Properties of Polynomial Functions

Cubic Functions:

  1. Using a graphing calculator, adjust the window settings so that the intervals are and on the axes. Sketch the graphs in the space provided below each equation.

a) b) c) d)

  1. How are these four graphs similar?______
  1. How are these four equations the same?______
  1. Using a graphing calculator, adjust the window settings so that the intervals are and on the axes. Sketch the graphs in the space provided below each equation.

a) b) c) d)

  1. How are these four graphs similar?______
  1. How are these four equations the same?______
  1. Complete the table below by referring to the graphs in #1 and #4.

Function / Degree / Degree: even or odd / Number of Turning Points / Leading Coefficient:
+ or - ? / End behaviour:
as / End behaviour:
as
1. a) / /
1. b)
1. c)
1. d)
4. a)
4. b)
4. c)
4. d)
  1. Describe how the graphs of cubic functions for which a is positive differ from those for which

a is negative.______

Quartic Functions:

  1. Using a graphing calculator, adjust the window settings so that the intervals are and on the axes. Sketch the graphs in the space provided below each equation.

a) b) c) d)

  1. How are these four graphs similar?______
  1. How are these four equations the same?______
  1. Using a graphing calculator, adjust the window settings so that the intervals are and on the axes. Sketch the graphs in the space provided below each equation.

a) b) c) d)

  1. How are these four graphs similar?______
  1. How are these four equations the same?______
  1. Complete the table below by referring to the graphs in #9 and #12.

Function / Degree / Degree: even or odd / Number of Turning Points / Leading Coefficient:
+ or - ? / End behaviour:
as / End behaviour:
as
9. a) / /
9. b)
9. c)
9. d)
12. a)
12. b)
12. c)
12. d)
  1. Describe how the graphs of quartic functions for which a is positive differ from those for which a is negative.

______

  1. Using a graphing calculator, adjust the window settings so that the intervals are and on the axes. Sketch the graphs in the space provided below each equation and then complete the table below.

a)b)c)

d) e) f)

g)h)

Function / Degree / Degree: even or odd / Number of Turning Points / Leading Coefficient:
+ or - ? / End behaviour:
as / End behaviour:
as
17. a) / /
17. b)
17. c)
17. d)
17. e)
17. f)
17. g)
17. h)
  1. The maximum number of turning points in the graph of a polynomial function with degree 8 is

______. The maximum number of turning points in the graph of a polynomial function

with degree 9 is ______. The maximum number of turning points in the graph of a

polynomial function of degree n is ______.

  1. Polynomials with EVEN degree have end behaviours that are ______

Polynomials with ODD degree have end behaviours that are ______

  1. State the end behaviours of a function with a degree that is:

a)even and has a positive leading coefficient

b)even and has a negative leading coefficient______

c)odd and has a positive leading coefficient______

d)odd and has a negative leading coefficient______

  1. Using a graphing calculator, adjust the window settings so that the intervals are and on the axes. Graph each function and complete the chart on the next page.

a)b)c)

d) e) f)

g)h)

Function / Degree of Polynomial / Number of Zeroes
a)
b)
c)
d)
e)
f)
g)
h)
  1. Complete the following chart stating the minimum and maximum number of zeroes possible for a polynomial function with each given degree.

Degree / Minimum number of zeroes / Maximum number of zeroes
5
6
7
8
n (odd)
n (even)
  1. Refer to the graphs of the following polynomial functionsto complete the chart below.

a)b)c)

Function / Cubic/Quartic? / Leading Coefficient:
+ or - ? / End behaviour as / End behaviour as / Number of Turning Points
23. a) / /
23. b)
23. c)
  1. Describe the end behaviour of each polynomial function by referring to the degree and the leading coefficient.

Function / End behaviour: as / End behaviour: as
a) / /
b)
c)
d)
e)
f)

Quintic Functions:

  1. Using a graphing calculator, adjust the window settings so that the intervals are and on the axes. Sketch the graphs in the space provided below each equation.

a)b)

c)d)

  1. How is the graph of a quintic function similar to the graph of a cubic function?

______

  1. Complete the table below by referring to the graphs in #25.

Function / Degree / Degree:
even or odd / Number of Turning Points / Leading Coefficient:
+ or - ? / End behaviour:
as / End behaviour:
as
25. a) / /
25. b)
25. c)
25. d)
  1. Describe how the graphs of quintic functions for which a is positive differ from those for which a is negative.

______