Graphing Polynomial Functions:

Standard Form for Linear Functions:

f(x) = ax + b

Standard Form for Quadratic Functions:
f(x) = ax²+ bx + c

The linear and quadratic functions above are special case polynomial functions.

Standard Form for Polynomial Functions:

f(x) =

This is a polynomial function with degree n.

n is a nonnegative integer.

a_n is a real number greater than zero.
a_n, a_n-1 ,. . . , a₁, a₀ are real numbers.

Consider the polynomial function:

f(x) = axⁿ

If n = 1, then it is an equation of a line that passes through the origin.
If n = 2, then it is a parabola with vertex at the origin and symmetric about the y-axis.
If a ¹ 1 and n = 3 ( f(x) = x³) the graph looks like:

Graphs of the function f(x) = axⁿ were n > 2 and a ¹ 1 are:

If n is even the curve is symmetrical about the y-axis.
If n is odd the curve is symmetrical about the origin.

see translation of axes:

Graphing Factored Polynomial Functions

ex.
f(x) = (x + 1)(x - 2)(x + 3)

find all the zero points of the graph:

f(x) = 0 = (x + 1)(x - 2)(x + 3)

x + 1 = 0 or x - 2 = 0 or x + 3 = 0
x = -1 or x = 2 or x = -3

Substitute values of x from each interval to the function to see what the graph will look like. For each value of x, f(x) is either always positive or always negative. Meaning that the graph is either above the x-axis if positive and below the x-axis if negative.