Functions Defined by Graphs
Functions defined by graphs is a Grade 7 math skill from Yoshiwara Intermediate Algebra where a curve on the coordinate plane represents a function. Students use the vertical line test to confirm that a graph represents a function, then read values, domain, and range directly from the graph.
Key Concepts
Property We can also use a graph to define a function. The input variable is displayed on the horizontal axis, and the output variable on the vertical axis. For a graph to represent a function, every vertical line drawn on the graph can intersect the curve at most one time. This is known as the vertical line test.
Examples A graph showing the temperature in a city over a 24 hour period represents a function, because at any specific time (input), there is only one temperature (output).
The graph of a parabola opening upwards or downwards, like $y = x^2$, is a function because any vertical line will only cross the graph once.
Common Questions
How do you tell if a graph defines a function?
Use the vertical line test: if any vertical line crosses the graph more than once, it is not a function. A function has exactly one y-value for each x-value.
How do you read f(2) from a graph?
Locate x = 2 on the horizontal axis, move vertically to the graph, and read off the corresponding y-value.
What is the domain of a function defined by a graph?
The domain is the set of all x-values for which the graph exists — the horizontal extent of the graph.
Can a circle be a function defined by its graph?
No. A full circle fails the vertical line test, so it does not define a function. However, the upper or lower semicircle alone can define a function.