This document discusses how to describe the shape of a cubic function by listing it in standard form, describing the end behavior of the graph, determining the possible number of turning points using a table of values, and determining the increasing and decreasing intervals. It explains that to describe the shape, you identify the sign of the leading coefficient to determine the end behavior and the number of turning points, which is one less than the possible degree. The document also discusses using differences of consecutive y-values in a table to determine the least degree of the polynomial function that could generate the data, with constant first differences indicating linear, constant second differences indicating quadratic, and constant third differences indicating cubic.

1. 5.1 Polynomial Functions
Part 2

2. Describing the Shape of a Cubic
Function
1. List the function in standard form
2. Describe the end behavior of the
graph
3. Determine the possible number of
turning points
4. Use a Table to plot points
5. Determine increasing and
decreasing intervals

3. Example: Describe the Shape of
a Cubic Function
x y
-2
-1
0
1
2

4. Example: Describe the Shape of
a Cubic Function
x y
-2
-1
0
1
2

5. Determine the sign of the leading
coefficient and the degree of the
polynomial.
1. Identify the end behavior
This tells you whether the leading coefficient a
is positive or negative
2. Identify the number of turning points
# of turning points + 1 = possible degree of the
polynomial

6. Determine the sign of the leading
coefficient and the least possible
degree of the polynomial.

7. Determine the sign of the leading
coefficient and the least possible
degree of the polynomial.

8. Using differences to determine
degree
 Given a table of values (or a set of
ordered pairs)
 Analyze the differences of consecutive y
– values, to determine the least-degree
polynomial function that could generate
the data
◦ If the first differences are constant, the
function is linear
◦ If the second differences are constant, the
function is quadratic
◦ If the third differences are constant, the
function is cubic
◦ And so on…

9. Example: Determine the degree
of the polynomial function with
the given data.

10. Example: Determine the degree
of the polynomial function with
the given data.

11. Homework
 P 285 #32 – 39, 41 – 49all, 58