Q&A #820

Teachers' Lounge Discussion: Explaining the relationship between the degree of a function and its shape

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From: David Richards

To: Teacher2Teacher Public Discussion
Date: 2007121713:48:57
Subject: The shape of a function and its degree

All even degree polynomials graph a parabola. They all enter in one
direction and leave in the opposite direction. All odd degree
polynomials graph a line. They enter and exit in the same direction.
The leading term determines the shape and the constant determines the
y-intercept. All the junk in between determines when and how many
times the polynomial will bounce in the middle. Every polynomial has a
line of reflection. In an odd polynomial you can draw a line through
the reflection point parallel to the legs of the graph. The graph will
be above or below the line to the left of the reflection point and
will flip to the other side of the line on the right. In an even
polynomial the reflection point is the central vertex of the parabola.
The graph will be increasing or decreasing on the left of the
reflection point and will flip directions on the right. I'm not sure
how to find the reflection points. In a second degree polynomial the
x-coordinate is -b/(2a). I don't know how to find it for higher degree
polynomials, but I'm sure someone has figured it out. This is the
elementary stuff I always point out to my Algebra students when we
start graphing polynomials. You know what the general shape of the
polynomial will be just from looking at the degree. The sign on the
leading term tells you if the polynomial is increasing or decreasing.
If it's positive for an even degree polynomial you get a U, if it's
negative you get an upside down U. If the polynomial is odd and the
leading term is positive you get an increasing line, if it's negative
you get a decreasing line. You can always plot the y-intercept just
 from looking at the constant or lack thereof (if there is no constant
the polynomial goes through the origin). You can tell how many times
it will bounce by counting the changes in sign as you read the
polynomial from left to right.

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