Polynomial Functions

Functions of the form:

f(x) = a_nxⁿ + a _n-1x ^n-1 + … + a₂x² + a₁x + a₀

· Characteristics of Polynomial functions:

1. The graphs are unbroken, smooth curves with no “corners” for polynomials of degree 2.

2. An nth degree polynomial has a most n-1 turning points (i.e. points where the graph changes from rising to falling or vice versa).

3. As |x| gets very large (->), the |y| gets very large as well (->) (For non-constant functions, such as x = a or y = b.. In fact as |x| grows larger the value of y most closely resembles f(x) = .

A polynomial of degree 1 is linear and the graph is the equation of a line.

A polynomial of degree 2 is quadratic and the graph is a parabola.

· x-intercept(s) (let y = 0 and solve for x) play a significant role in determining behavior of a polynomial. It is best to look at f(x) in its factored form.

· Text version of the behavior of a polynomial near an x-intercept

Let f(x) be a polynomial and suppose that (x-a)ⁿ is a factor of f(x). [Furthermore, assume that none of the other factors of f(x) contains (x-a).] Then, in the immediate vicininty of the x-intercept at a, the graph of y = f(x) closely resembles that of y = A(x-a)ⁿ.

· Helpful Hint for behavior near the x-intercepts of polynomials:

If the x-intercept comes from a factor that has an even exponent, then the graph, will touch the x-intercept, but not cross the graph at that point. If the x-intercept comes from a factor that has an odd exponent, then the graph will cross the x-axis at that x-intercept

· y-intercept (let x = 0 and solve for y).