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Lesson 2.2 Polynomial Functions of Higher Degree

Lesson 2.2 Polynomial Functions of Higher Degree. Polynomial Characteristics Continuous graphs – no holes or jumps Curves – no “v’s”. Y = x 3 : Cubic Function Odd Function Origin Symmetry Increasing on :. Leading Coefficient Test Function: Even : ends go in same direction

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Lesson 2.2 Polynomial Functions of Higher Degree

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  1. Lesson 2.2Polynomial Functions of Higher Degree

  2. Polynomial Characteristics Continuous graphs – no holes or jumps Curves – no “v’s” Y = x3 : Cubic Function Odd Function Origin Symmetry Increasing on :

  3. Leading Coefficient Test Function: • Even: ends go in same direction • Odd: ends go in opposite direction • Sign of first term determines how they start Even Functions Odd Functions

  4. Example: • Discuss the end behavior of each function. Check each with your calculator. • f(x) = -x3 + 4x • f(x) = x4 - 9x2 +3x + 1 • f(x) = x5 – 3x

  5. Polynomial Zeros, Roots, Factors, X-intercepts • For a polynomial function f with degree n : • has at most n real zeros • has at most n – 1 relative maxima or minima (humps) • x = a is a zero of the function • x = a is a solution when f(x) = 0 • (x – a) is a factor of f • (a, 0) is an x-intercept Calculator – Zero Function 2nd – Calc – “Zero” – Left Bound – Right Bound – Guess Use table to find x-intercept (a, 0)

  6. Repeated Zeros • A function with repeated factors • If k is odd → graph crosses x-axis at x = a • If k is even → graph touches x-axis at x = a, does not cross

  7. Intermediate Value Theorem If f is continuous on an interval [a, b], then f takes on every value in between a and b. Mainly used to test for “zeros”: If f(a) > 0 and f(b) < 0, then there must be a value where the function is 0.

  8. Example • Graph and find the zeros of • Graph, find the zeros, find relative extrema (max & min) • Find a polynomial with the zeros

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