Polynomial Functions on a Graphing Calculator



Properties of Polynomial Functions

Cubic Functions: [pic]

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

a) [pic] b) [pic] c) [pic] d) [pic]

2. How are these four graphs similar?____________________________________________

3. How are these four equations the same?________________________________________

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

a) [pic] b) [pic] c) [pic] d) [pic]

5. How are these four graphs similar?____________________________________________

6. How are these four equations the same?________________________________________

7. Complete the table below by referring to the graphs in #1 and #4.

| | |Degree: even |Number of Turning |Leading Coefficient: | | |

| | |or odd |Points | |End behaviour: |End behaviour: |

|Function |Degree | | |+ or - ? |as [pic] |as [pic] |

|1. a) | | | | |[pic] |[pic] |

|1. b) | | | | | | |

|1. c) | | | | | | |

|1. d) | | | | | | |

|4. a) | | | | | | |

|4. b) | | | | | | |

|4. c) | | | | | | |

|4. d) | | | | | | |

8. Describe how the graphs of cubic functions for which a is positive differ from those for which

a is negative._____________________________________________________________

Quartic Functions: [pic]

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

a) [pic] b) [pic] c) [pic] d) [pic]

10. How are these four graphs similar?____________________________________________

11. How are these four equations the same?________________________________________

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

a) [pic] b) [pic] c) [pic] d) [pic]

13. How are these four graphs similar?____________________________________________

14. How are these four equations the same?________________________________________

15. Complete the table below by referring to the graphs in #9 and #12.

| | | | |Leading Coefficient: | | |

| | |Degree: even or odd |Number of Turning Points| |End behaviour: |End behaviour: |

|Function |Degree | | |+ or - ? |as [pic] |as [pic] |

|9. a) | | | | |[pic] |[pic] |

|9. b) | | | | | | |

|9. c) | | | | | | |

|9. d) | | | | | | |

|12. a) | | | | | | |

|12. b) | | | | | | |

|12. c) | | | | | | |

|12. d) | | | | | | |

16. Describe how the graphs of quartic functions for which a is positive differ from those for which a is negative.

______________________________________________________________________

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

a) [pic] b) [pic] c) [pic]

d) [pic] e) [pic] f) [pic]

g) [pic] h) [pic]

| | | | |Leading Coefficient: | | |

| | |Degree: even or odd |Number of Turning Points| |End behaviour: |End behaviour: |

|Function |Degree | | |+ or - ? |as [pic] |as [pic] |

|17. a) | | | | |[pic] |[pic] |

|17. b) | | | | | | |

|17. c) | | | | | | |

|17. d) | | | | | | |

|17. e) | | | | | | |

|17. f) | | | | | | |

|17. g) | | | | | | |

|17. h) | | | | | | |

18. 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 __________.

19. Polynomials with EVEN degree have end behaviours that are _________________________

Polynomials with ODD degree have end behaviours that are __________________________

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

a) even and has a positive leading coefficient [pic]

b) even and has a negative leading coefficient ______________________________________

c) odd and has a positive leading coefficient ______________________________________

d) odd and has a negative leading coefficient ______________________________________

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

a) [pic] b) [pic] c) [pic]

d) [pic] e) [pic] f) [pic]

g) [pic] h) [pic]

|Function |Degree of Polynomial |Number of Zeroes |

|a) [pic] | | |

|b) [pic] | | |

|c) [pic] | | |

|d) [pic] | | |

|e) [pic] | | |

|f) [pic] | | |

|g) [pic] | | |

|h) [pic] | | |

22. 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) | | |

23. Refer to the graphs of the following polynomial functions to complete the chart below.

a) b) c)

[pic] [pic] [pic]

| | |Leading Coefficient: | | | |

| | |+ or - ? |End behaviour as [pic] |End behaviour as [pic] |Number of Turning Points |

|Function |Cubic/Quartic? | | | | |

|23. a) | | |[pic] |[pic] | |

|23. b) | | | | | |

|23. c) | | | | | |

24. Describe the end behaviour of each polynomial function by referring to the degree and the leading coefficient.

| |End behaviour: as [pic] |End behaviour: as [pic] |

|Function | | |

|a) [pic] |[pic] |[pic] |

|b) [pic] | | |

|c) [pic] | | |

|d) [pic] | | |

|e) [pic] | | |

|f) [pic] | | |

Quintic Functions: [pic]

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

a) [pic] b) [pic]

c) [pic] d) [pic]

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

______________________________________________________________________

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

| | | | |Leading Coefficient: | | |

| | |Degree: |Number of Turning Points|+ or - ? |End behaviour: |End behaviour: |

|Function |Degree |even or odd | | |as [pic] |as [pic] |

|25. a) | | | | |[pic] |[pic] |

|25. b) | | | | | | |

|25. c) | | | | | | |

|25. d) | | | | | | |

28. Describe how the graphs of quintic functions for which a is positive differ from those for which a is negative.

______________________________________________________________________

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