How do you find the polynomial funtion that best models data givn in a table for x and f(x). The only example in my testbook shows how to do it on a calculator but I cannot find any instructions with my calculator to tell me how to do it.
The data for x=-5, -4,-3, -2, -1, 0, 1, 2, 3, 4, and 5
The data for f(x)= 4, 0, 0, 0, 0, 1, 1, 0, 0, 1, 4
Thanks.
This is a tricky one. I eventually managed to get an answer out of it, but I have a feeling that the original problem might have been mis-specified (unless of course I've got it wrong myself).
The zeros of this function are easy: all you need are the products of all the factors as follows:
(X+4)(X+3)(X+2)(X+1)(X-2)(X-3)
If you feed in X = -4, -3, -2, -1, +2 or +3 into the above, you'll get f(X)=0 as required, so that's six of the original eleven function values taken care of. However, if you feed the remaining (non-zero) X values into this function, you'll get the following:
f(-5) = 1344 = 48*28;
f(0) = 144 = 48*3;
f(1) = 144 = 48*5;
f(4) = 3360 = 48*70;
f(5) = 18144 = 48*378.
(Just inspecting the five numbers above shows that all of them have a common factor of 48, so we can scale them all to make the numbers a bit easier to handle.)
These figures however aren't the ones we want: what we ACTUALLY want are:
f(-5) = 4; f(0) = 1; f(1) = 1; f(4) = 1; f(5) = 4.
Okay, so let's assume we want to multiply the factorization we got earlier by another polynomial, say g(X), in such a way as to deliver the correct function values. The function values of g(X) will need to be given by:
g(-5) = 4/28 = 1/7;
g(0) = 1/3;
g(1) = 1/5;
g(4) = 1/70;
g(5) = 4/378;
after which we'll have to scale everything by a factor of 48. That is, we're dividing the function values we want by the ones we're actually getting at the moment.
There are five points here, so we can get a perfect fit to these points with a fourth degree polynomial, i.e. a quartic. If you set this up correctly either on your calculator or in a spreadsheet, you'll end up with an equation that goes through all the points. Take that equation, divide it by 48 and multiply it by the factorization for all the zeros that we got earlier. You should find that the resulting product delivers the right answer for every one of your original eleven points.
Having said all that, I reckon it might be an idea to check to see whether you've got the original problem specified correctly!
Correction: f(1) = 240 = 48*5. I got the factorization right, but copied down the original number wrong. Whoops.
I checked and it is copied down correctly.
The answers I have to choose from are:
a) y=0.02x^4-0.25x^2+0.11x+0.84
b) y=0.2x^4+0.25x^2+0.11x+0.84
c) y=0.2x^4-0.25x^2+0.11x+0.84
d) y=0.02x^4+0.25x^2+0.11x+0.84
Thanks for any help you can give.
Ah - you didn't mention that you were selecting the answer from a list of options - that would have saved a lot of time! Having said that, I've just checked all four, and none of them produces the f(X) values listed. Not surprisingly, because the function values listed describe a quite irregular pattern - certainly not something that a quartic would be expected to fit exactly. But then I see now that you're not TRYING to fit the points exactly; just to get the best quartic equation that you can find - in which case the answer is a), as you can verify by just working out the function values for all four of the options available, and comparing them with the list of function values. The easiest way to do that would probably be with a spreadsheet like Excel, if you have access to one; failing that, the Windows calculator will do.
Thanks bunches
To find the polynomial function that best models the given data, you can use the method of polynomial regression. Although your calculator might not have a specific function for this, you can still calculate it manually by following these steps:
Step 1: Write down the data points in a table, with x values in one column and f(x) values in another column.
x | f(x)
--------------
-5 | 4
-4 | 0
-3 | 0
-2 | 0
-1 | 0
0 | 1
1 | 1
2 | 0
3 | 0
4 | 1
5 | 4
Step 2: Determine the degree of the polynomial you want to fit to the data. In this case, since you want to find the polynomial function that best models the data, you need to determine the degree of the polynomial.
Step 3: Choose the degree of the polynomial that best represents the data. In this case, inspecting the data points, it seems like a quadratic (degree 2) polynomial would be a suitable choice.
Step 4: Use the method of least squares to fit the polynomial to the data. This involves finding the coefficients of the polynomial that minimize the sum of the squared differences between the predicted values and the actual values.
To do this, set up a system of equations using the given data points. Since we are looking for a quadratic polynomial, the equation will have the form: f(x) = ax^2 + bx + c.
Using the data points, you can substitute the x values into the equation and set them equal to the corresponding f(x) values. For example, using the first data point (-5, 4), you get the equation:
4 = a(-5)^2 + b(-5) + c
Step 5: Set up a system of equations using all the data points mentioned. In this case, you would set up 11 equations, one for each data point.
Step 6: Solve the system of equations to find the values of the coefficients a, b, and c. This can be done through various methods, such as matrix operations or using online solvers.
Once you have obtained the values for a, b, and c, you can plug them back into the equation f(x) = ax^2 + bx + c to find the polynomial function that best models the given data.
Note: Calculators with specific polynomial regression functions can simplify these steps, but when such functions are not available, the manual method described above is the way to go.