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

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

Transformations of Monomial Functions y x –2 2 Example 1: The degree is odd, the negative coefficient reflects the graph on the x-axis, this graph is similar to

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

Transformations of Monomial Functions Example 3: (Goes through points (0,1),(-1,0),(-2,1)) y x –2 2 The degree is even, and shifts the graph of 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 When n is odd: If the leading coefficient is negative The graph rises to the left and falls to the right y x –2 2 If the leading coefficient is positive The graph falls to the left and rises to the right

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

Even Odd + -

Applying the leading coefficient test Use the leading coefficient test to describe left and right hand behavior of the graph y x –2 2 Example 1 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 x –2 2 Example 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 x –2 2 Example 3 Verify the answer on you calculator

Review of Yesterday Given: Degree?______ Leading Coefficient Test Right side___ Left Side___ Does it shift?________ Draw the graph!

Mulitiplicity of Zeros How many zeros do you expect? How many zeros do you expect? How many zeros do you get? How many zeros do you get? What do those zeros look like? What do those zeros look like?

The possible number of zeros in any quadratic rely on what?

How many zeros do you expect? How many zeros do you expect? How many zeros do you get? How many zeros do you get? What do those zeros look like? What do those zeros look like?

Name the number of zeros and multiplicities in this 6th degree function:

Homework p. 108-109 #1-8, 9-33x3 p. 109 #36-75x3

#36 Calculate the zeros to three decimal places. Also find the zeros algebraically.

#43

#54 Find a polynomial function that has the given zeros: 0,2,5

#57 Find a polynomial function that has the given zeros:

#61 Sketch a graph showing all zeros, end behavior, and important test points.

#72 Sketch a graph showing all zeros, end behavior, and important test points.

Zeros of Polynomial functions For a polynomial function f of degree n The function f has at most n real zeros. The graph of f has at most n-1 relative extrema (relative minima or maxima) For example: 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: Has at most ______ real zeros Has at most ______ relative extrema Solution: Write the original function Substitute 0 for y Remove common factor Factor Completely 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: Has at most ______ real zeros Has at most ______ relative extrema Solution: So, the real zeros are: x=____________ And, the corresponding x-intercept are: ( , ),( , )

The Intermediate Value Theorem Let a and b be real numbers such that a<b. If f is a polynomial function such that , then in the interval [a,b], f takes on every value between and

#73 Use the Intermediate Value Theorem and a graphing calculator to list the integers that zeros of the function lie within.

The End Of Section 2.2