Question

What is the end behavior of `f(x) = (x - 2)^4(x + 1)^3`?

Answer 1

For any polynomial function that is factored, use the Zero Product Property to solve for the zeros (x-intercepts) of the graph. For this function, x = 2 or -1.

For factors that appear an even number of times like `(x - 2)^4`, the number is a point of tangency for the graph. In other words, the graph approaches that point, touches it, then turns around and goes back in the opposite direction.

For factors that appear an odd number of times, the function will run right through the x-axis at that point. For this function, x = -1.

If you multiply the factors out, your term of highest degree will be `x^7`. The leading coefficient is +1, and the degree is odd. The end behavior will resemble that of other odd powered functions like f(x) = x and f(x) = `x^3`. Left end will point downward, right end will point upward. Written like: as `xrarr\infty, y rarr\infty` and as `xrarr\-infty, yrarr\-infty`.

Here is the graph: