Consequences in functions graphs?

Question

Hello. I would like to know about how graphs are affected from the characteristics of the polynomials in the function (an equation), specifically: 
  * what is the maximum of cuts on the x axe in relation with the equation (should I take into consideration the degree on the function, if the term is negative/positive, if there is an independent term, etc.)?
  * how could I identify the degree and main characteristics of a polynomial function only by looking at the graph?
  * If I had to propose a polynomial function that had to cut “m” times the x axe and that starts down and ends up, how could I do it? And if the graph had to have “U” shape?

? · Accepted Answer

There's a beautiful and actually obvious pattern for polynomials.

For a linear other than y = k it must cross the x -axis once so 1 solution always.

For a quadratic that's a U or n shape.  In either case it can miss the x-axis altogther or have a double 
root by touching at one point or at most two intersections.

For a cubic that adds another wiggle like a sideways s so it must cut the axis once or twice with a second double root or at most 3 times.

And so you go on.  A quartic is like a w or m so it can miss the x axis altogether or cut at most 4 times.

In general odd order functions including the linear will have at least 1 solution.  Even functions can have no solutions - a solution is where it cuts the x axis.

Hope that's clear and enought to get you started.

? · Answer

It seems in basic terms like a corner. Mathematically, a corner happens while the superb and bounds at the instant are not equivalent, yet neither are constructive or adverse infinity (which would be a cusp or a vertical tangent)

Consequences in functions graphs?

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