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Advanced Math/Trigonometry: Polynomial Functions

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Expert: - 6/7/2009


Question
Determine the equation of the given function

f(x)=k(2x-3)(x+4)(3x-5)(x-10) ?

Also, if I am given a graph, and told to determine the equation of the cubic funtio shown, how do I figure that out? I am aware that a cubic function is f(x)=x^3+bx^2+cx+d ..

Thank you so much!

Answer

Values
The graph would cross the axis every where x is 0.
This can be seen by looking at each set of parenthesis.
For each root, just one set of parenthisesis needs to be 0.
So the roots can be found at 2x-3=0, x+4=0, 3x-5=0, and x-10=0.

They are at, in order, when 2x=3, x=-4, 3x=5, and x=10.

For the 1st and 3rd equations, divide by the constant in front of the x.  For equation 1, divide by 2, and get x = 3/2.
For equation 3, divide by 3 and get x = 5/x.
Note that beyond the last point, they are all positive, so the product is a positive number times k.  When x is extremely negative, each of the valuess for each set of parenthesis is negative, but the are four different values.  The product of four negative numbers is positive, so the function is still a positive * k.

I would put each of them in a separate column of Excel (B-E) and then put the product in the last column (F).  In column A, put down the x values between -10 and 10 by 1/5.  The first few rows of formulas in the sheet would look like

0.2   2x-3   x+4   3x-5   x-10   Formula
-5   2*A3-3   A3+4   3*A3-5   A3-10   B3*C3*D3*E3
A2+$A$1   2*A3-3   A3+4   3*A3-5   A3-10   B3*C3*D3*E3
A3+$A$1   2*A3-3   A3+4   3*A3-5   A3-10   B3*C3*D3*E3
A4+$A$1   2*A3-3   A3+4   3*A3-5   A3-10   B3*C3*D3*E3

Note that A1 contains the value that is added on in the rows below.
This can be copied down to over a hundred rows.

The first few rows with calculation look like
0.2   2x-3   x+4   3x-5   x-10   Formula
-5   -13   -1   -20   -15   3900
-4.8   -12.6   -0.8   -19.4   -14.8   2894.1696
-4.6   -12.2   -0.6   -18.8   -14.6   2009.1936
-4.4   -11.8   -0.4   -18.2   -14.4   1237.0176

Now these are spread apart here in both views of the spreadsheet,
but the translation probably runs them together.  Since that is the case, I will send an image of the sheet.

I don't see the cubic here - it's to the fourth power.

Scott A Wilson

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