Algebra Know-It-ALL

226 The Cartesian Plane

Are you confused?

The coordinate planes in Figs. 14-1, 14-2, and 14-3 show values only to up to ±6 for each variable. If we

want to show graphs “far out,” we can increase the numbers on one or both scales. Instead of going from

−6 to 6 in increments of 1 unit per division, we can go from −60 to 60 in increments of 10 units per

division, or from −3,000 to 3,000 in increments of 500 units per division. If we want to graph something

“close in,” we can make the numbers on the scales smaller. We might go from −0.6 to 0.6 in increments

of 0.1 unit per division, or from −0.0012 to 0.0012 in increments of 0.0002 unit per division! Our use

of 6 increments on each of the four scales is arbitrary. We can have more or fewer, as long as we draw the

coordinate system so it’s easy to read.

Here’s a challenge!

Imagine an ordered pair (x,y). You have plotted its point on the Cartesian plane. Neither x nor y is equal

to 0, so the point does not fall on either axis. What will happen to the location of the point if you multiply

bothx and y by −1?

Solution

The point will move diagonally to the opposite quadrant. In other words, it will go “kitty-corner” across

the coordinate plane, as follows:

• If it starts out in the first quadrant, it will move to the third.


• If it starts out in the second quadrant, it will move to the fourth.


• If it starts out in the third quadrant, it will move to the first.


• If it starts out in the fourth quadrant, it will move to the second.

If you have trouble envisioning this, draw a Cartesian plane on a piece of graph paper. Then plot a specific

point or two in each quadrant. Calculate how the x and y values change when you multiply both of them

by −1, and then plot the new points.

Three Relations

To graph a relation in Cartesian coordinates, we can pick a few numerical values for the

independent variable, calculate the resulting values for the dependent variable, and plot the

ordered pairs as points. When we’ve plotted enough points so we’re reasonably sure we know

what the graph will look like, we can connect the points with a smooth line or curve. This line

or curve is the actual graph.

Add 2 to the input

Figure 14-4 shows some points plotted in Cartesian coordinates for the following relation,

which was the first one we evaluated in Chap. 13:

y=x+ 2

These points lie along a straight line. We can plot more points for the relation, and they will

always lie along the same straight line. The line is the graph of the relation.