Lesson 2.2Polynomial Functions of Higher

Degree

Polynomial Characteristics

Continuous graphs – no holes or jumps

Curves – no “v’s”

Y = x3 : Cubic Function

Odd FunctionOrigin SymmetryIncreasing on :

)0,0(:,:,:

InterceptRD

,

Leading Coefficient TestFunction:• Even: ends go in same direction• Odd: ends go in opposite direction• Sign of first term determines how they start

Even Functions Odd Functions

Example:Discuss the end behavior of each function. Check each with your calculator.

1) f(x) = -x3 + 4x

2) f(x) = x4 - 9x2 +3x + 1

3) f(x) = x5 – 3x

Polynomial Zeros, Roots, Factors, X-intercepts

For a polynomial function f with degree n :

1) has at most n real zeros2) has at most n – 1 relative maxima or minima (humps)3) x = a is a zero of the function4) x = a is a solution when f(x) = 05) (x – a) is a factor of f6) (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)

Repeated Zeros

A function with repeated factors

1) If k is odd → graph crosses x-axis at x = a2) If k is even → graph touches x-axis at x = a, does not cross

kax

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

4) Graph and find the zeros of

5) Graph, find the zeros, find relative extrema (max & min)

6) Find a polynomial with the zeros

ttt 96 35

343 23 xxx

36,36