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. t (time) v (velocity, or speed) 1 second 50 ft/sec 3 seconds 18 ft/sec

Obviously, the ball is slowing down as it travels upward. Based on these two data points, what function v ( t ) 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: y = m x + b 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 ( 3 ) + 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. m + b = 50 → b = 50 - m 3 m + b = 18 → 3 m + ( 50 - m ) = 18 2 m + 50 = 18 2 m = – 32 m = - 16 b = 50 - ( - 16 ) = 66 So we have found m and b . Since these are the unknowns the in the equation y = m x + b , the equation we are looking for is: y = – 16 x + 66 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.