An oblique linear asymptote of the form $y = mx + b$ that occurs in a rational function when the degree of the numerator is exactly 1 more than the degree of the denominator ($N = D + 1$), found using long division.

Zero of a Rational Function

An $x$-value where the numerator equals 0 AND the denominator is NOT equal to 0.

Hole (Removable Discontinuity)

A feature of a rational function that occurs where the denominator equals 0 AND the factor cancels out with the numerator (the numerator has an equal or larger multiplicity).

Vertical Asymptote

A feature of a rational function that occurs where the denominator equals 0 AND the factor does NOT cancel out with the numerator (the denominator has a larger multiplicity).

Pascal's Triangle

A triangular numerical pattern used to find the coefficients when expanding powers of binomials via the Binomial Theorem.