Vertical Line Test

Last Updated : 25 Sep, 2026

The vertical line test is used to determine whether the given equation is a function or not.

  • It states that a graph represents a function if every vertical line intersects the graph only once.
  • If even one vertical line intersects the graph at more than one point, the graph does not represent a function.
  • The intersection of a vertical line and the curve represents the solution at that particular value of the input (x).
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Applications

The applications of the vertical line test are as follows:

  • Graphical Analysis: It helps in analyzing the behavior of different functions and their graphs.
  • Curve Sketching: It helps determine whether a given curve sketch represents a function.

Working

The Vertical Line Test is used to determine whether a graph represents a function. A graph represents a function if every input value x has exactly one output value y.

To apply the Vertical Line Test, follow these steps:

  1. Look at the graph you want to test.
  2. Imagine drawing a vertical line from top to bottom at different values of x.
  3. Observe how many times the vertical line intersects the graph.
  4. If every vertical line intersects the graph at most once, the graph represents a function.
  5. If any vertical line intersects the graph more than once, the graph does not represent a function.

Solved Example

Example 1: Determine whether the graph of y = x2 passes the Vertical Line Test.

Solution:

The graph of y = x2 is shown below:

Screenshot-2024-08-25-010311
y = x2

In this graph, a vertical line drawn for any value of x intersects the curve y=x2 at only one point. This confirms that y=x2 is a function, as it passes the Vertical Line Test.

Example 2: Determine whether the graph of x2+ y2 = 1 passes the Vertical Line Test.

Solution:

To determine if the curve x2+ y2 = 1 passes the Vertical Line Test, the graph is shown below:

Screenshot-2024-08-25-011908
x2+y2=1

In this graph, vertical lines intersect the curve at two points in many places, indicating that the graph does not represent a function. Therefore, the equation x2 + y2 = 1 fails the Vertical Line Test.

Although a vertical tangent line touches the circle at exactly one point, this does not make the circle a function. The Vertical Line Test must be satisfied for every vertical line, not just the tangent line.

Practice Problems

Problem 1: Examine the graph of the relation y = |x|. Does it pass the Vertical Line Test

Problem 2: Given the graph of a circle with the equation x2 + y2 = 25, determine if the graph represents a function using the Vertical Line Test.

Problem 3: Consider the graph of y = x3. Apply the Vertical Line Test to check if it is a function.

Problem 4: Look at the graph of the equation y=±√x​. Does this relation pass the Vertical Line Test

Problem 5: For the graph of the equation y=1/x​, test if it satisfies the Vertical Line Test.

Problem 6: Use the Vertical Line Test on the graph of y = sin⁡(x). Does this graph represent a function

Problem 7: Evaluate the graph of x = y2 + 3 using the Vertical Line Test. Is this graph a function

Problem 8: Given a straight line graph of y = 3x + 2, determine if it passes the Vertical Line Test.

Problem 9: Look at the graph of the relation y = x2/4 + 2. Can it pass the Vertical Line Test

Problem 10: Examine the graph of a horizontal line y = 5. Does this line pass the Vertical Line Test

Answer Key

    • The graph of y = ∣x∣ passes the Vertical Line Test. It is a function.
    • The graph of the circle x2 + y2 = 25 fails the Vertical Line Test. It is not a function.
    • The graph of y = x3 passes the Vertical Line Test. It is a function.
    • The graph of y = ±√x​ fails the Vertical Line Test. It is not a function.
    • The graph of y = 1/x passes the Vertical Line Test. It is a function.
    • The graph of y = sin⁡(x) passes the Vertical Line Test. It is a function.
    • The graph of x = y2 + 3 fails the Vertical Line Test. It is not a function.
    • The graph of y = 3x + 2 passes the Vertical Line Test. It is a function.
    • The graph of y = x2/4+2 passes the Vertical Line Test. It is a function.
    • The graph of the horizontal line y = 5 passes the Vertical Line Test. It is a function.
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