Mastering Functions and Graphing for Quantitative Reasoning

Quantitative Reasoning on the GRE assesses your ability to understand, interpret, and analyze quantitative information. Functions and their graphical representations are fundamental to many problems you'll encounter. This module will equip you with the knowledge and skills to confidently tackle these questions.

Understanding Functions

A function is a relationship between a set of inputs (domain) and a set of possible outputs (range) where each input is related to exactly one output. Think of it as a machine: you put something in, and it produces a specific output.

What is the defining characteristic of a function regarding its inputs and outputs?

Each input must correspond to exactly one output.

Representing Functions: The Coordinate Plane

The coordinate plane is our primary tool for visualizing functions. The horizontal axis (x-axis) typically represents the input (domain), and the vertical axis (y-axis) represents the output (range). A function is graphed as a set of points $(x, y)$ where $y = f(x)$ .

The graph of a function is a visual representation of all the ordered pairs $(x, f(x))$ that satisfy the function's rule. To determine if a graph represents a function, we use the Vertical Line Test. If any vertical line intersects the graph at more than one point, then the graph does not represent a function because it means there's an input ( $x$ -value) with multiple outputs ( $y$ -values). Common function graphs include lines (linear functions), parabolas (quadratic functions), and curves. Understanding the shape of these graphs can reveal properties of the function, such as its slope, intercepts, and turning points.

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Text-based content

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Key Function Types and Their Graphs

Function Type	General Form	Graph Shape	Key Feature
Linear Function	f(x) = mx + b	Straight Line	Constant rate of change (slope, m)
Quadratic Function	f(x) = ax^2 + bx + c	Parabola	Symmetrical U-shape (opens up or down)
Absolute Value Function	f(x) = \|x\|	V-shape	Minimum point at the vertex

Interpreting Graphs for GRE Problems

GRE questions often present functions through their graphs. You'll need to extract information like:

Function Values: What is $f(x)$ for a given $x$ ? (Find the $x$ on the axis, go to the graph, then find the corresponding $y$ on the axis).
Roots/Zeros: For what $x$ is $f(x) = 0$ ? (Where the graph crosses the x-axis).
Intercepts: Where does the graph cross the x-axis (x-intercept) and y-axis (y-intercept)?
Increasing/Decreasing Intervals: Where is the function's value going up or down as $x$ increases?
Maximum/Minimum Values: What are the highest or lowest points on the graph (local or global extrema)?
Rate of Change: How steep is the graph at a certain point (slope for linear functions, instantaneous rate of change for curves)?

Remember the Vertical Line Test! If a graph fails this test, it's not a function, and GRE questions will typically specify this or ask you to identify why it's not.

Transformations of Functions

Understanding how basic transformations (shifts, stretches, reflections) affect a graph is crucial. For a base function $f(x)$ :

$f(x) + c$ : Vertical shift up by $c$ units.
$f(x) - c$ : Vertical shift down by $c$ units.
$f(x+c)$ : Horizontal shift left by $c$ units.
$f(x-c)$ : Horizontal shift right by $c$ units.
$c \cdot f(x)$ (where $c>1$ ): Vertical stretch.
$c \cdot f(x)$ (where $0<c<1$ ): Vertical compression.
$f(c \cdot x)$ (where $c>1$ ): Horizontal compression.
$f(c \cdot x)$ (where $0<c<1$ ): Horizontal stretch.
$-f(x)$ : Reflection across the x-axis.
$f(-x)$ : Reflection across the y-axis.

How does the graph of

f(x-3)

compare to the graph of

f(x)

It is shifted 3 units to the right.

Practice Strategies

To excel in functions and graphing on the GRE:

Master the basics: Ensure you understand the definition of a function and the coordinate plane.
Visualize: Draw graphs, even simple ones, to understand relationships.
Practice the Vertical Line Test: Apply it consistently.
Recognize common graphs: Be familiar with the shapes of linear, quadratic, and absolute value functions.
Work through GRE-specific examples: Focus on how functions are presented and tested in actual exam questions.
Understand transformations: Practice identifying how changes in the function's equation alter its graph.

Learning Resources

Khan Academy: Functions and Graphs(tutorial)

Comprehensive video lessons and practice exercises covering function notation, domain, range, and graphing basic functions.

GRE Math Review: Functions(documentation)

Official GRE math review from ETS, covering function definition, notation, and graphical interpretation relevant to the exam.

Math is Fun: Functions(wikipedia)

An accessible explanation of functions, including domain, range, and how to graph them, with interactive examples.

Purplemath: Introduction to Functions(tutorial)

Detailed explanations and step-by-step examples for understanding function notation, evaluation, and basic graphing.

YouTube: GRE Quantitative Reasoning - Functions & Graphs(video)

A video tutorial specifically addressing functions and graphing concepts as they appear on the GRE Quantitative Reasoning section.

The Art of Problem Solving: Functions(documentation)

A more advanced look at functions, suitable for building a deeper understanding of their properties and applications.

Varsity Tutors: GRE Math - Functions(tutorial)

Covers key concepts of functions, including evaluation, domain, range, and graphing, with practice questions.

Brilliant.org: Functions(tutorial)

Interactive lessons and problems that build intuition for functions and their graphical representations.

Paul's Online Math Notes: Functions(documentation)

A thorough overview of functions, including domain, range, and graphical properties, with clear examples.

GRE Prep Club: Functions and Graphs Forum(blog)

A community forum where GRE test-takers discuss and solve problems related to functions and graphing, offering diverse perspectives and solutions.