Instructional Unit 4

Fitting Functions to Parabolic Data

by Sandy Cederbaum


Our goal in Unit 4 will be to try to find a rule for a given parabola or given data that is parabolic in nature. We will write this rule in vertex form which we studied in Unit 2 and Unit 3. This form of a quadratic equation is called vertex form because the coordinate (h,k) is the vertex of the parabola.

You will need a TI-83 graphing utility and you may need to do a little guess and check in order to find the best fitting rule. We will go through one example of how we might attack this problem and then we will load a program on the TI-83 graphing calculator that will enable you to practice this skill.

Example: Find the rule for the function that best fits the following graph:

If you had any trouble with this example, here is some strategy you can try.

Now we are ready to load a program onto your graphing calculator. The program is called PARAB. I will have a copy in class that you may download using the graphing link. This program can also be found on the Texas Instruments website. The following link gives you information about how to download programs from this site. At the top of the page is a link to the program archives. This is a useful website that you might want to bookmark and explore on your computer.

Once you have downloaded the program to you graphing calculator, go to your program screen and select program PARAB. The instructions on the calculator are pretty thin, but when you press enter, the calculator will show you a graph of a parabola in the Cartesian Plane. There is a grid to help you determine the vertex and other coordinates that might give you a hint as to how the graph has been stretched or compressed. Once you have an idea of what the rule should look like for the given parabola in vertex form, go to the y = screen and input your rule. When you hit the graph button, the calculator will show you the given graph and your new graph. If necessary, make adjustments to your rule so that the two graphs coincide (or are very close). If you enter 2nd MODE (QUIT), the calculator will give you an opportunity to replay or quit the program.

Replay the program as many times as it takes for you to get a rule that approximates a given parabola in less than one minute.

Refer back to Unit 1 again. We were given data that appeared to be parabolic in shape when plotted in a Cartesian Plane. In the sciences, it is a useful skill to be able to find the best fit curve for a given set of data. The catch here is that you may find that your model will not necessarily go through every data point given. In fact, it may not go through any of the data points given. This might be due to some error in the data collection equipment, the technique by which the data is collected, or some natural phenomenon inherent in the experiment. However, we can still find a decent fitting rule if the plotted data appears parabolic in shape. We did this as the final task in Unit 2 by eyeballing what we thought would be a good rule for the data given. If you compare your rule to several others of your classmates, you may find them all to be slightly different. In the study of Statistics, there are mathematical processes that enable us to find the "best fitting" rule. We will reserve these more rigorous processes for a later course.

Move on to Unit 5 - Finding X and Y-intercepts