Polynomial Functions Algebra III, Sec. 2.2

Objective

Use transformations to sketch graphs. Use the Leading Coefficient Test to determine the end behaviors of graphs.Find and use the real zeros. Use the Intermediate Value Theorem to help locate the real zeros.

Important Vocabulary

Continuous – the graph has no breaks, holes, or gaps

Repeated Zero – the zeros from recurring factors

Test Intervals – intervals used to choose x-values to determine the value of the polynomial fn

Intermediate Value Theorem – given two endpoints [a, b] of a continuous fn, there is at least one y value for every x between a and b

Graphs of Polynomial FnsName two basic features of the graphs of polynomial functions.

1. Continuous

2. Only smooth, rounded turns

Example (on your handout)

Will the graph of look more like the

graph of or the graph of ?

Odd degree similar to x3

Example 1Sketch the graph of each function.

a)

Even degree similar to x2

Vertical shrink

Shift right 8

Example 1Sketch the graph of each function.

b)

Even degree similar to x2

Shift down 5

The Leading Coefficient TestFor odd degree polynomials,

If the LC is positive ______________________________

If the LC is negative ______________________________

For even degree polynomials,

If the LC is positive ______________________________

If the LC is negative ______________________________

Example 2Describe the right-hand and left-hand behavior of the graph of each function.

a)

Even degree

Negative LC

So the graph falls to the left and to the right

Example 2Describe the right-hand and left-hand behavior of the graph of each function.

b)

Odd degree

Positive LC

So the graph falls to the left and rises to the right

Example (on your handout)

Describe the right-hand and left-hand behavior of the graph of each function.

Zeros of Polynomial Fns

On the graph of a polynomial function,

turning points are

________________________________________________

________________________________________________

________________________________________________

the relative minima or relative maxima…

points at which the graph changes between increasing and decreasing…

Zeros of Polynomial Fns

Let f be a polynomial function of degree n.

The graph of f has, at most, __________ turning

points.

The function f has , at most, __________ real zeros.

n

n – 1

Zeros of Polynomial Fns

Let f be a polynomial function and let a be a real number. Four equivalent statements about the real zeros of f.

1.

2.

3.

4.

x = a is a zero of the function f

x = a is a solution of the polynomial equation

(x – a) is a factor of the polynomial f(x)

(a, 0) is an x-intercept of the graph of f

Example 3

Find all the real zeros of f(x). Then determine the number of turning points of the graph of the function.

If a polynomial function f has a repeated zero

x = 3 with multiplicity 4, the graph of f

___________ the x-axis at x = 3.

If a polynomial function f has a repeated zero

x = 4 with multiplicity 3, the graph of f

___________ the x-axis at x = 4.

touches

crosses

Example 4Sketch the graph of

Example (on your handout)

Sketch the graph of

Intermediate Value Theorem

If (a, f(a)) and (b, f(b)) are two points on the graph of a polynomial function, then for any number d between f(a) and f(b) there must be a number c between a and b such that f(c)=d.

Intermediate Value Theorem

The Intermediate Value Theorem helps you locate the real zeros of a polynomial function.

If you can find a value x=a at which a polynomial function is positive, and another value x=b at which it is negative, you can conclude that the function has at least one real zero between these two values.

Example 6Use the Intermediate Value Theorem to approximate the real zero(s) of the function.

First, create a table of function values. x -x5+3x3-2x+2 f(x)

-2

-1

0

1

2

sign change a real zero

must be

between 1 &

2

x -x5+3x3-2x+2 f(x)

-2 -(-2)5+3(-2)3-2(-2)+2

-1

0

1

2

x -x5+3x3-2x+2 f(x)

-2 -(-2)5+3(-2)3-2(-2)+2 14

-1

0

1

2

x -x5+3x3-2x+2 f(x)

-2 -(-2)5+3(-2)3-2(-2)+2 14

-1 -(-1)5+3(-1)3-2(-1)+2

0

1

2

x -x5+3x3-2x+2 f(x)

-2 -(-2)5+3(-2)3-2(-2)+2 14

-1 -(-1)5+3(-1)3-2(-1)+2 2

0

1

2

x -x5+3x3-2x+2 f(x)

-2 -(-2)5+3(-2)3-2(-2)+2 14

-1 -(-1)5+3(-1)3-2(-1)+2 2

0 -(0)5+3(0)3-2(0)+2

1

2

x -x5+3x3-2x+2 f(x)

-2 -(-2)5+3(-2)3-2(-2)+2 14

-1 -(-1)5+3(-1)3-2(-1)+2 2

0 -(0)5+3(0)3-2(0)+2 2

1

2

x -x5+3x3-2x+2 f(x)

-2 -(-2)5+3(-2)3-2(-2)+2 14

-1 -(-1)5+3(-1)3-2(-1)+2 2

0 -(0)5+3(0)3-2(0)+2 2

1 -(1)5+3(1)3-2(1)+2

2

x -x5+3x3-2x+2 f(x)

-2 -(-2)5+3(-2)3-2(-2)+2 14

-1 -(-1)5+3(-1)3-2(-1)+2 2

0 -(0)5+3(0)3-2(0)+2 2

1 -(1)5+3(1)3-2(1)+2 2

2

x -x5+3x3-2x+2 f(x)

-2 -(-2)5+3(-2)3-2(-2)+2 14

-1 -(-1)5+3(-1)3-2(-1)+2 2

0 -(0)5+3(0)3-2(0)+2 2

1 -(1)5+3(1)3-2(1)+2 2

2 -(2)5+3(2)3-2(2)+2

x -x5+3x3-2x+2 f(x)

-2 -(-2)5+3(-2)3-2(-2)+2 14

-1 -(-1)5+3(-1)3-2(-1)+2 2

0 -(0)5+3(0)3-2(0)+2 2

1 -(1)5+3(1)3-2(1)+2 2

2 -(2)5+3(2)3-2(2)+2 -10

Example 6Use the Intermediate Value Theorem to approximate the real zero(s) of the function.

Now, divide the interval into tenths & evaluate…

Example 6Use the Intermediate Value Theorem to approximate the real zero(s) of the function.

You can also evaluate the graph…