# Vertical Line Test: Definition and Examples

*Ray is a licensed engineer in the Philippines. He loves to write about mathematics and civil engineering.*

## What Is the Vertical Line Test?

The vertical line test is a graphical test method used to determine whether a graph is the graph of a function. The vertical line test states that the graph of a set of points in a coordinate plane is the function's graph if every vertical line intersects the graph in at most one point.

We often attach the label y = f(x) to a sketch of the graph. If P(a,b) is a point on the function's graph, y-coordinate b is the function value f(a), as illustrated in the figure below. The figure shows the function f's domain, which is the set of possible values of x, and the range, which is the corresponding values of y. Although the domain and range are closed intervals, they may be infinite intervals or other sets of real numbers.

Since there is exactly one value f(a) for each variable "*a*" in the domain of function f, only one point on the graph of f has x-coordinate a. In general, we may use the vertical line test to identify functions. Thus, every vertical line intersects the graph of a function at most one point.

Consequently, the graph of a function cannot be a figure such as a circle or an ellipse, in which a vertical line may intersect the graph at more than one point. The x-intercepts of the graph of a function f are the solutions of the equation f(x) = 0. These numbers are called the zeros of the function. The y-intercept of the graph is f(0) if it exists.

## How to Perform Vertical Line Test

The vertical line test is a simple method to use in determining if a curve is a graph of a function or not. Observe from the figure shown below. The vertical line test is helpful for graphical purposes only. In simple terms, it is a visual way to check if a curve is a graph of a function. Take note that the vertical line test shall pass the following:

- The graph shall only have one output of y for every input of x.
- Draw a vertical line and check if it intersects a curve on an XY plane more than once. If so, the curve is not a function graph and therefore does not represent a function. This condition causes the relationship to be not qualified. Otherwise, if all drawn vertical lines intersect the curve most once, it means the graph is a function.

## Example 1: Identifying if a Given Equation Is a Function

Graph the line with an equation y^{2} = -5x and show it is a function.

**Solution**

By observing the equation itself, we can infer that the graph of the function is a parabola opening to the left. Now, to make sure of it, let us solve for the values of y-values for some x-values. Try to substitute x = 0, x = -5, x = -10, x = -25, and x = -40.

x-values | y-values |
---|---|

0 | 0 |

-5 | 5, -5 |

-10 | 5√2, -5√2 |

-25 | 5√5, -5√5 |

-40 | 10√2, -10√2 |

Then, draw a vertical line across the sketch of the graph using the obtained values of x and y. As you can observe from the diagram shown below, the vertical line passes through two points. Therefore, the vertical line test fails, and the line equation does not represent a function.

**Final Answer**

The line y^{2} = -5x does not pass the vertical line test and does not represent a function.

## Example 2: Application of the Vertical Line Test

Which of the following graphs represent a function y = f(x)?

**Solution**

There are three graphs of functions given that we need to identify. Recall that the vertical line test states that if any vertical line intersects a graph more than once, the relation of the graphical sketch is not a function. Draw a vertical line on each of the diagrams and check how many points the vertical line intersects.

For (a), the graph sketched is a parabolic curve. As you can observe from the vertical line passing through the function, there is only one point of intersection. Therefore, we can conclude that the parabolic curve represents a function.

The same scenario applies for graph (b), as shown. The second graph is linear. Most linear graphs, when passing through a vertical line, satisfies the vertical line test. Therefore, the straight graph represents a function.

For graph (c), as you can observe from the vertical line drawn across the sketch of the circular function, it passes through two distinct points. No matter where you put a vertical line, it passes through two points. Therefore, it does not conform to the vertical line test definition. At most x-values in the circle, it would intersect two or more y-values. Thus, the third graph does not represent a function.

**Final Answer**

Therefore, using the vertical line test, the given parabolic graph and linear graph represent a function, and the circular graph does not.

## Example 3: Identifying if a Relation Is a Function

Given the following relations, determine if they are functions using the vertical line test.

- A:{(2,4) (3,6) (4,8) (5,10)
- B:{(3,4) (-2,0) (3, -2)

**Solution**

The vertical line test is easier to use given ordered pairs of a particular graph. It is a simple plotting of the points in the cartesian coordinate system and checks if ordered pairs overlap vertically on the diagram. The primary consideration in this kind of vertical line test problem is to make sure the inputs are different and the outputs can be the same.

