Graphing equations gives us a visual, geometric representation of the equation. By looking at the graph of an equation we can see trends in the data our equation represents.
Let's look at the equation 3x + 2 = y.
The first thing we want to do is set up a t-table. A t-table is a table that has a set of values for x and a set of values for y corresponding to the given x-value. To generate the table we can pick random x values and substitute them in for x to find the y value. Here is an example of a t table for this equation:
|
x
|
y = 3x + 2 |
| -3 |
-7 |
| -2 |
-4 |
| -1 |
-1 |
| 0 |
2 |
| 1 |
5 |
| 2 |
8 |
| 3 |
11 |
Once we have our t-table, we can graph our ordered pairs on a cartesian plane:
By graphing the ordered pairs we see a trend that this equation is a straight line:
Unfortunately just by choosing a few points and plotting them we can't be 100% certain of the shape of the graph, however there are ways to tell the general shape of a graph by just looking at the equation it represents. If you don't yet know how to tell the shape of a graph by looking at the equation, it is important to use a wide range of x values to tell what the graph is going to do. Let's look at another equation as an example:
Here we will look at the equation y = |x|
First we will do our t-table, but let's only user positive x values:
|
x
|
y =|x| |
| 0 |
0 |
| 1 |
1 |
| 2 |
2 |
| 3 |
3 |
| 4 |
4 |
| 5 |
2 |
| 6 |
6 |
Now that we have some values, let's plot them.
We can see a trend here, it looks like we have another straight line.
Remember we didn't user any negative x values though. so what happens when we check for x = -3. According to our graph, when x = -3 we go to the left 3 and look to see what the y value is on the graph:
According to our graph, the |-3| is -3, however we know that that cannot be true. The absolute value of -3 should be 3. We must include some negative numbers for our x values in our t-table:
|
x
|
y =|x| |
| 0 |
0 |
| 1 |
1 |
| 2 |
2 |
| 3 |
3 |
| 4 |
4 |
| 5 |
2 |
| 6 |
6 |
| -1 |
1 |
| -2 |
2 |
| -3 |
3 |
| -4 |
4 |
| -5 |
2 |
| -6 |
6 |
Now when we plot all the values:
We can now see that the graph makes a V shape. This makes more sense as we know the absolute value of any number x has to be positive.
This example shows the importance of choosing a wide range of x values so that you get the best overall view of the graph.