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. Step 1: Understand the vertical line test. The vertical line test is a method used to determine if a graph represents a function. If any vertical line intersects the graph at more than one point, then the graph does not represent a function.
2. Step 2: Observe the graph provided. The graph shows an orange curve that appears to open outward, resembling a sideways parabola.
3. Step 3: Apply the vertical line test. Imagine drawing vertical lines at various x-values across the graph. For example, at x = 5, x = 10, and x = 15, check if the vertical line intersects the graph at more than one point.
4. Step 4: Analyze the intersections. In this graph, vertical lines at certain x-values intersect the curve at two points, indicating that for a single x-value, there are multiple corresponding y-values.
5. Step 5: Conclude based on the test. Since some vertical lines intersect the graph at more than one point, the graph does not represent 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 intersects the graph at more than one point, the graph does not represent a function of x. This test is crucial for identifying whether y is a function of x, as it ensures that each input (x-value) corresponds to exactly one output (y-value).

Function Definition

A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. In mathematical terms, for a relation to be a function, it must pass the vertical line test. Understanding this definition is essential for analyzing graphs and determining if they represent functions.

Graph Interpretation

Graph interpretation involves analyzing the visual representation of mathematical relationships. It requires understanding how the axes represent different variables and how the shape of the graph indicates the nature of the relationship between those variables. This skill is vital for applying the vertical line test and determining if a graph depicts a function.