Plotting a Relationship

The most frequent use of the coordinate plane is to display a relationship between x and y. Often, this relationship is expressed this way: if you tell me x, I can tell you y.

As an equation, this sort of relationship looks like this:

y = some expression involving x Another way of saying this is “We have y in terms of x.”
Examples: If you plug in a number for x in any of these equations, you can calculate a value for y.

For example, take y = 2x + 1. You can generate a set of y’s by plugging in various values of x. Start by making a table:

x y = 2x + 1
-1 y = 2(−1) + 1 = −1
0 y = 2(0) + 1 = 1
1 y = 2(1) + 1 = 3
2 y = 2(2) + 1 = 5

Now that you have some values, see what you can do with them. You can say that when x equals 0, y equals 1. These two values form a pair. You express this connection by plotting the point (0, 1) on the coordinate plane. Similarly, you can plot all the other points that represent an x-y pair from your table:

You might notice that these points seem to lie on a straight line. You’re right—they do. In fact, any point that you can generate using the relationship y = 2x + 1 will also lie on the line:

This line is the graphical representation of y = 2x + 1.

So now you can talk about equations in visual terms. In fact, that’s what lines and curves on the coordinate plane are—they represent all the (x, y) pairs that make an equation true. Take a look at the following example:

 y = 2x + 1
5 = 2(2) + 1
The point (2, 5) lies on the line y = 2x + 1 If you plug in 2 for x in y = 2x + 1, you get 5 for y.

You can even speak more generally, using variables:

 y = 2x + 1
b = 2(a) + 1
The point (a, b) lies on the line y = 2x + 1 If you plug in a for x in y = 2x + 1, you get b for y.

Check Your Skills

  1. True or False? The point (9, 21) is on the line y = 2x + 1.

  2. True or False? The point (4, 14) is on the curve y = x2 – 2.