From: David Richards
To: Teacher2Teacher Public Discussion
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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