11.2 Finding a linear function for any two points

Finding a linear function for any two points

In an earlier unit, we did a great deal of work with the equation for the height of a ball thrown straight up into the air. Now, suppose you want an equation for the speed of such a ball. Not knowing the correct formula, you run an experiment, and you measure the two data points.

Obviously, the ball is slowing down as it travels upward. Based on these two data points, what function $v\left(t\right)$ might model the speed of the ball?

Given any two points, the simplest equation is always a line. We have two points, (1,50) and (3,18). How do we find the equation for that line? Recall that every line can be written in the form:

If we can find the m and b for our particular line, we will have the formula.

Here is the key: if our line contains the point (1,50) that means that when we plug in the x-value 1, we must get the y-value 50.

Similarly, we can use the point (3,18) to generate the equation $18=m\left(3\right)+b$ . So now, in order to find $m$ and $b$ , we simply have to solve two equations and two unknowns! We can solve them either by substitution or elimination: the example below uses substitution.

So we have found $m$ and $b$ . Since these are the unknowns the in the equation $y=mx+b$ , the equation we are looking for is:

Based on this equation, we would expect, for instance, that after 4 seconds, the speed would be 2 ft/sec. If we measured the speed after 4 seconds and found this result, we would gain confidence that our formula is correct.