lesson 2.2 polynomial functions of higher degree

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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 - PowerPoint PPT Presentation

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Lesson 2.2 Polynomial Functions of Higher Degree

Lesson 2.2Polynomial Functions of Higher DegreePolynomial Characteristics

Continuous graphs no holes or jumps

Curves no vs

Y = x3 : Cubic Function

Odd FunctionOrigin SymmetryIncreasing on :

Leading Coefficient TestFunction:Even: ends go in same directionOdd: ends go in opposite directionSign of first term determines how they start

Even Functions

Odd FunctionsExample: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

Polynomial Zeros, Roots, Factors, X-intercepts

For a polynomial function f with degree n :

has at most n real zeroshas at most n 1 relative maxima or minima (humps)x = a is a zero of the functionx = a is a solution when f(x) = 0(x a) is a factor of f(a, 0) is an x-interceptCalculator Zero Function

2nd Calc Zero Left Bound Right Bound Guess

Use table to find x-intercept (a, 0)Repeated Zeros

A function with repeated factors

If k is odd graph crosses x-axis at x = aIf k is even graph touches x-axis at x = a, does not cross

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.

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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