Master Study Guide: Functions, Graphs & Transformations

Vocabulary-style flashcards defining core terms, notations, graph types, and transformation rules from the lecture notes.

Function

A rule where every allowed input has exactly one output, meaning one $x$ cannot produce two different $y$-values.

Vertical Line Test

A visual test stating that if any vertical line hits a graph more than once, the graph is not $y$ as a function of $x$.

Function Notation

A naming convention $f(x)$ for output $y$, where whatever appears inside parentheses replaces $x$ everywhere in the function rule.

Domain

Every possible input $x$ for a function.

Range

Every possible output $y$ for a function.

Interval Notation

A compact way to describe sets of numbers, using parentheses $( \nu )$ to exclude an endpoint and brackets $[ \nu ]$ to include an endpoint.

Linear Function

A function that graphs as a straight line and can be written as $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Transformation Formula

The general function form $y = a f(x-h) + k$, where $a$ controls vertical stretch/compression and reflection, $h$ controls horizontal movement, and $k$ controls vertical movement.

Vertical Stretch

A transformation occurring when $|a| > 1$ in $y = a f(x)$, making the graph steeper and narrower.

Vertical Compression

A transformation occurring when $0 < |a| < 1$ in $y = a f(x)$, making the graph flatter and wider.

Vertex Form of a Quadratic

The equation $y = a(x-h)^2 + k$, where $(h,k)$ is the vertex of the parabola.

Sideways Parabola

A parabola formed when $y$ is squared instead of $x$, written in the form $x = a(y-k)^2 + h$ with vertex $(h,k)$, which fails the vertical line test.

Absolute-Value Transformation Form

The equation $y = a|x-h| + k$, representing a V-shaped graph with vertex $(h,k)$.

Square-Root Transformation Form

The equation $y = a \times \text{sqrt}(x-h) + k$ or $y = a \nu \text{sqrt}(x-h) + k$, which starts at an endpoint of $(h,k)$ and extends to the right.

Difference Quotient

The expression $\frac{f(x+h)-f(x)}{h}$ where $h \neq 0$, measuring the average rate of change and serving as the foundation for derivatives.