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Section 2.2

Section 2.2. Polynomial Functions of A High Degree. Learning Goal. I will be able to determine the number of rational and real zeros of polynomial functions and find the zeros. In your groups…. Complete Quick Graphs of Polynomials. Translations of Cubics and Quartics.

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Section 2.2

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  1. Section 2.2 Polynomial Functions of A High Degree

  2. Learning Goal I will be able to determine the number of rational and real zeros of polynomial functions and find the zeros.

  3. In your groups… Complete Quick Graphs of Polynomials

  4. Translations of Cubics and Quartics • Polynomial Function: Cubic y = x3 Domain: Range:

  5. Translations of Cubics and Quartics • Polynomial Function: Cubic y = x4 Domain: Range:

  6. Translations Start with the graph y = x4. Complete a table of points, on the left. Reflect it across the x-axis. Stretch it by a factor of 3 in the y direction. Translate it 2 units to the right and 1 unit up. Write the equation of the new function. Translate each point from your first table and record them in the 2nd table. Then graph the translated function only. Identify the domain and range of the translated function. Domain: Range:

  7. Leading Coefficient Test • When n is odd: • If the leading coefficient is positive the graph falls to the left and rises to the right • If the leading coefficient is negative the graph rises to the left and falls to the right

  8. Leading Coefficient Test • When n is even: • If the leading coefficient is positive the graph rises to the left and right • If the leading coefficient is negative the graph falls to the left and right

  9. Applying the Leading Coefficient Test Use the Leading Coefficient Test to describe the right-hand and left-hand behavior of the graph of each polynomial function. a) f(x) = -x3 + 5x b) g(x) = x4 – 2x2 + 6 Rises to the left and falls to the right Rises to the left and right

  10. Zeros of Polynomial Functions • The function f has at most n real zeros. • The graph of f has at most (n – 1) relative extrema (relative maxima or minima). • A zero of a function f is a number x for which f(x) = 0. • Recall that zero, solution, factor, root, and x-intercept all refer to the same number on a polynomial graph.

  11. Factoring • By pattern • Difference of squares: a2 – b2 = (a + b)(a – b) • Sum of cubes: a3 + b3 = (a + b)(a2 – 2ab + b2) • Difference of cubes: a3 – b3 = (a – b)(a2 + 2ab + b2)

  12. Factoring • By grouping f(x) = x3 – 25x2 – 4x + 100 f(x) = x2(x – 25) – 4(x – 25) f(x) = (x – 25)(x2 – 4) f(x) = (x – 25)(x + 2)(x – 2)

  13. Finding Zeros of a Polynomial Function Find all zeros of f(x) = 6x3 – x2 – 2x Zeros: (0,0), (2/3,0) and (-1/2,0)

  14. Analyzing a Polynomial Function Find all real zeros and relative extrema of f(x) = -6x4 + 6x2. Relative Maxima: (-0.707,0) and (0.707,0) Relative Minima: (0,0) Zeros: (-1,0), (0,0), and (1,0).

  15. Finding Zeros of a Polynomial Function Find all real zeros of f(x) = x5 - 2x3 – x2 – 5x – 1. Zeros: (-1.710,0), (-0.205,0), and (1.978,0).

  16. Write an equation. Roots: -3, 2 and 6 Factors: (x+3), (x-2) and (x-6) Factored Form: f(x) = a(x+3)(x-2)(x-6) Find a using (1, 80). a = 4 so f(x) = 4(x+3)(x-2)(x-6) passes through the point (1, 80) Polynomial Form: f(x) = (4x+12)(x2 – 8x + 12) = 4x3 – 20x2 – 48x + 144

  17. Write an equation. Zeros: -1, 0, 2 Changes signs at roots -1 and 2, but not at 0. Starts up and ends up.

  18. What I Learned Today

  19. Gateway Problem Sketch the graph of f(x) = x2(x – 4)(x + 3)3.

  20. Gateway Solution

  21. Homework • Successful: P. 108 – 111 • Vocab Check (1, 4, 5) • Exercises (1, 3, 5, 13, 15, 17, 21, 25, 31, 53, 65, 85) • Unsuccessful: P. 108 – 111 • Vocab Check (1, 4, 5) • Exercises (1, 3, 5, 13, 15, 17 – 33 odds, 51, 53, 55, 61, 65, 71, 85)

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