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Vary the coefficient d
in an
inflection trace of Y=ax4+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 two vertical relations (lines) when you vary the coefficient
d 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 d varies, then
the locus will be the vertical lines
which is also
because the x-coordinates of the inflection points do not depend
on the value of d.
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 14, 2000