This document discusses key concepts for graphing polynomial functions including: - Making a table of x and f(x) values to graph a function such as f(x) = -x3 - 4x2 + 5. - The graph of this cubic function will cross the x-axis 3 times, which is indicated by looking at the degree of the polynomial. - Finding relative maxima and minima by making a table of x and f(x) values and looking for the highest and lowest points when graphing. For the function f(x) = x3 – 4x2 + 5, the relative max is near x = 0 and the relative min is near x = 3.

1. 6.4 Graphing Polynomial
Functions
Relative maximum
Relative minimum

2. Making a table to graph
Graph the function
f(x) = -x3 - 4x2 + 5
The graph will cross the
x axis 3 or less times
since the degree is 3.
x
-4
-3
-2
-1
0
1
2
3
4
f(x)
5
-4
-3
2
5
0
-19
-58
-123

3. Making a table to graph
Graph the function
f(x) = -x3 - 4x2 + 5
The graph x f(x)
-4
5
crosses
-3
-4
-2
-3
the x axis
-1
2
0
5
3 times
1
2
3
4
0
-19
-58
-123

4. Find the area where the function
equals zero
f(x) = x4 – x3 - 4x2 + 1
Make a table

5. Find the area where the function
equals zero
f(x) = x4 – x3 - 4x2 + 1
Make a table
x
f(x)
( -2,
9)
( -1,
-1)
( 0,
1)
(1,
-3)
( 2,
-7)
( 3,
19)
( 4,
129)

6. Find the area where the function
equals zero
f(x) = x4 – x3 - 4x2 + 1
Make a table
Where are the
x
f(x)
( -2,
9)
zeros?
( -1,
-1)
between
( 0,
1)
-2 and -1
(1,
-3)
( 2,
-7)
- 1and 0
( 3,
19)
0 and 1
( 4,
129)
2 and 3

7. What are the Relative Maximum or
Relative Minimum
It is an area where the highest or lowest
points happen.
We make a table.
Graph the points and look for the highest
and lowest points.

8. What are the Relative Maximum or
Relative Minimum
Make the table for
f(x) = x3 – 4x2 + 5
x
f(x)
( -2,
-19)
(-1,
0)
( 0,
5)
( 1,
2)
( 2,
-3)
( 3,
-4)
( 4,
5)
( 5,
30)

9. What are the Relative Maximum or
Relative Minimum
Make the table for
f(x) = x3 – 4x2 + 5
x
f(x) Relative Max
( -2, -19)
near x=0
(-1,
0)
( 0,
5)
( 1,
2)
( 2,
-3)
( 3,
-4)
( 4,
5)
( 5,
30)
Relative Min
near x = 3
.

10. Graph the function
Between what numbers are the zeros of
the function?

11. Graph the function
Between what numbers are the zeros of
the function?
Between
-7 and -6
-4 and -3
3 and 4

12. Graph the function
Between what numbers are the Relative
Max. and Min.
of the function?
Between
-6 and -5 R. Min
0 and 2 R. Max

13. Homework
Page 356
# 13 – 25 odd

14. Homework
Page 356
# 14 – 26 even