Determining Linear Functions

Hi. In this math post, I go over determining linear functions from a table of values.

Pixabay Image Source

Topics

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• The Slope-Intercept Form Of A Line
• Common Difference Of Numbers As Slope
• Examples In Finding The Slope & Y-Intercept For The Equation Of A Line

The Slope-Intercept Form Of A Line

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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.

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Common Difference Of Numbers As Slope

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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.

Pixabay Image Source

Examples In Finding The Slope & Y-Intercept

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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.

Desmos Screenshot

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.

Desmos Screenshot

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.