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Polynomials Day 2

Polynomials Day 2 . Inverse/compositions Even and odd functions Synthetic Division Characteristics. Inverses. MA2A2. Students will explore inverses of functions. Discuss the characteristics of functions and their inverses, including one-to-oneness, domain, and range.

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Polynomials Day 2

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  1. Polynomials Day 2 Inverse/compositions Even and odd functions Synthetic Division Characteristics

  2. Inverses MA2A2. Students will explore inverses of functions. Discuss the characteristics of functions and their inverses, including one-to-oneness, domain, and range. Determine inverses of linear, quadratic, and power functions and functions of the form , including the use of restricted domains. Explore the graphs of functions and their inverses. Use composition to verify that functions are inverses of each other.

  3. A function f is one-to-one if for each x in the domain of f there is exactly one y in the range and no y in the range is the image of more than one x in the domain. A function is not one-to-one if two different elements in the domain correspond to the same element in the range.

  4. Theorem Horizontal Line Test If horizontal lines intersect the graph of a function f in at most one point, then f is one-to-one.

  5. Use the graph to determine whether the function is one-to-one. Not one-to-one.

  6. Use the graph to determine whether the function is one-to-one. One-to-one.

  7. The inverse of a one-one function is obtained by switching the role of x and y

  8. Let and Find

  9. g is the inverse of f.

  10. Domain of f Range of f

  11. Theorem The graph of a function f and the graph of its inverse are symmetric with respect to the line y = x.

  12. y = x (0, 2) (2, 0)

  13. Finding the inverse of a 1-1 function Step1: Write the equation in the form Step2: Interchange x and y. Step 3: Solve for y. Step 4: Write for y.

  14. Find the inverse of Step1: Step2: Interchange x and y Step 3: Solve for y

  15. Even and Odd Functions MA2A3. Students will analyze graphs of polynomial functions of higher degree. b. Understand the effects of the following on the graph of a polynomial function: degree, lead coefficient, and multiplicity of real zeros. c. Determine whether a polynomial function has symmetry and whether it is even, odd, or neither. d. Investigate and explain characteristics of polynomial functions, including domain and range, intercepts, zeros, relative and absolute extrema, intervals of increase and decrease, and end behavior.

  16. Even functions A function f is an even function if for all values of x in the domain of f. Example: is even because

  17. Odd functions A function f is an odd function if for all values of x in the domain of f. Example: is odd because

  18. Determine if the given functions are even or odd

  19. The graph of an even function is symmetric with respect to the y-axis. The graph of an odd function is symmetric with respect to the origin. Graphs of Even and Odd functions

  20. Determine if the function is even or odd?

  21. Determine if the function is even or odd?

  22. Determine if the function is even or odd?

  23. Synthetic Division MA2A3. Students will analyze graphs of polynomial functions of higher degree. b. Understand the effects of the following on the graph of a polynomial function: degree, lead coefficient, and multiplicity of real zeros. c. Determine whether a polynomial function has symmetry and whether it is even, odd, or neither. d. Investigate and explain characteristics of polynomial functions, including domain and range, intercepts, zeros, relative and absolute extrema, intervals of increase and decrease, and end behavior.

  24. 3x 3x x

  25. Synthetic Division Summary • Set denominator = 0 and solve (box number) • Bring down first number • Multiply by box number and add until finished • Remainder goes over divisor Notes of Caution • ALL terms must be represented (even if coefficient is 0) • If box number is a fraction, must divide final answer by • the denominator • To evaluate a function at a particular value, you may EITHER: • Substitute the value and simplify OR • Complete synthetic division…the remainder is your answer

  26. 3 1 1 2 1 -4 2 -5 -3 -3 3 -1 -8 -1 2 -1 2 -3 1 3 2 3 0 1

  27. 3 5 4 2 4 -3 -8 4 27 57 12 19 61 9 2 -5 -28 14 25 -15 10 -3 -1 5

  28. 2 3 16 1 16 -32 -81 162 -162 32 0 0 0 -81 1 -2 -1 1 3 6 3 2 7 1

  29. 3 4 1 1 1 0 -5 2 12 3 9 14 3 4 1 0 -17 0 16 -16 16 -4 4 -1 -4 4 0

  30. 2/3 2 6 3 -1/2 2 6 -4 3 -2 2 4 0 0 0 3 2 5 4 5 2 -2 -2 -1 -1 4 4 2 0

  31. 1/4 4 4 4 -1 -4 1 -1 1 0 0 0 -4

  32. 4 -1/2 2 4 0 -13 -6 1 6 -2 0 -2 -12

  33. 3 1 1 0 -5 2 12 3 9 14 3 4

  34. 4 1 1 0 -17 0 16 -16 16 -4 4 -1 -4 4 0

  35. Direct Substitution • A fancy term for plug in and find f(x).

  36. Homework- Math 3 Book • Page 69 • 1-19 all, skip 8 – 10 and don’t do part e on 17-19.

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