SOLUTION: Can you please explain the vertical line test in algebra, so I can help my son understand it better. I am truly ready to pullmy hair out because I absolutely do not understand how

Question 3185: Can you please explain the vertical line test in algebra, so I can help my son understand it better. I am truly ready to pullmy hair out because I absolutely do not understand how to do it. I even went and bought a cliffsnote book and still don't understand it.
Thank you
Answer by drglass(89) (Show Source): You can put this solution on YOUR website!
One of the most important concepts in mathematics is the function. Functions have the property that each value of x will produce at most one value for y. Some functions may produce no values for y. If some relation produce more than one y value for a given x value, then the relation is not a function.

Consider two relations, and . In the first relation, I know, regardless of the x value I choose, the relation will produce only one y value, therefore, this relation is a function. In the second relation, if I pick x = 4, the relation can have to y values, 2 and -2. Therefore, the second relation is not a function.

Think about what this means in terms of a graph. Imagine any relation. I can draw a vertical line (that is, a line with the equation x = b) on the same graph as the relation and check for an intersection between them. If the vertical line has the equation x = b and the two graphs intersect when y = a, then the point (b, a) is in the relation. If the same vertical line intersects the relation again, say when y = c, then the point (b, c) is in the relation. This means the relation has two y values for x = b, which means the relation fails the definition of a function. If the line does not intersect the relation again, then we really can't conclude anything.

For the test to be conclusive, we must check all points on the relation or at least convince ourselves that we've check enough points. In essence, we have to redraw the line in enough different locations to verify that the vertical line will intersect the graph in at most one unique point for each drawing.

To make this definition a little more concrete, let's consider two examples. First, let's look at the graph of :

Regardless of where I place the line, there will be only one point of intersection:

,

therefore, this relation is a function.

Now consider the relation, . To do this, I will plot , but, rotating the graph clockwise by 90 degrees, is equivalent to graphing :

,

If can place a vertical line on this graph at the location x = 2, the line will intersect the relation at two points (again, you must rotate the graph to get the desired effect):

,

,

Since the graph intersects the relation at two points for a given value of x, the relation is not a function.

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