The graph of a function is the visual representation of every input-output pair. It turns abstract formulas into pictures that reveal domain, range, behavior, and much more at a glance. This section explains how to construct and read graphs of functions.
Quick Reference
| Concept | Description |
|---|---|
| Graph of |
Set of all points |
| Table of values | Selected |
| Domain (from graph) | Projection of the graph onto the |
| Range (from graph) | Projection of the graph onto the |
| Smooth curve | Connect plotted points with a smooth, free-hand line |
What Is a Graph?
If we represent the independent variable by
The graph of a function
The graph gives a complete picture of the function's behavior: where it is positive or negative, where it rises or falls, what happens at the boundary of the domain, and more.
The graph of a function
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From this graph we can read off the following facts:
: the graph crosses the -axis at . -
x > 1 f(x) 0 < x < 1 f(x) x 0 f (0, \infty) f x y = f(x) (x, y) f(x) = x^3/2 x \in [-2, 2] x -2 -1.5 -0.5 0 0.5 1 1.5 2 y = x^3/2 -4 -1.6875 -0.0625 0 0.0625 1 1.6875 4 y = x^3/2 y = x^3/2 (x, y) f x x f y y x x x y y y y = x^2 y = x^2 y = ax^2 + bx + c a \neq 0$) is a parabola.
