2.2 polynomial functions of higher degree
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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