Polynomial Functions Lesson 9.2

Power Function Definition Recall from the chapter on shifting and stretching, what effect the k will have? Vertical stretch or compression for k < 1

Special Power Functions Parabola y = x2 Cubic function y = x3 Hyperbola y = x-1

Special Power Functions y = x-2

Special Power Functions Most power functions are similar to one of these six xp with even powers of p are similar to x2 xp with negative odd powers of p are similar to x -1 xp with negative even powers of p are similar to x -2 Which of the functions have symmetry? What kind of symmetry?

Polynomials Definition: The sum of one or more power function Each power is a non negative integer

Polynomials General formula a0, a1, … ,an are constant coefficients n is the degree of the polynomial Standard form is for descending powers of x anxn is said to be the “leading term”

Polynomial Properties Consider what happens when x gets very large negative or positive Called “end behavior” Also “long-run” behavior Basically the leading term anxn takes over Compare f(x) = x3 with g(x) = x3 + x2 Look at tables Use standard zoom, then zoom out

Polynomial Properties Compare tables for low, high values

Polynomial Properties Compare graphs ( -10 < x < 10) The leading term x3 takes over For 0 < x < 500 the graphs are essentially the same

Zeros of Polynomials We seek values of x for which p(x) = 0 We have the quadratic formula There is a cubic formula, a quartic formula

Zeros of Polynomials We will use other methods Consider What is the end behavior? What is q(0) = ? How does this tell us that we can expect at least two roots?

Methods for Finding Zeros Graph and ask for x-axis intercepts Use solve(y1(x)=0,x) Use zeros(y1(x),x) When complex roots exist, use cSolve() or cZeros()

Practice Given y = (x + 4)(2x – 3)(5 – x) What is the degree? How many terms does it have? What is the long run behavior? f(x) = x3 +x + 1 is invertible (has an inverse) How can you tell? Find f(0.5) and f -1(0.5)

Assignment Lesson 9.2 Page 400 Exercises 1 – 29 odd