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Last updated on September 20, 2025
The vertical line test is used to check whether the graph represents a function by checking whether a vertical line intersects the graph at more than one point. This article explains the vertical line test and how it works.
One can verify whether a curve represents a function in the coordinate plane by drawing a vertical line parallel to the y-axis. The curve assigns precisely one y-value to each x-value if this vertical line intersects the curve at only one point for each x-value. Then the graph represents a function. On the other hand, the curve is not a function if the line crosses it more than once, resulting in multiple y-values for the same x-value.
To perform the vertical line test:
The first graph (y = x²) has only one intersection with a vertical line at x = 1. Therefore, it passes the test and is a function. The second graph, which is the circle with x² + y² = 1, does not represent a function and fails the test because it has two intersection points with a vertical line at x = 0.5.
To graphically apply the Vertical Line Test, sketch several vertical lines over the curve parallel to the y-axis. A graph can be considered a function if a line crosses the curve once; if it crosses twice or more, it is not.
Learn how to use the Vertical Line Test to find functional relationships in a variety of real-world situations.
Here are common mistakes and ways to avoid them while using the Vertical Line Test, including misaligned lines, domain gaps, and confusion.
Solve the Linear Function (y = 2x + 1)
The equation y = 2x + 1 represents a function.
Step 1: We will draw a straight line.
Step 2: Insert vertical lines at x = -2, 0, and 2.
Step 3: Every vertical line should make one contact with the graph.
In conclusion, the Vertical Line Test → Function is passed. We will understand it better by the figure given below:
The quadratic function (y = x²)
The equation (y = x²) passes the function.
For a given x = c, y = c².
For every 𝑐, exactly one 𝑦-value.
No vertical line ever makes multiple strikes.
Passes test ⇒ Function at last.
Circle (x² + y² = 4)
The equation (x² + y² = 4) does not pass the function.
For x = c, y² = 4 − c². So y = ±√(4 − c²).
Any ∣𝑐∣ < 2 has two real 𝑦-values—one “lower” and one “upper.”
So, for many values of 𝑐, the vertical line intersects the graph at two points.
Therefore, if fails the test, ⇒ Not a function at last.
Cubic Function (y = x³)
Figure shown below.
Step 1: Draw the S-curve.
Step 2: Add vertical lines all across the domain.
Consequently, the ending shows every line crossing once. Lastly, the function is passed.
Sideways Parabola (x = y²)
Shown in the figure below.
The first step is to plot the sideways U-curve.
Step 2: Draw vertical lines in step two at x = 1 and 2. Each vertical line intersects the sideways parabola at two points, so it fails the test.
Step 3: Every vertical line intersects at two spots.
So it fails → Not a Function, at last.