PrecalH.sec2.2a.polyfunctions.notebook
February 24, 2015
Polynomial Functions
If n is a non­negative integer such that, an, an­1, an­2.... a2, a1, a0
are real numbers, with an ≠ 0, then,
f(x) = anxn + an­1xn­1 + an­2xn­2.... a2x2+ a1x + a0
is a polynomial function of x with degree n.
Functions are classified by their degree:
Type
Degree
constant
0
linear
1
quadratic
2
cubic
3
Example
f(x) = 15
f(x) = x­5
f(x) = 3x2 + 2x ­6
f(x) = x3 ­ 5x +1
PrecalH.sec2.2a.polyfunctions.notebook
February 24, 2015
Graphs of Polynomials
All polynomial graphs (regardless of its degree) have some things in common:
1) The natural domain of all polynomial functions is all reals.
2) All polynomial functions are continuous. Continuous graphs have no break in the graph along its domain.
3) All polynomial functions have smooth curves (no sharp points).
4) A polynomial function of degree n has at MOST n zeros. This means the graph has x­intercepts numbering less than or equal to the degree. How many zeros can a parabola have?
How many zeros could f(x) = x4 + 3x3 ­ 2x ­6 have?
4) A polynomial function of degree n has at MOST (n­1) turning points.
PrecalH.sec2.2a.polyfunctions.notebook
February 24, 2015
Power functions
Power functions are polynomial functions of the form y = xn.
2
If n is even, the graph is similar to that of y = x . As n gets larger, the graph flattens out more around the origin.
If n is odd, the graph is similar to that of y = x3. As n gets larger, the graph flattens out more around the origin.
PrecalH.sec2.2a.polyfunctions.notebook
Sketch the following transformations of the Power Functions:
a) y = x +1
3
b) y = ­(x­2)
4
c) y = 1/2 (x+1)5
February 24, 2015
PrecalH.sec2.2a.polyfunctions.notebook
February 24, 2015
All polynomial functions behave at the "extremes" as follows:
1) Lim f(x) = ∞ or Lim f(x) = -∞
x ⇒∞
x ⇒∞
(x goes to the right)
2) Lim f(x) = ∞ or Lim f(x) = -∞
x ⇒-∞
x ⇒-∞
(x goes to the left)
What does this mean??
As the x values get very large in the positive or negative direction, the f(x) (or y) values are also getting very large in the negative or positive direction.
What specifically happens in a given polynomial depends on its degree and the coefficient of the leading term.
Leading Coefficient Test:
To determine whether f(x) = anxn + an­1xn­1 + an­2xn­2.... a2x2+ a1x + a0 rises or falls on left or right, first determine if the degree, (n), is odd or even.
If n is odd:
if an is positive, the graph falls to the left and rises to the right.
if an is negative, the graph rises to the left and falls to the right.
If n is even:
if an is positive, the graph rises to the left and rises to the right.
if an is negative, the graph falls to the left and falls to the right.
PrecalH.sec2.2a.polyfunctions.notebook
Examples:
Find the left and right hand behavior of:
1. f(x) = ­2x3 + 4x
2. f(x) = ­5x4 ­ 5x + 4
3. y = ­3(­2x8 ­ 3x7 + 5x4 ­ 3x3 + x2 + 2)
February 24, 2015