Inflection Points and Roots
The Microworld…
The Story…
We use
the term inflection point here to denote the point on the graph of a
polynomial of degree where the derivative of the polynomial becomes 0, that
is, where that derivative "changes sign." In the case of a quadratic,
it is where the derivative changes sign. In the case of a cubic, it is where
the second derivative (which indicates concavity) changes sign. In the case of
a quartic (degree 4 polynomial) it is where the third derivative changes sign.
We do not have an informal name for what the third derivative describes.
On this
page, we experiment with the fact that for polynomials in general, if all roots
are real, then the abscissa of the inflection point is the average value of the
roots. If some roots are not real, then the inflection point is not the average
of the real roots. It is in general the average of the real parts of the
roots. We will discuss that idea in the next section when we consider complex
numbers and roots of unity.
Use this
exploration exactly as you did the previous one. You may construct a polynomial
up to degree 9, and the system will graph it. If you choose a polynomial all of
whose roots between x = -50 and x = 50 are real, then the system prints some
information.
First it
lists the points you chose (integer or tenths approximation). For example, it
might print:
Point
1 is [-13, -7]
Point 2 is [-6, 5]
Point 3 is [5, -7]
Point 4 is [12, 3]
Then it
describes the exact polynomial so determined.
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Next, it tells you
what the exact value of the abscissa of the inflection point is, and draws the
inflection as a large yellow dot on the graph. It also draws a yellow vertical
line on the graph to indicate this.
The abscissa of the inflection point is: or as decimal -0.2764227
Finally,
it reports the values of the roots, and their average value. This average may
be compared with the abscissa of the inflection point. The roots themselves are
draw with red dots on the graph and the average is indicated with a short light
blue vertical line.
The 3 roots between -50 and 50 are:
solution 1 ) is -10.9541855
solution 2 ) is -1.0272204
solution 3 ) is
11.1521378
The average of the roots is -0.2764228
So the final picture might look like:
You should experiment a bit with cubics (Point(s) = 4)
And you may be asking yourself by now just what guarantees that there will be 3 real roots for a cubic? We will discuss and explore that condition in When do Cubics Have Real Roots?.
