2. 2.2 Graphing Polynomial functions of
higher degree
1. Polynomial functions are continuous
No breaks , no holes, or gaps

3. 2.2 Graphing Polynomial functions of
higher degree
2. Polynomial functions are SMOOTH curves
- No sharp corners.

4. Exploration
• We will explore the most simplest polynomial:
n
xy
Odd powers Even Powers

7. More complicated polynomials
61425
235
xxxy
How do we graph this??

8. Review - Characteristic of a graph

9. Leading coefficient test
Tells us what is happening at the ends of the graph
(left and right behavior)

10. Let’s try a few examples
61425
235
xxxy
45
24
xxy
xxxy 2
2

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

12. 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.

13. xxxy 2
2

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

15. 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

16. 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.

17. To sketch the graph of a polynomial
function
1. Apply the leading coefficient test.
2. Find the zeros and y - intercepts
3. Plot a few points:
1. A few to the left of the zeros
2. A few to the right of the zeros
4. Complete the graph.

18. Sketch the graph by hand
34
43)( xxxf

19. Sketch the graph by hand
23
2)( xxxf

20. Sketch the graph by hand – more
complicated example
xxxxf
2
9
62)(
23