Lesson Vertical Line Test

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About drglass: Ph.D. in Computer Science, BA in Physics, volunteer tutor for low income families K - 12, tutor for hire for college level math, computer science, chemistry, physics and biology
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, y+=+3x+%2B+2 and y%5E2+=+x. 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 y+=+3x+%2B+2:

graph%28300%2C+200%2C+-5%2C+5%2C+-5%2C+5%2C+x+%2B+2%2C+%28200000%2Ax+-+400000%29%29

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

graph%28300%2C+200%2C+-5%2C+5%2C+-5%2C+5%2C+x+%2B+2%2C+%28200000%2Ax+-+600000%29%29,

therefore, this relation is a function.

Now consider the relation, y%5E2+=+x. To do this, I will plot y%0D%0A=+x%5E2, but, rotating the graph clockwise by 90 degrees, is
equivalent to graphing y%5E2+=+x:

graph%28300%2C+200%2C+-5%2C+5%2C+-5%2C+5%2Cx%5E2%29,

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):

graph%28300%2C+200%2C+-5%2C+5%2C+-5%2C+5%2Cx%5E2%2C+2%29,

graph%28300%2C+200%2C+-5%2C+5%2C+-5%2C+5%2Cx%5E2%2C+3%29,

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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