Properties of Polynomial Functions
Cubic Functions: y=ax3+bx2÷cx÷d
1. Using a graphing calculator, adjust the window settings so that the intervals are —5 x 5 and
—10 y 10 on the axes. Sketch the graphs in the space provided below each equation.
d) y=x3—3x2 ÷3x—1
b) v=x3+2x2—4x—8
c) y=x3—2x2—2x—3
a) y=x3+2x2—3x—4
2.
How are these four graphs similar?_____
3.
How are these four equations the same?.
Using a graphing calculator, adjust the window settings so that the intervals are —5 x 5 and
—10 y 10 on the axes. Sketch the graphs in the space provided below each equation.
d) y=—x3+3x2—3x+1
b)y=—x3÷3x2—5x+6 c)y=—x3+x2+5x+3
a) y=—x3—3x2+x÷3
4.
5.
How are these four graphs similar?_____
6.
How are these four equations the same?
Complete the table below
Degree:
even or
Function Degree odd
7.
La)
1.b)
1. c)
1.d)
4. a)
4. b)
4.c)
4.d)
by referring to the graphs in #1 and #4.
Number of Leading
Turning
Coefficient: End behaviour:
Points
as x —>
+ or
J
—
-
f(x)—>
End behaviour:
as x
—
8.
bescribe how the graphs of cubic functions for which a is positive differ from those for which
a is negative..
Quartic Functions: y = ax4 ÷bx3 +cx2 +dx÷e
9. Using a graphing calculator, adjust the window settings so that the intervals are —5 x 5 and
—10 y 10 on the axes. Sketch the graphs in the space provided below each equation.
a) y=x4 —5x2 +2x+2
b) y=x4 .i.3x3 —x—3
d) y=x4 —2x2 +1
c) y=x4 +2x3 +2x÷6
.
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 —5 x 5 and
—10 y 10 on the axes. Sketch the graphs in the space provided below each equation.
a) y=—x4+5x2÷4
b) y=—x4÷x3±3x2—2x—5 c) y=—x4—5x3—5x2+5x+6 d) y=—x4+3x3÷3x2—7x—6
.
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
Degree:
Number of
Coefficient; End behaviour:
Function begree even or odd Turning Points + or ?
as x
-
9.a)
—,
—
End behaviour:
as x —÷ +
1(x)—*
9b)
9. c)
9.d)
12. a)
12. b)
12.c)
12. d)
——l
16. Describe how the graphs of quartic functions for which a is positive differ from those for
which a is negative.
Using a graphing calculator, adjust the window settings so that the intervals are —5 x 5 and
—40 y 20 on the axes. Sketch the graphs in the space provided below each equation and
then complete the table below.
a) f(x)=9x2 —8x—2
b) f(x)=—x4 —3x3 +3x2 +8x+5
c) f(x)=2x6 —13x4 +15x2 +x—17
d)
f(x)=-2x4 -4x3+3x2 +6x+9
g)
f(x)=—x7÷8x5—16x3+8x
Function
17.a)
17. b)
17. c)
17. d)
17. e)
17. f)
17. g)
17. h)
Degree
Degree:
even or odd
e) f(x)=x3 -5x2 ÷3x+4
h)
Number of
Turning Points
f) f(x)=2x5 +7x4 -3x3 -18x2 -20
f(x)=—2x3+8x2 —5x+3
Leading
Coefficient:
+ or -?
End behaviour:
as x
—,
—
End behaviour;
as x
—,
±z
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
The maximum number of turning points in the graph of a
with degree 9 is
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
as x —*
—,
f(x)
—,
as x
—,
+,
f(x)
—*
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 —10 x 10
and —10 y 10 on the axes. Graph each function and complete the chart on the next page.
c)f(x)=x3+2x2—3x-5
a) f(x)=x3—2x2—4x÷6
b) f(x)=x3÷x2—2x—7
d)
f(x)=x4+2x3—x2 —2x
g)
f(x)=—x4—x3÷3x2+x—2
e) f(x)=—x4 ÷2x3÷x2 ÷2x
Function
h)
f) f(x)=2x4 —6x3 +x2 +4x÷5
f(x)=x4+x3+x+1
Degree of Polynomial
Number of Zeroes
a) f(x)=x3—2x2—4x+8
b) f(x)=x3+x2—2x—7
c) f(x)=x3+2x2—3x—5
d) f(x)=x4÷2x3-x2-2x
e) f(x)=—x4+2x3+x2÷2x
.
f) f(x)=2x4—6x3÷x2+4x+5
g) f(x)=—x4—x3+3x2+x—2
h) f(x)=x4+x3+x+1
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
S
7
8
n_(odd)
n_(even)
23. Refer to the graphs of the following polynomial functions to complete the chart below.
a)
c)
x
Function Cubic/Quartic?
Leading
Coefficient:
+ or
2
-
23. a)
End behaviour
as x —> —c
End behaviour
as x —> n
f(x)
f(x)
—>
Number of
Turning Points
—*
23. b)
23. c)
24. Describe the end behaviour of each polynomial function by referring to the degree and the
leading coefficient.
Function
a) f(x)=2x2—3x+5
b) f(x)=—3x3+2x2÷5x+1
c) f(x)=5x3—2x2—2x÷6
d) f(x)= —2x4 ±5x3 —2x2 +3x—t
e) f(x)=O.5x4+2x2—6
f) f(x)=—3x5÷2x3—4x
End behaviour:
as x——
End behaviour:
as x—>±
Quintic Functions: y = ax5 + bx4 +cx3 ÷dx2 +ex + I
25. Using a graphing calculator, adjust the window settings so that the intervals are —5 x
—5 y 5 on the axes. Sketch the graphs in the space provided below each equation.
a)
y=x5—5x3+4x
b)
y=—x5÷2x3—2x2
c)
y=x5+x4—5x3—5x2+4x+4
d)
y=—x5—2x4+2x3÷4x2—x—2
5 and
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
Degree even or odd Turning Points + or ?
Function
-
25. a)
()
End behaviour:
as x —+
End behaviour:
as x —> +
1(x)->
1(x)->
—
25. b)
25. c)
25.d)
I_________
28. Describe how the graphs of quintic functions for which a is positive differ from those for
which a is negative.
.