Obj. 7 Polynomial Functions and Their Graphs (Presentation)

Obj. 7 Polynomial Graphs

Unit 2 Quadratic and Polynomial Functions

Concepts and Objectives

� Objective #7

� Identify and interpret vertical and horizontal

translations

� Identify the end behavior of a function

� Identify the number of turning points of a function

� Use the Intermediate Value Theorem and the

Boundedness Theorem to locate zeros of a function

� Use the calculator to approximate real zeros

Graphing Polynomial Functions

� If we look at graphs of functions of the form ,

we can see a definite pattern:( ) =

nf x ax

( ) =2f x x ( ) =

3g x x

( ) =4h x x ( ) =

5j x x

� For a polynomial function of degree n

� If n is even, the function is an even function.

� An even function has a range of the form (–∞, k] or

[k, ∞) for some real number k.

� The graph may or may not have a real zero

(x-intercept.)

� If n is odd, the function is an odd function.

� The range of an odd function is the set of all real

numbers, (–∞, ∞).

� The graph will have at least one real zero

(x-intercept).

� Compare the graphs of the two functions:

( ) =2f x x

( ) = −2 2g x x

( ) =2h x x

( ) ( )= −2

1j x x

� Vertical translation

� The graph of is shifted k units up if

k > 0 and |k| units down if k < 0.

� Horizontal translation

� The graph of is shifted h units to the

right if h > 0 and |h| units to the left if h < 0.

( ) = +nf x ax k

( ) ( )= −n

f x a x h

� Example: Write the equation of the function of degree 3

graphed below.

This is an odd function.

The vertex is at (2, 3).

The vertex has been shifted up 3

units and to the right 2 units.

So, it’s going to be something like:

( ) ( )= − +3

32f x a x

� Example (cont.):

To determine what a is, we can pick

a point and plug in values:

( ) =3 4f

( ) ( )= − + =3

3 3 2 3 4f a

+ =3 4a

( ) ( )3

2 3f x x= − +

( ) ( )= − +3

2 3f x a x

Multiplicity and Graphs

� What is the multiplicity of ?

The zero 4 has multiplicity 5

� The multiplicity of a zero and whether the function is

even or odd determines what the graph does at a zero.

� A zero of multiplicity one crosses the x-axis.

� A zero of even multiplicity turns or “bounces” at the

x-axis .

� A zero of odd multiplicity greater than one crosses

the x-axis and “wiggles”.

( ) ( )= −5

4g x x

Turning Points and End Behavior

� The point where a graph changes direction (“bounces”

or “wiggles”) is called a turning point of the function.

� A function of degree n will have at most n – 1 turning

points, with at least one turning point between each

pair of adjacent zeros.

� The end behavior of a polynomial graph is determined by

the term with the largest exponent (the dominating

term).

� For example, has the same end

behavior as .( ) = − +

32 8 9f x x x

( ) =32f x x

End Behavior

� Example: Use symbols for end behavior to describe the

end behavior of the graph of each function.

( ) = − + + −4 22 8f x x x x even function

opens downward

( ) = + − +3 22 3 5g x x x x odd function

increases

( ) = − + +5 32 1h x x x odd function

decreases

Intermediate Value Theorem

� This means that if we plug in two numbers and the

answers have different signs (one positive and one

negative), the function has to have crossed the x-axis

between the two values.

If f(x) defines a polynomial function with only real

coefficients, and if for real numbers a and b, the

values f(a) and f(b) are opposite in sign, then there

exists at least one real zero between a and b.

� Example: Show that has a real

zero between 2 and 3.

You can either plug the values in, or you can use

synthetic division to evaluate each value.

Since the sign changes, there must be a real zero

between 2 and 3.

( ) = − − +3 22 1f x x x x

− −2 1 2 1 1

− −3 1 2 1 1

� If f(a) and f(b) are not opposite in sign, it does not

necessarily mean that there is no zero between a and b.

Consider the function, , at –1 and 3:( ) = − −2 2 1f x x x

f(–1) = 2 > 0 and f(3)= 2 >0

This would imply that there is no

zero between –1 and 3, but we can

see that f has two zeros between

those points.

Boundedness Theorem

Let f(x) be a polynomial function of degree n ≥ 1 with

real coefficients and with a positive leading coefficient.

If f(x) is divided synthetically by x – c, and

(a) if c > 0 and all numbers in the bottom row are

nonnegative, then f(x) has no zeros greater than c;

(b) if c < 0 and the numbers in the bottom row

alternate in sign, then f(x) has no zero less than c.

Boundedness Theorem

� Example: Show that the real zeros of

satisfy the following conditions,

a) No real zero is greater than 1

Since the bottom row numbers are all ≥ 0, f(x) has

no zero greater than 1.

( ) = + + −4 25 3 7f x x x x

−1 1 0 5 3 79

Boundedness Theorem

� Example: Show that the real zeros of

satisfy the following conditions,

b) No real zero is less than –2

Since the signs of the bottom numbers alternate, f(x)

has no zero less than –2.

( ) = + + −4 25 3 7f x x x x

− −2 1 0 5 3 730

Approximating Real Zeros

� Example: Approximate the real zeros of

Step 1: Enter the function into o