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7.1 Polynomial Functions

Definition. Recall that a polynomial consists of the sum of one or more monomialsRecall that a monomial is a constant, a variable, or a product of constants and variablesWe say the expression

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7.1 Polynomial Functions

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    1. 7.1 Polynomial Functions Definition / features Evaluating for a specific x value Graphs / end behavior

    2. Definition Recall that a polynomial consists of the sum of one or more monomials Recall that a monomial is a constant, a variable, or a product of constants and variables We say the expression “3r2 – 3r + 1” is a polynomial in one variable We generally express polynomials in the form “aoxn + a1xn-1 + a2xn – 2 +…an-1x + an” In this expression a0, a1, a2, …an represent coefficients

    3. Features of the polynomial The DEGREE of a polynomial in one variable is the greatest exponent of its variable The LEADING COEFFICIENT is the coefficient of the term with the highest degree For certain degree values, we have names for the corresponding polynomial

    4. Types of Polynomials NAME EXAMPLE DEGREE L.C. Constant 9 0 9 Linear 5x – 2 1 5 Quadratic 3x2 – 4x + 6 2 3 Cubic x3- 8 3 1 Quartic 7x4 + 2x2 – 3x 4 7 Quintic x5 + 111 5 1

    5. Example 1-1a

    6. Example 1-1b

    7. Example 1-1c

    8. Example 1-1d

    9. Example 1-1e

    10. Example 1-1f

    11. Example 1-2a

    12. Example 1-2d

    13. Example 1-3a

    14. Example 1-3b

    15. Example 1-3c

    16. Example 1-3d

    17. Example 1-3e

    18. Graphs of polynomials Let’s look at some example graphs on the board We’ll see if we can make some observations about the number of zeroes (a.k.a. x-intercepts), as well as which way the arrows point

    19. End behavior Our work with which way the arrows are pointing will help us to describe the end behavior of the function The end behavior is the behavior of the graph as x approaches positive infinity (i.e., as the graph heads off to the right) and as x approaches negative infinity (i.e., as the graph heads off to the left) We represent infinity and negative infinity using the symbols 8 and -8 To say “x approaches positive infinity” we write: x ? 8

    20. Analyzing end behavior End behavior is determined by two factors: the degree of the polynomial, and the sign of the lead coefficent Look at the graphs for polys with degree 1, 3, and 5: do you notice anything about the arrows on the graph? They point in opposite directions Look at the graphs for polys with degree 2 and 4: what do you notice? The arrows point in the same direction (both up or both down) Whether the arrows both point up or both point down is determined by the sign of the lead coefficient (think in terms of how the sign of “a” affects a parabola) The LC has a similar effect for odd degree polys

    21. FAR-END SUMMARY See the board for a summary of far end behavior

    22. Example 1-4a

    23. Example 1-4b

    24. Example 1-4c

    25. Example 1-4d

    26. Example 1-4e

    27. Example 1-4f

    28. HOMEWORK ? ? ? Pg. 350 16 – 44 evens 46 – 48

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