Function Graph — Definition, Formula & Examples
A function graph is the visual representation of a function on the coordinate plane, where each input value x is paired with its output f(x) to form a plotted point (x, f(x)).
The graph of a function is the set , consisting of all ordered pairs where the second coordinate equals the function evaluated at the first coordinate.
How It Works
To graph a function, you choose several input values, compute the corresponding outputs, plot the resulting points, and connect them with a smooth curve or line segments as appropriate. You can read the graph to find outputs for given inputs, identify where the function increases or decreases, and locate key features like intercepts, maxima, and minima. A curve on the coordinate plane represents a function if and only if it passes the vertical line test — every vertical line crosses the curve at most once.
Worked Example
Problem: Graph the function f(x) = 2x − 3 by plotting points for x = 0, 1, 2, and 3.
Step 1: Compute f(x) for each input value.
Step 2: List the ordered pairs to plot.
Step 3: Plot these four points on the coordinate plane and draw a straight line through them, since f is a linear function.
Answer: The graph is a straight line with slope 2 and y-intercept −3.
Why It Matters
Reading and drawing function graphs is central to algebra, precalculus, and calculus — you will interpret graphs to find zeros, analyze rates of change, and solve optimization problems. In fields like physics and economics, graphs translate abstract equations into pictures that reveal trends at a glance.
Common Mistakes
Mistake: Reversing the coordinates by plotting (f(x), x) instead of (x, f(x)).
Correction: Always place the input on the horizontal axis and the output on the vertical axis. The point is (x, y) where y = f(x).