Functions & Graphs Master Study Guide Flashcards

Vocabulary flashcards generated from the Functions & Graphs Master Study Guide, covering core terms, notation, formulas, and definitions.

Function

A rule that assigns exactly one output to each allowed input.

Function Notation $f(x)$

Represents the output of $f$ when the input is $x$; it does not mean $f$ times $x$.

Vertical-Line Test

A visual test where a graph represents $y$ as a function of $x$ if and only if no vertical line intersects the graph more than once.

Domain

The set of all allowed $x$-inputs for a function.

Range

The set of all possible $y$-outputs for a function.

Difference Quotient

An expression measuring the average rate of change over a tiny input step $h$, defined as $\frac{f(x+h)-f(x)}{h}$ where $h \neq 0$.

Core Transformation Rule (Outside vs. Inside)

Outside modifications control the output/$y$ and behave normally, while inside modifications control the input/$x$ and behave oppositely.

Slope ($m$)

The constant rate of change of a line, defined as $m = \frac{y_2-y_1}{x_2-x_1} = \frac{\text{rise}}{\text{run}}$.

Slope-Intercept Form

The linear equation form $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.

Point-Slope Form

The linear equation form $y - y_1 = m(x - x_1)$, used when given a point $(x_1, y_1)$ and slope $m$.

Standard Form of a Line

The linear equation form $Ax + By = C$, usually written with integer coefficients where $A$ is positive.

Extrapolation

Predicting values far outside the range of observed data, which carries a higher risk of unreliability.

Interpolation

Predicting values between observed data points.

Standard Form of a Quadratic

The quadratic form $y = ax^2 + bx + c$, which reveals the $y$-intercept $c$ and is used in the quadratic formula.

Vertex Form of a Quadratic

The quadratic form $y = a(x - h)^2 + k$, which reveals the vertex $(h, k)$, axis of symmetry $x = h$, minimum/maximum $k$, and range.

Factored Form of a Quadratic

The quadratic form $y = a(x - r_1)(x - r_2)$, which directly reveals the $x$-intercepts $r_1$ and $r_2$.

Discriminant

The value $D = b^2 - 4ac$, which determines the number of real $x$-intercepts: $D > 0$ gives two, $D = 0$ gives one, and $D < 0$ gives none.

Half-and-Square Rule

The method used to complete the square for $x^2 + bx$ by adding $\frac{b}{2}^2$.

Polynomial Function

A function using nonnegative whole-number exponents, where the degree is the largest exponent after simplifying.

Rational Function

A quotient of polynomials $\frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$.

Hole (in a Rational Function)

A point discontinuity created when a factor in the denominator cancels with a factor in the numerator.

Vertical Asymptote

A vertical line created by an uncanceled zero in the denominator of a rational function.

Exponential Function

A function of the form $y = a \times b^x$, where $b > 0$ and $b \neq 1$, with the variable located in the exponent.

Logarithm

The operation answering what exponent is needed to produce a value, defined by $y = \text{log}_b(x) \rightleftharpoons b^y = x$.

Product Law of Logarithms

The logarithmic identity $\text{log}_b(MN) = \text{log}_b(M) + \text{log}_b(N)$.

Quotient Law of Logarithms

The logarithmic identity $\text{log}_b(\frac{M}{N}) = \text{log}_b(M) - \text{log}_b(N)$.

Power Law of Logarithms

The logarithmic identity $\text{log}_b(M^p) = p \times \text{log}_b(M)$.

Piecewise Function

A function that uses different formulas depending on the specific input interval.

Equilibrium Point

The point where demand and supply functions are equal, solved by setting $D(x) = S(x)$.