This function modeled all of the first 5 data points, but this equation doesn’t model the scatter plot’s decreasing values.

Next, I used the last 5 data points from the data to create an equation.

Using the same method as the first 5 data points, I created the following equation:

y= .0041005051a4+ -.1625963516b3+ 2.380130469c2 + -15.47352983d + 42.01355157

, and entered the equation to graph:

This equation models the points of the last 5 data, but is very close for modeling points 2 and 5.

Subsequently after using matrices, I realized that using polynomial equations won’t help model the given data since both polynomial equations create a parabola not a decaying model. So I decided that using matrices wasn’t a good way to model the data. I began looking at ways to form to model decay. Using the GCD, I started with using PwrReG.

To find the formula using PwrReg, press STAT, go to CALC, scroll down to A, and enter again. This will bring the calculator to the main screen with PwrReG, so now press enter again. Then the screen will give the values of “a” and “b”. Plug the information into (y=) and press (GRAPH).

Looking at the graph, this function shows decay and the equation is very close to all of the data, but only goes through the 2nd and 8th point. The regression of the equation is -.95.

After that, I used the calculator again to find ExpReG. Using the method as the PwrReG, I got the equation:

Y=21.37241705 x (.8358432605x)

Using the given function I plugged in the equation and got:

This equation does model decay, but it needs to be adjusted since the function doesn’t go through any of the points.

With what I know from the exponential regression formula, I wanted to modify the equation to fit the data more accurately. So I started with the general exponential formula:

Y= a • bx-c +d

I subbed in “a” and “b” from the exponential regression formula:

Y= 21.37241705 • (.8358432605 x-c) +d

C= the horizontal translation

D= vertical translation

So I look back to my graph with the ExpReG and made some changes based on what I knew from above to tried making the function model all the data as it possible could and the closest I got was:

Y= 21.37241705 • (.8358432605 4x-5) +4

Graph:

This is my best equation that models the data of the large nuts because it goes through 6 points. This method of finding a function combines using technology, a bit of exploration of how graphs work, and analyze how exponential decay work.

Medium nuts:

Small Nuts:

Graph:

Observing the graph and data, I see that not all of the points are consistent in decreasing the number of drops when the height is higher. This graph still shows exponential decay, although the points: (8, 5.8) for the medium nuts and (10, 13.3) for the small nuts.

A parameter in this investigation is that the heights aren’t constant in all of the data from the different sized nuts. In order to compare how the first model fits with the small and medium nuts, the heights must be the same, so I created a new set of data: using the formula: Y=21.37241705 x (.8358432605x)

Ex: x=3

y=21.37241705 x (.83584326053)

=21.37241705 x .5839483675

12.48

After subbing in each x, to find the y-value, the data table became:

Using the new data points, I used the calculator again to find the ExpReG:

Y=20.040386 • (.8360116108x)

I began to modify the equation to fit the data more precisely, since the function doesn’t go through any of the points, so to dilate it higher I added a vertical translation to the equation. The attuned equation that models the above data is:

Y=20.04386 • (.836011610x)+ .5

The graph that models the equation is:

Now that I have an equation that models the data, I want to compare how well that equation fits with the small and medium sized nuts.

By looking at the graph above, I am able to see that the higher the drops, the lesser amount of drops are needed. Before being able to compare data, I need to find the equation for both medium and small size nuts.

To find the equation that models medium nuts, I used the GDC to plot in the data points to find ExpReG. The equation is:

Y= 30.46145561 • (.8540785676x)

After graphing the data, I noticed the above equation only goes through the 7th point, so I began altering the equation using the following information:

F(x-h)= h is a constant that moves the function h spaces to the right

F(x)+h= h is a constant that moves the function h spaces up.

c f(x)= c is a constant that moves function taller by c.

After many attempts of modifying the equation, the formula that best models the medium nuts is:

y= 30.46145561 x (.8540785676x+.015)-1.2

Graph:

This was the best equation to model the data of the medium nuts, since this equation goes through points 4 and 6.

Using the same method as the medium nuts, I found the ExpReG using the data of the small nuts. The equation is:

y= 37.67398021 x (.8949115919x)

This equation doesn’t go through any of the points, and so once again it needs to be modified. After, amending the equation a few times, the function that models the data the best is:

y= 37.67398021 x (.894911559191.5x + 2.5) +4

Graph:

Even after modification, this function only goes through the points 3 and 4.

Looking at the two equations from medium and small size nuts, I see that both model exponential decay. Each of the equations only touches 2 points, but gets very close to the other points. I also find that creating a function to model the medium and small nuts is a bit harder because the decay isn’t constant, so finding a model to fit all the points seems to be impossible.

Graph of all different size functions and data:

Model one from Large nuts: Y= 21.37241705 • (.8358432605 4x-5) +4

Large Nuts: Y=20.04386 • (.836011610x) + .5

Medium Nuts: y= 30.46145561 x (.8540785676x+.015)-1.2

Small Nuts: y= 37.67398021 x (.894911559191.5x + 2.5) +4

Also, by inferring to the equations above, I see that the medium and small nuts have greater number of drops in order to break open than the large nuts.

My first model: Y=20.04386 • (.836011610x) + .5 doesn’t replicate the data of medium and small size nuts because the data from the large nuts had a different heights than both of the small and medium size nuts. The decay equation from the first model is much steeper than the other two. In order to adjust the model to fit the smaller sizes, I first began by making the heights the same, and then compared the ExpReG of each. Even though the heights are the same, the numbers of drops for each of the different sizes of nuts aren’t very similar, so making the first function be able to model the medium and small nuts isn’t very likely. In addition, the regressions of the medium and small nuts aren’t very strong, given that they only model 2 of the 8 points from the data. The limitation in this investigation is: there just can’t be any size for a nut. For example, there can’t be a negative height, some heights aren’t able to break nuts, and the size of nuts must be too large or too small so a seed can’t be 10 pounds.