We are going to try to use a graphing calculator to take a set of data, and not only graph it but show us an equation to approximate values of the equation.
The first thing to do is set diagnostics on. To do that, hit the catalog button (to do this, hit 2nd and 0), and scroll down with the arrows on the top right to "DiagnosticOn." To speed up this process, you can jump to the Ds by pressing whatever button has a small blue D to the upper right of it. The x^-1 button, also known as the reciprocal button, has the D next to it, so that will make things quicker. Click diagnostics on, then hit enter. It will say "Done" underneath after you have turned it on.
Now, we can put our data into the calculator. To do this, click the "Stat" button, then click edit. This will bring up a chart with L1 and L2 as the first columns. Fill in L1 with your x-values and L2 with your y-values. To keep things simple, we will use the same data:
L1: 1, 2, 3, 4, 5
L2: 1, 3, 6, 10, 15
Now, we will go fix up our window. Last week, we had the following:
Xmin = -10
Xmax = 10
Xscl = 5
Ymin = -10
Ymax = 10
Yscl = 5
Xres = 1
I'd like you to change it to the following, just for this data:
Xmin = 0
Xmax = 7
Xscl = 1
Ymin = 0
Ymax = 30
Yscl = 5
Xres = 1
Now, we will decide what we want our data to look like on the graph. To do this, we first hit statplot (2nd, Y=), then hit enter for plot one. Make sure that your type of graph is the first one, and Xlist is L1 and Ylist is L2. For mark, put whatever you prefer.
If you hit the graph button, you should see the data all graphed. I think that is pretty cool alone. However, we can do better!
This part is a little complicated and has to be done perfectly to work. First, hit stat again, and this time, move right to calc. Now, choose the type of equation it is. To do this, we can use some of the skills we learned from sequences in July and August, by finding common differences. In this sequence, the equation would be a quadratic.
1 3 6 10 15
2 3 4 5
1 1 1
If you don't know this process, don't worry. You can learn it at this post: http://coolmathstuff123.blogspot.com/2011/07/problem-of-week-day-2-week-of-717-723.html. However, it isn't necessary for the graphing calculator. You can either take an educated guess, or if you really don't know, hit PwrReg.
Since this is a quadratic, hit QuadReg, then enter. This should come up with this screen:
y = ax^2 + bx + c
a = .5
b = .5
c = 0
R^2 = 1
This looks like nonsense, but I'll bet we can interpret most of it. We have already mentioned that the base quadratic equation is ax^2 + bx + c. Well, we have that up top, then a =, b =, and c =. These are actually the values that are a, b, and c in the equation. If you want, you can plug them in to get the following:
y = 0.5x^2 + 0.5x
The R^2 button is a little complicated, but I will explain it briefly. It is called the correlation coefficient, and tells you how close the equation is to being a quadratic, or whatever regression you chose previously. In this case, your value is 1 because it is a quadratic. If it were something like .94386, this would be an equation that was close to a quadratic, but not quite there. The closer to 1 it is, the better the data is. If it were .32894, this would be bad data, and you might want to recollect your data.
But it can do better! For the next step, click stat calc again, and hit QuadReg. Now, we need to put in a little code to tell the calculator what we want it to do with the equation. First, hit L1 (2nd, 1). Then, put a comma, and hit L2 (2nd, 2). Now, put another comma, and hit Y1. To do this, you have to hit Vars, move over to Y-vars, then hit function, and Y1. Now, hit enter.
Wait, it just deleted it! It's okay, it gets better. Click Y=. You should see an equation present; the exact same one that it told us! Let's graph it. To do it, just hit graph. We already set ourselves up.
At this point, we should use this equation to approximate, or in this case determine, more data for the equation. Let's say we want to know the 6th value. To figure it out, do what we would have done before. Hit calc, just above trace, then hit value, and type in 6. It tells you that Y = 21.
In school, we learned about graphing calculators, but I would never have expected that they could do something as crazy as this! Considering that some of my friends who don't really like math refer to my graphing calculator as a "game," I think there must be something cool about it.
Since the algebra has been getting a little heavy over the past few weeks, I will bring it down a notch and prove to you that my friends are correct in calling it a game.