Hi. In this math post, I go over determining linear functions from a table of values.
When you have two points, a line can be created that passes these two points. On the Cartesian plane a line has a slope and a y-intercept. When x = 0 the corresponding y-value with x = 0 is the y-intercept. The equation of the line is of the form:
where m is the slope of the line and b is the y-intercept.
The slope of a line can be viewed as rise over run. This rise over run is the change in y-values divided by change in x-values.
From a table a values the slope is also the common difference.
Example
| x | y = x + 1 |
|---|---|
| -2 | -1 |
| -1 | 0 |
| 0 | 1 |
| 1 | 2 |
| 2 | 3 |
From the above example, the slope of the line is just 1. As x increases by 1, the value of y increases by 1 each time. The slope would be 1 divided by 1 which is just 1. This value of 1 is the slope which is also the difference in y-values as x increases by 1 each time.
Example Two
Determine the equation of the line from the following table.
| x | y |
|---|---|
| 0 | -2 |
| 1 | 1 |
| 2 | 4 |
| 3 | 7 |
| 4 | 10 |
From here the value of y goes up by 3 for each 1 unit increase of x. This value of 3 is the slope for the liine that passes the (x, y) points from the table of values.
There are cases where a y-intercept is not given from a table of values. Let's look at some examples.
Example One
| x | y |
|---|---|
| 2 | -3 |
| 4 | 0 |
| 6 | 3 |
| 8 | 6 |
In this table of values we do not have a y-intercept (y-value when x = 0). For every increase of x by 2 we increase y by 3. This is a slope of 3 divided by 2.
So far we have the equation of a line as:
To find the y-intercept, we can use a (x, y) pair from the table of values to help solve for the y-intercept represented by b. I will use the point (4, 0) as my (x, y) point.
The equation of the line for the table of values in this example is y = 1.5x - 6.
Example Two
| x | y |
|---|---|
| 1 | 8 |
| 4 | 2 |
| 5 | 0 |
| 10 | -10 |
This table of values does not have equal spacing with the x-values. Do be mindful of this and check that the slope is the same between any two points here. For the purpose of this exercise I have made the table of values such that the slope is the same between any two points.
For computing the slope I use the points (4, 2) and (5, 0).
Now we solve for b in y = mx + b. The point (1, 8) is used for (x, y) to find b.
Example Three
I present here a more technical example. Look for how much y goes up by each time x increases by 1. Also the y-intercept is given.
| x | y |
|---|---|
| 0 | pi (π) |
| 1 | 3π |
| 2 | 5π |
| 3 | 7π |
| 4 | 9π |
From this table of values the y-intercept is pi π. When increases by 1, y increases by 2π. The slope here is 2π.
The equation of the line here is y = 2πx + π. Desmos screenshot below.
Thank you for reading.
