Vary the coefficient a in an
inflection trace of Y=ax
4+bx3+c2+dx+e
SOLUTION
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View a Geometer's SketchPad file with this construction

Conjecture
The red path above suggests that the inflection trace will form a cubic function when you vary the coefficient a in the equation Y=ax4+bx3+cx2+dx+e. We can verify this conjecture by finding an association beteen the x and y-coordinates of the inflection points of a quartic function.

Method
You can find the coordinates of the inflection point of a quartic function with differential calculus. The location of the inflection point can be found by identifying the moment when the second derivative changes sign (when the concavity changes direction).

Location of Inflection Point
In this case, we have:

Take the derivative and second derivative.


Set the second derivative equal to zero and find the location where the second derivative changes sign.


Using the quadratic formula, you will find the inflection points of the quartic function are at

which simplifies to

Since radicals can sometimes be messy, I will make a substitution. Let

Therefore, the extrema of the cubic function will occur at

The y-coordinate of the extrema can be found by substituting this x-value back into the function.

simplifies to

Therefore, the extrema of any cubic function will be located at

Proof
Any trace of the extrena must relate the x and y-coordinates in a true statement. If the coefficient a varies, then the locus will be the cubic function

We can verify this equation by substituting in
, ,
and
into the quartic relation and show that the x-coordinate will map directly onto the y-coordinate.


simplifies to

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This is part of the solution to the inflection trace of a cubic function investigatation. You can explore this phenomenon by playing with the interactive applet and answering the questions that follow.

Last revised: July 13, 2000