Identify the end behavior of a function Use the Intermediate Value Theorem and the Boundedness Theorem to locate zeros of a function.

1. 3.4 Polynomial Graphs
Chapter 3 Polynomial and Rational Functions

2. Concepts and Objectives
 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

3. Graphing Polynomial Functions
 If we look at graphs of functions of the form ,
we can see a definite pattern:
  n
f x ax
  2
f x x   3
g x x
  4
h x x   5
j x x

4. Graphing Polynomial Functions
 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).

5. Graphing Polynomial Functions
 Compare the graphs of the two functions:
  2
f x x
  
2
2
g x x
  2
h x x
   
 
2
1
j x x

6. Graphing Polynomial Functions
 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.
  
n
f x ax k
   
 
n
f x a x h

7. Graphing Polynomial Functions
 Example: Write the equation of the function of degree 3
graphed below.

8. Graphing Polynomial Functions
 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
3
2
f x a x


9. Graphing Polynomial Functions
 Example (cont.):
To determine what a is, we can pick
a point and plug in values:
 
3 4
f
   
   
3
3 3 2 3 4
f a
 
3 4
a
1
a
   3
2 3
f x x
  
•
   
  
3
2 3
f x a x

10. Multiplicity and Graphs
 What is the multiplicity of ?
The zero 4 has multiplicity 5
 The multiplicity of a zero and whether the multiplicity 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
4
g x x

11. Turning Points
 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.

12. End Behavior
 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 .
   
3
2 8 9
f x x x
  3
2
f x x

13. End Behavior
 Example: Use symbols for end behavior to describe the
end behavior of the graph of each function.
1.
2.
3.
     
4 2
2 8
f x x x x
    
3 2
2 3 5
g x x x x
    
5 3
2 1
h x x x

14. End Behavior
 Example: Use symbols for end behavior to describe the
end behavior of the graph of each function.
1.
2.
3.
     
4 2
2 8
f x x x x even function
opens downward
    
3 2
2 3 5
g x x x x odd function
increases
    
5 3
2 1
h x x x odd function
decreases

15. 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.

16. Intermediate Value Theorem
 Example: Show that has a real
zero between 2 and 3.
    
3 2
2 1
f x x x x

17. Intermediate Value Theorem
 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 2
2 1
f x x x x
 
2 1 2 1 1
1
2
0
0
–1
–2
–1
 
3 1 2 1 1
1
3
1
3
2
6
7

18. Intermediate Value Theorem
 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 1
f 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.

19. 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.

20. Boundedness Theorem
 Example: Show that the real zeros of
satisfy the following conditions,
a) No real zero is greater than 1
b) No real zero is less than –2
    
4 2
5 3 7
f x x x x

21. 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 2
5 3 7
f x x x x

1 1 0 5 3 7
9
9
6
6
1
1
1
1 2

22. 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 2
5 3 7
f x x x x
 
2 1 0 5 3 7
30
–15
–18
9
4
–2
–2
1 23

23. Approximating Real Zeros
 Example: Approximate the real zeros of
     
3 2
8 4 10
f x x x x

24. Approximating Real Zeros
 Example: Approximate the real zeros of
 Open desmos.com/calculator
 Type the function into the table
 Evaluate the zeros as marked.
     
3 2
8 4 10
f x x x x

25. Classwork
 3.4 Assignment (College Algebra)
 Page 352: 22-28, 48-56 (even); page 338: 32-44 (4);
page 326: 42-54 (even)
 3.4 Classwork Check
 Quiz 3.3