2.2 polynomial functions of higher degree

2.2 Graphing Polynomial functions of higher degree

1. Polynomial functions are continuous No breaks , no holes, or gaps

2.2 Graphing Polynomial functions of higher degree

2. Polynomial functions are SMOOTH curves - No sharp corners.

Exploration

• We will explore the most simplest polynomial:nxy

Odd powers Even Powers

More complicated polynomials

61425 235 xxxy

How do we graph this??

Review - Characteristic of a graph

Leading coefficient testTells us what is happening at the ends of the graph (left and right behavior)

Let’s try a few examples

61425 235 xxxy

45 24 xxy

xxxy 22

Middle behavior

• To get the middle behavior we need more information:– X Intercepts (zeros)– Max/min points (extrema)

Zeros of polynomials

• Zeros are the same as x-intercepts.• Zeros happen when f(x) = 0

Zero = solution = factor

If x = c is a zero of polynomial, then x – c is a factor of the polynomial.

xxxy 22

Extrema

• Each polynomial of degree, n, has at most n-1 relative extrema

Practice

• Describe the end behavior• Find the zeros of the function (manually)• Graph the function on the graphing calculator

and find the relative extrema.

24 22)( xxxf

The Fundamental Theorem of Algebra

• Every polynomial of degree, n, has exactly n roots.

• Repeated roots:– An even number of repeats will touch the x-axis.– An odd number of repeats will cross the x-axis.

To sketch the graph of a polynomial function

1. Apply the leading coefficient test.2. Find the zeros and y - intercepts3. Plot a few points:

1. A few to the left of the zeros2. A few to the right of the zeros

4. Complete the graph.

Sketch the graph by hand

34 43)( xxxf

Sketch the graph by hand

23 2)( xxxf

Sketch the graph by hand – more complicated example

xxxxf2

962)( 23