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2.2 polynomial functions of higher degree

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2.2 polynomial functions of higher degree

1. 1. 2.2 Graphing Polynomial functions of higher degree 1. Polynomial functions are continuous No breaks , no holes, or gaps
2. 2. 2.2 Graphing Polynomial functions of higher degree 2. Polynomial functions are SMOOTH curves - No sharp corners.
3. 3. Exploration • We will explore the most simplest polynomial: n xy Odd powers Even Powers
4. 4. More complicated polynomials 61425 235 xxxy How do we graph this??
5. 5. Review - Characteristic of a graph
6. 6. Leading coefficient test Tells us what is happening at the ends of the graph (left and right behavior)
7. 7. Let’s try a few examples 61425 235 xxxy 45 24 xxy xxxy 2 2
8. 8. Middle behavior • To get the middle behavior we need more information: – X Intercepts (zeros) – Max/min points (extrema)
9. 9. 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.
10. 10. xxxy 2 2
11. 11. Extrema • Each polynomial of degree, n, has at most n-1 relative extrema
12. 12. 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
13. 13. 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.
14. 14. 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.
15. 15. Sketch the graph by hand 34 43)( xxxf
16. 16. Sketch the graph by hand 23 2)( xxxf
17. 17. Sketch the graph by hand – more complicated example xxxxf 2 9 62)( 23