Polynomial Graphs

ZHero · 2010-04-21 05:09:00

I was just going through >>this!<<
Note the part in bold .... This extra contribution of the cubic makes it impossible ....

now I'm more curious to know the effect of change of A, B, C and D on the graph of y=Ax^3+Bx^2+Cx+D
Effect of D is Self Evident and some light is thrown from >>this!

I want some "Exact Results" or "Works" upon "Effects of Change of Coefficients on Graphs of Cubics or Higher Degree Polynomials"..... if any!

JaneFairfax · 2010-04-21 10:58:24

Obviously varying D just shifts the curve up and down, but varying A can also be fairly easily visualized. If A is positive and you make it more positive, or if its negative and you make it more negative, you squash the curve sideways, making it narrower. This would be because the curve tends to ±∞ more and more quickly as you increase the absolute value of the leading coefficient.

For B and C, things may be more complicated. I experimented with some cubic curves but my results are not very conclusive what actually happens may very well depend on the sign of A. For B this could be what might happen:

Increasing the absolute value of B appears to increase the vertical distance between the local maximum and minimum points. In addition: (a) If A > 0 then (i) if B > 0 and you make it more positive, the curve tends to be shifted in a NW direction, and (ii) if B < 0 and you make it more, the curve tends to be shifted in a SE direction; (b) If A < 0 then (i) if B > 0 and you make it more positive, the curve tends to be shifted in a NE direction, and (ii) if B < 0 and you make it more, the curve tends to be shifted in a SW direction.

Last edited by JaneFairfax (2010-04-21 11:03:41)

JaneFairfax · 2010-04-21 11:26:02

For C, I think you can try this: Given

then

Thus

(1) If A > 0, increasing C will make the curve more and more strictly increasing, i.e. make it look more and more like a positively sloped straight line. The curve will have local maximum and minimum points if

and decreasing C further will widen the horizontal distance between the local max and min points.

(2) If A < 0, decreasing C will make the curve more and more strictly decreasing, i.e. make it look more and more like a negative sloped straight line. The curve will have local maximum and minimum points if

and increasing C further will widen the horizontal distance between the local max and min points.

Last edited by JaneFairfax (2010-04-22 00:35:34)

JaneFairfax · 2010-04-21 12:11:53

Okay, more ideas. Let

(1) If D varies and the other coefficients are fixed, then [1] is obviously the general solution to the differential equation

(2) If C varies and the other coefficients are fixed, then

and so [1] is the general solution to the differential equation

(3) If B varies and the other coefficients are fixed, then using the same trick in (2) shows that [1] is the general solution to the differential equation

(3) And if A varies and the other coefficients are fixed, then [1] is the general solution to the differential equation

And, for more fun, you can try varying some of the coefficients and keeping the others fixed. For instance, varying two of the coefficients and keeping the other two fixed will result in a second-order differential equation.

Last edited by JaneFairfax (2010-04-22 00:35:10)

ZHero · 2010-04-21 17:54:38

That's a really nice <Experimentation+Result>!!

Thanks for going about with it in detail...

ZHero · 2010-04-21 18:50:52

Here's something else i'd like to know...

Math Is Fun Forum

#1 2010-04-21 05:09:00

Polynomial Graphs

#2 2010-04-21 10:58:24

Re: Polynomial Graphs

#3 2010-04-21 11:26:02

Re: Polynomial Graphs

#4 2010-04-21 12:11:53

Re: Polynomial Graphs

#5 2010-04-21 17:54:38

Re: Polynomial Graphs

#6 2010-04-21 18:50:52

Re: Polynomial Graphs

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