The vertical line test

0:00How can you tell from a graph whether it shows a function?

0:04Every graph shows a relation: a set of points, each with an x and a y.

0:10A relation is a function when each input x has exactly one output y.

0:16The vertical line test sweeps a vertical line across the graph, from left to right.

0:22If the line ever meets the graph more than once, the graph is not a function.

0:28The sloped line y equals 2x plus 1 is only ever met once, so it passes.

0:36The U-shaped curve y equals x squared minus 1, called a parabola, is also met once, so it passes.

0:44The circle x squared plus y squared equals 9 is met twice, above and below, so it fails.

0:53The sideways U, x equals y squared, is also met twice, so it fails.

1:00Why does meeting twice break the rule? Stop the line at x equals 4.

1:06It meets the curve at 4, 2 and at 4, negative 2.

1:11Check: 2 squared is 4, and negative 2 squared is also 4.

1:17Same x, different y-values: the input 4 has two outputs, so x equals y squared is not a function.

1:27A common mistake is to sweep a horizontal line instead.

1:31The horizontal line y equals 3 meets the parabola at negative 2, 3 and 2, 3.

1:39Check: 2 squared minus 1 is 3, and so is negative 2 squared minus 1.

1:46These points have the same y but different x-values, so the parabola is still a function.

1:52A graph shows a function when no vertical line meets it more than once.