For the first ordered pairs A, we can see that no inputs are the same. It means that vertical lines to pass through the graph of the given ordered pairs will pass through at precisely one point on the graphical sketch.

For function B, there are two ordered pairs with the same input of 3 and have different ranges y. Therefore, this relation is not a function. Take a look at the graph of the given ordered pairs below.

**Final Answer**

The relations A passes the vertical line test and, therefore, a function. On the other hand, the ordered pair B is not a function since the vertical line test fails and two of its pairs have the same inputs or x-values.

## Example 4: Graphing Functions and Using the Vertical Line Test

Graph the following lines and identify if they are functions using the vertical line test.

- f(x) = x + 2
- f(x) = 2x
^{3} - x = 5y
^{2} - x
^{2}+ y^{2}= 81

**Solution**

There are two significant steps that you can perform in determining if the given line equation represents a function. First, given the equation f(x), solve for y at any value of x. Then, tabulate the results and graph them.

For f(x) = x + 2, solve for the values of y at x = 0, x = 1, x = -1, x = 5, and x = -5. After tabulating, plot the points in the cartesian coordinate plane and perform the vertical line test.

x-values | y-values |
---|---|

0 | 2 |

1 | 3 |

-1 | 1 |

5 | 7 |

-5 | -3 |

Solve for the y-values of the cubic equation f(x) = 2x^{3} at x = 0, x = 1, x = -1, x = 5, and x = -5. Plot the obtained points in the plane system and do the vertical line test.

x-values | y-values |
---|---|

0 | 0 |

1 | 2 |

-1 | -2 |

5 | 250 |

-5 | -250 |

For the parabolic function x = 5y^{2}, solve for the y-values for x = 0, x = 1, x = -1, x = 5, and x = -5. Plot the points and graph the function. See if it passes the vertical line test.

x-values | y-values |
---|---|

0 | 0 |

1 | √5/5, -√5/5 |

-1 | √5i/5, -√5i/5 |

5 | 1, -1 |

-5 | i, -i |

Lastly, for the given equation x^{2} + y^{2} = 81, solve the y-values at x = 0, x = 1, x = -1, x = 9, and x = -9. Then, plot the points in the cartesian coordinate system and perform the vertical line test. Based on the obtained values, we can easily infer that the equation does not represent a function because of the recurring values on the column of y-values. Therefore, the pairs are unique for all x-values.

x-values | y-values |
---|---|

0 | 9, -9 |

1 | 4√5, -4√5 |

-1 | 4√5, -4√5 |

9 | 0 |

-9 | 0 |

**Final Answer**

The graph of the equations f(x) = x + 2 and f(x) = 2x^{3} pass the vertical line test since the drawn vertical lines pass through only one point. A graph with one point of intersection with the vertical line is a sign that the graph represents a function. On the other hand, the graph of the equations x = 5y^{2} and x^{2} + y^{2} = 81 fails the vertical line test since vertical lines pass through or intersect two points on their graphs.

## Example 5: Identifying if a Graph Is a Function

Sketch the graph of the function x = y^{4} - y^{2} + 2 and utilize the vertical line test if it is a function.

**Solution**

Let us start the solution by tabulating all x and y points of the given equation. Solve for the values of y using random values of x. Say, solve values of y at x = 2, x = 3, x = 5, and x = 10.

x-values | y-values |
---|---|

2 | 0 |

14 | 2 |

74 | 3 |

Sketch the graph of the given equation and perform the vertical line test. As you can observe from the diagram shown below, no matter how many vertical lines you draw, they intersect the graph more than once.

**Final Answer**

The graph of the equation x = y^{4} - y^{2} + 2 fails the vertical line test since two or more points of the function's graph intersect the vertical line.

## Example 6: Using the Vertical Line Test

Consider the graph shown with multiple points and identify if it is a function.

**Solution**

Let us perform the vertical line test by drawing a vertical line to all the points shown in the graph. You can observe that no vertical lines pass through two points on the diagram twice or more. Always note that points (-2, -3) and (1, -5) area open points and are not on the graph. So the vertical lines x = 1 and x = -2 pass through only one spot of each line segment on the cartesian coordinate plane.

**Final Answer**

Since no vertical lines pass through the line segments twice, the composite graph is a relation or a function.

*This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.*

**© 2021 Ray**