Creating an equation to model a curve

I’m doing a mathematical investigation about crows dropping nuts. The info given is:
Height of drop 1.7 2.0 2.9 4.1 5.6 6.3 7.0 8.0 10.0 13.9
Number of drops 42.0 21.0 10.3 6.8 5.1 4.8 4.4 4.1 3.7 3.2

We are supposed to plot the data on a graph and then figure out the type of function that models it, and then create an equation that fits the graph. The creating an equation is the part I find difficult.

sydneyjs said:

I’m doing a mathematical investigation about crows dropping nuts. The info given is:
Height of drop 1.7 2.0 2.9 4.1 5.6 6.3 7.0 8.0 10.0 13.9
Number of drops 42.0 21.0 10.3 6.8 5.1 4.8 4.4 4.1 3.7 3.2

We are supposed to plot the data on a graph and then figure out the type of function that models it, and then create an equation that fits the graph. The creating an equation is the part I find difficult.

Click to expand...

What did you learn in the class about modeling a curve?

There are numerous ways to tackle this problem - and you need to guide us about it.

Well im in grade 12 IB math so i know a lot about functions but since its an investigation the guidelines are non-existent...the exact question is "using technology, plot the data points on a graph. Define all variables used and state any parameters clearly. What type of function models the behaviour of the graph> Explain why you chose this function. Create an equation (a model) that fits the graph." And then theres more but basically we have to create an equation and then compare that to the original graph, and then use it to compare to other sizes of nuts being dropped etc.

Excel is very good at doing these. Do you have Excel?.

Yeah i do have excel but i dont know how to apply this to it, and im afraid that if i use it i wont be able to explain what i did and thats really what im getting marked on! i just need something to start me off and then i figure out how it came to that conclusion.

I will try to explain the steps, then you can play around and see what fits best.

Open Excel and enter your values in the first two columns A and B.
In the Toolbar, click on Insert
Then click on Chart
Then XY-Scatter
Click all the Next and Finish tabs and your chart will appear.
Then, in the toolbar click on Chart.
Add Trendline
Options(you should have 6 different regression models, log, exponential, linear, polynomial up to 6 degree, etc.)
Check the boxes that say "Display Equation in chart" and Display R^2

I tried several and the Power is as close as any I tried with an R^2 of .8965
The closer R^2 is to 1 the more accurate it is.

wow ok thanks a lot.
but as for explaining choosing what type of function models the behaviour of the graph, what should i say? because its not a certain type of function, its just a curve...?

Ok i tried that in Excel and it worked well! But its just not what i'm supposed to do, unfortunately! because i need to find an equation that can be applied to other models, and i need to somehow find that equation using logic and kind of do-it-yourself math not in the computer, so I just don't know! Thanks a lot though

No one does these by hand. Linear regressions are cumbersome enough to do manually, let alone the others.

Most calculators do regressions as well. Just play with Excel and find the best fit you can. As I said before, the Power regression gives about the best fit.

Here is what you have to do, according to the criteria,(your teacher will have given you a list of the criteria) for this task:
1. After you have plotted the points, and looked at the shape of the curve (you can use Excel or download a free graphing program called "Graph 4.3" ("google" this, and you will find the home page to download it) which will do this nicely) you have to decide what type of function you think it looks like - using your knowledge about functions which have this shape.
2. You must then conjecture the general function with parameters, a,b,c etc (you will find most of these standard functions in your text book).
3. Decide on a way of using your data to solve for the parameters (the a,b,c etc). You need some consistency in the solutions and you need to decide which points in your data you are going to use to find the solutions.
4. Choose values for the parameters from your solutions

This is the part in the criteria which refers to "analysis". To obtain a 3 on criterion C. If you just do curve fitting, then you may only get 2 for criterion C.

5. Graph this model and see how well it fits the data.
6. Amend the model. This you can do either by a "guess and check proces" (trying to change the parameters) and graphing the results, or by using other graphing programs (Geogebra is good for this - because it lets you set up sliders which can change). Use your knowledge of transformations of curves to decide how to change the parameters (stretch, translation and so forth)
7. You can then check your results for accuracy and compare your results to regression models. You can use the regression tools on your calculator or Graph 4.3, which does regression. BUT it is important to remember, that regression is just a tool itself - it doesn't necessarily give answers!!! and you have to decide whether you accept or reject a model, and give reasons for it.

At all times, you have to keep your data context in mind. For example, you would be unlikely to use a quadratic model for population growth. If you look at the longer problems in the various chapters of your text book, you will see the types of situations that use different models (e.g. problems in the exponential section of your text book talks about growth of algae in fish ponds, growth of bacteria, populations and so on - these are the usual models used for this type of problem, whereas throwing a ball in the air would be a quadratic model etc etc)

Just remember to make sure you do all the following:
a)Keep the context of the data in mind
b)Define ALL your variables clearly, and state their constraints (you can't have negative values for distances etc)
c)Analysis - using your maths knowledge
d) Checking with technology afterwards
e) Making critical comments on the way
f)Testing for accuracy
g)Writing it up in good mathematics and using appropriate terminology

so im confused... what is the equation of the graph? and how do we find it? please help me!!!!!!!!

I ran this through Excel and tried various regression options (power, exponential, logarithmic, polynomial)

Letting x=height and y=number dropped.

The Power gave me \(\displaystyle y=46.09x^{-1.11536}\) with an \(\displaystyle R^{2}=.8965\)

The closer R^2 is to 1, the more accurate it is.

A 6 degree polynomial(as high as Excel goes) is:

\(\displaystyle y=.0054x^{6}-.2393x^{5}+4.1495x^{4}-36.237x^{3}+167.77x^{2}-390.72x+365.9\)

It has an \(\displaystyle R^{2}=.9779\)

These are probably abut as close as you'll get.

Good calculators will do regressions as well. Do you have a good one?. If not, Excel is always a good bet at these things. It's what I use to do them because it's easy and makes a nice graph as well as adding the trendline(equation of graph)

Perhaps if we switch a x and y around. Would that be better?. May get even closer than before.

woah. so thats all we have to do for finding the equation? that seems too simple =S
but omg, thankyou!your a lifesaver!!

fairytale101 said:

woah. so thats all we have to do for finding the equation? that seems too simple =S
but omg, thankyou!your a lifesaver!!

Click to expand...

If I were 12 grade IB math teacher - teaching functions and expecting an invesigative approach - I might look for something of a different response.

It almost looks like a y = f(1/x) type of relation with assymptotes at x ~ 1.5 and y ~3.

This choice would depend on other information in the story - if available.

Anyway - with above observation then a good choice of curve would be

\(\displaystyle \: y \: = \: 3 \: + \: \frac{A}{x-1.5}\)

Now try different values of 'A' to see what value comes "close" to the values given.