## Monday, December 19, 2011

### Graphs of Polynomials with Repeated Roots and How I Helped My Friend

Graphing polynomials is a tricky subject. I have a friend who asked me what the graph of $f(x)=(x+7)^7$ looks like. He explained to me that he knows what happens on the graph when there is a double root, such as $f(x)=(x+7)^2$. But what does the graph of a septa root look like?

Just for some context - he teaches a class in preparing for the CSET, which is the subject matter test to get a teacher credential in that subject. He often asks me some questions on the subtler details of mathematics.

The basic polynomial graphs that are emphasized over and over in school are $x^2, x^3$.

The graphs of these look like:

Where $x^2$ is a parabola and $x^3$ becomes recognizable as the "cubic curve."

The interesting thing about powers, that is, functions of the form of $x^n$, where the exponent n is an integer, is that there is a nice property regarding whether the exponent is an even or odd integer. Recall that an even power makes gives a positive output for any positive input, and that an odd power will have a positive output for positive input and negative output for negative input. Let's look at some power functions on the same graph:

Notice that all the functions with even powers are making a nice U shape, and all the functions with odd powers are making a shape similar to the cubic curve. But as the power increases, while the shape is similar, the graphs are getting steeper. This is because we are multiplying the same number together more times, so we are getting a bigger number at the same input value, or x-value.

So, to answer my friend's question, we will shift the graph over 7:

You can think of these as shifted power functions, or polynomials with one root repeated. Notice that these graphs have the same shape as before, but they are shifted to the left 7 because of the +7.

To summarize, a polynomial with only one root can be thought of a shifted power function. Because of that, it has the shape of either a parabola or a cubic curve, but steeper as the exponent increases.

That answers the question about $f(x)=(x+7)^7$, and about polynomial functions with the same root repeated many times (in this case the the root is -7, repeated 4 times).

You might be wondering about polynomial functions with multiple roots. For a discussion on graphing polynomials in general, I recommend a very long but helpful discussion on Purple Math's website:
http://www.purplemath.com/modules/polyends.htm

Graphs in this post are from:
www.wolframalpha.com