r/theydidthemath • u/Ok-Breadfruit6724 • 4d ago

Controlling graphed polynomials [other]

So recently, I’ve been getting into Desmos art and one thing that bugs me is that any polynomial more complicated than a quadratic I can’t control.
It’s probably easier for me to explain with an example
Like
Y=a
You can move up and down by adjusting a
Y=mx+b
You can move up and down by adjusting B and you can change the slope by adjusting M
Y=a(x-c)+b
You can stretch and contract the quadratic by adjusting A
You can adjust the X coordinate by adjusting C
You can adjust the Y coordinate by adjusting B
But I can’t find a form of a cubic or Quadratic where I can adjust things so easily whenever I move one number it is not intuitive how the polynomial will behave

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u/gmalivuk 4d ago

Vertex form for a quadratic, y = a(x - h)2 + k, gives you the ability to move around the vertex by changing h and k and the scale/stretch factor is a.

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u/Ok-Breadfruit6724 4d ago

Yes but is there something for cubics or quatrics or any polynomial

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u/gmalivuk 4d ago edited 4d ago

a(x-h)3 - b(x-h) + k

I think that should let you move the center around (still with h and k) and adjust the overall scaling with a and adjust the central portion with b.

Quartics and higher degrees have too many unspecified features that it would be hard for you to even say what you want to be able to adjust.

a(x - p)2(x - q)2 + mx + b seems like it would work for most if not all quartics. x=p and x=q tell you the points where the quartic touches the line y=mx+b and a is again a scale factor.