Polynomial functions of Higher degree Chapter 2.2 You should be able to sketch the graphs of a polynomial function of: Degree 0, a Constant Function Degree 1, a Linear Function Degree 2, a Quadratic Function In this section you will learn how to recognize some of the basic features of graphs of polynomial functions. Using those features, point plotting, intercepts and symmetry you should be able to make reasonably accurate sketches by hand. Polynomial functions of Higher degree Chapter 2.2 Polynomial functions are continuous

y y y 2 2 2 x 2 2 Functions with graphs that are not continuous

are not polynomial functions (Piecewise) Graphs of polynomials cannot have sharp turns (Absolute Value) x 2 x Polynomial functions have graphs with smooth , rounded turns. They are continuous

Polynomial functions of Higher degree Chapter 2.2 The polynomial functions that have the simplest graphs are monomials of the form n f ( x) x , n 0 If n is even-the graph is similar to f ( x) x 2 2 If n is odd-the graph is similar to

y x 2 f ( x) x 3 For n-odd, greater the value of n, the flatter the graph near(0,0) Transformations of Monomial Functions y 2 Example 1: f ( x) x

x 2 5 The degree is odd, the negative coefficient reflects the graph on the x-axis, this graph is similar to f ( x ) x 3 Transformations of Monomial Functions Example 2: y 4

f ( x) x 1 2 (Goes through Point (0,1)) x 2 The degree is even, and has as upward shift of one unit of the graph of 4 f ( x) x Transformations of Monomial Functions

Example 3: h( x) ( x 1) y 4 2 (Goes through points x 2 (0,1),(-1,0),(-2,1)) The degree is even, and shifts the

graph of f ( x) x 4 one unit to the left. Leading Coefficient Test The graph of a polynomial eventually rises or falls. This can be determined by the functions degree odd or even) and by its leading coefficient If the When n is odd: y leading If the leading

coefficient coefficient is 2 is negative positive a n 0 x a 0 The graph falls to the left and rises to the right 2 n The graph rises to the

left and falls to the right Leading Coefficient Test When n is even: y If the leading coefficient is positive a 0 2 n The graph rises to the left and

rises to the right x 2 If the leading coefficient is negative an 0 The graph falls to the left and falls to the right

Even + - Odd Applying the leading coefficient test Use the leading coefficient test to describe left and right hand behavior of the graph y Example 1 3 f ( x) x 4 x

2 x 2 Verify the answer on you calculator Applying the leading coefficient test Use the leading coefficient test to describe left and right hand behavior of the graph y Example 2 4 2

f ( x) x 5 x 4 2 x 2 Verify the answer on you calculator Applying the leading coefficient test Use the leading coefficient test to describe left and right hand behavior of the graph y Example 3 5

f ( x) x x 2 x 2 Verify the answer on you calculator Review of Yesterday 5 f ( x )

( x 1 ) Given: Degree?______ Leading Coefficient Test Right side___ Left Side___ Does it shift?________ Draw the graph! Multiplicity of Zeros x 2 5x 6 0 How many zeros do you expect?

How many zeros do you get? What do those zeros look like? x 2 4 x 4 0 How many zeros do you expect? How many zeros do you get? What do those zeros look like? The possible number of zeros in any quadratic rely on what? b b 2 4ac x 2a x 3 9 x 2 27 x 27 0 How many zeros do you expect? How many zeros do you get?

What do those zeros look like? 3 2 x x 9x 9 How many zeros do you expect? How many zeros do you get? What do those zeros look like? Name the number of zeros and th multiplicities in this 6 degree function: Zeros of Polynomial functions

For a polynomial function f of degree n 1. The function f has at most n real zeros. 2. The graph of f has at most n-1 relative extrema (relative minima or maxima) 5 For example: y 3 x 2 x 7 Has at most ______ real zeros Has at most ______ relative extrema Finding Zeros of a Polynomial function (This is very similar to finding the x-intercepts) Example: 3

2 y x x 2x Has at most ______ real zeros Has at most ______ relative extrema Solution: 3 2 y x x 2x

Write the original function 3 2 0 x x 2x Substitute 0 for y 2 Remove common factor 0 x( x x 2) Factor Completely

0 x( x 2)( x 1) So, the real zeros are x=0,x=2, and x=-1 And, the corresponding x-intercepts are: (0,0),(2,0),(-1,0) Finding Zeros of a Polynomial function (This is very similar to finding the x-intercepts) Practice: 2 y x x 6 Has at most ______ real zeros Has at most ______ relative extrema Solution:

So, the real zeros are: x=____________ And, the corresponding x-intercept are: ( , ),( , ) Finding Zeros of a Polynomial Functions Find all zeros of graphically.

5 3 2 f ( x) x 3x x 4 x 1 Finding a Polynomial Function with Given Zeros Find polynomial functions with the following zeros. (there are many correct solutions). -1/2, 3, 3 Finding a Polynomial Function with Given Zeros

3, 2 11, 2 11 The Intermediate Value Theorem Intermediate Value Theorem Let a and b be real numbers such that a

graphing calculator to list the integers that zeros of the function lie within. 3 2 f ( x ) x 3x 3 Homework Page 112-114 1-79 odd The End Of Section 2.2