2.2 polynomial functions of higher degree
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2.2 polynomial functions of higher degree

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2.2 polynomial functions of higher degree 2.2 polynomial functions of higher degree Presentation Transcript

  • 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: n xy Odd powers Even Powers
  • More complicated polynomials 61425 235 xxxy How do we graph this??
  • Review - Characteristic of a graph
  • Leading coefficient test Tells 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 2 2
  • 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 2 2
  • 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 - 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.
  • Sketch the graph by hand 34 43)( xxxf
  • Sketch the graph by hand 23 2)( xxxf
  • Sketch the graph by hand – more complicated example xxxxf 2 9 62)( 23