In Exercises 55–64, use the vertical line test to identify graphs in which y is a function of x.

Verified step by step guidance

1. Recall the vertical line test: A graph represents y as a function of x if and only if no vertical line intersects the graph at more than one point.
2. Observe the given graph, which is a straight line sloping downward crossing both the x-axis and y-axis.
3. Imagine drawing vertical lines (lines parallel to the y-axis) at various x-values along the graph.
4. Check if any vertical line intersects the graph at more than one point. Since this is a straight line, each vertical line will intersect it at exactly one point.
5. Conclude that since every vertical line intersects the graph at only one point, the graph passes the vertical line test, so y is a function of x.

Key Concepts

Vertical Line Test

The vertical line test is a method used to determine if a graph represents a function. If any vertical line drawn through the graph intersects it at more than one point, the graph does not represent a function. This test ensures that each input (x-value) has only one output (y-value).

Definition of a Function

A function is a relation where each input value (x) corresponds to exactly one output value (y). This means no x-value can be paired with multiple y-values. Understanding this definition is crucial to identifying functions from graphs.

Graph of a Linear Function

A linear function is represented by a straight line on the coordinate plane. Since a line passes the vertical line test (no vertical line intersects it more than once), it always represents a function. The graph shown is a downward sloping line, indicating a linear function with a negative slope.