Final answer:
To determine if a graph represents a function, perform the vertical line test by checking if each x-value maps to only one y-value. Linear equations like y = mx + b always represent functions because they pass this test. If an x-value has multiple y-values, the graph is not a function.
Step-by-step explanation:
Understanding if a Graph is a Function
To determine if a graph represents a function, one must check if each x-value on the graph corresponds to exactly one y-value. In mathematical terms, this is known as the vertical line test. A simple way to perform this test is to imagine drawing vertical lines through each point on the graph. If any vertical line intersects the graph at more than one point, then some x-values map to multiple y-values, and the graph does not represent a function.
A common example of a function's graph is the graph of a linear equation such as y = mx + b. The slope (m) and the y-intercept (b) determine the shape of this line. For every x-value, there is only one corresponding y-value, which complies with the definition of a function.
Conversely, if a graph has an x-value with more than one corresponding y-value, as might be seen in a vertical line or a more complex curve, it fails the vertical line test and does not represent a function.