Advanced Algebra
Activity – Beyond Quadratics
Name:
____
Key Vocabulary
Conjecture –
Function –
End Behavior -
Goal: Describe the end behavior of f(x) = 2x3 – x2 – 7x + 5.
Enter f(x) into your graphing calculator. Choose a large window, and graph.
Notice that f(x) appears to rise for positive x-values.
Notice that f(x) appears to fall for negative x-values.
Activity #1
1)
Consider the functions g(x), h(x), and k(x).
In the space provided under each function, sketch a quick graph of each.
g(x) = 3x3 – 2x2 + x + 4
h(x) = 2x3 + 3x2 + x – 7
k(x) = x3 – 3x + 4
2)
What do the actual equations of g(x), h(x), and k(x) have in common?
3)
Describe the end behavior of g(x), h(x), and k(x).
4)
Make a Conjecture. What can you say about the end behavior of functions of the same type
as g(x), h(x), and k(x)?
Activity #2
1)
Consider the functions a(x), b(x), and c(x).
In the space provided under each function, sketch a quick graph of each.
a(x) = – 2x3 + x + 4
b(x) = - ½ x3 + x2 + x – 5
c(x) = - x3 + 3x - 4
2)
What do the actual equations of a(x), b(x), and c(x) have in common?
3)
Describe the end behavior of a(x), b(x), and c(x).
4)
Make a Conjecture. What can you say about the end behavior of functions of the same type
as a(x), b(x), and c(x)?
Activity #3
1)
Consider the functions g(x), h(x), and k(x).
In the space provided under each function, sketch a quick graph of each.
g(x) = x4 – 2x2 - 3
h(x) = 2x4 - 5x2 + x
k(x) = 3x4 - 3x3 – 2x + 4
2)
What do the actual equations of g(x), h(x), and k(x) have in common?
3)
Describe the end behavior of g(x), h(x), and k(x).
4)
Make a Conjecture. What can you say about the end behavior of functions of the same type
as g(x), h(x), and k(x)?
Activity #4
1)
Consider the functions a(x), b(x), and c(x).
In the space provided under each function, sketch a quick graph of each.
a(x) = –x4 + 3x2 - x
b(x) = - 3x4 + 2x3 + x + 3
c(x) = - 2x4 + 3x + 4
2)
What do the actual equations of a(x), b(x), and c(x) have in common?
3)
Describe the end behavior of a(x), b(x), and c(x).
4)
Make a Conjecture. What can you say about the end behavior of functions of the same type
as a(x), b(x), and c(x)?
Activity #5
1)
Consider the functions g(x), h(x), and k(x).
In the space provided under each function, sketch a quick graph of each.
g(x) = 2x5 + 6x2 - 5
h(x) = ½ x5 - 4x3 + 5x2 - 3
k(x) = 3x5 - x3 – 2x + 6
2)
What do the actual equations of g(x), h(x), and k(x) have in common?
3)
Describe the end behavior of g(x), h(x), and k(x).
4)
Make a Conjecture. What can you say about the end behavior of functions of the same type
as g(x), h(x), and k(x)?
Conclusions
1)
Based upon the conjectures you have made during the prior activities, can you predict the end
behavior of the following function?
f(x) = -3x5 + x4 + 2x3 - 5x
End Behavior: f(x) appears to
f(x) appears to
for positive x-values
for negative x-values
What are your conclusions based upon?
2)
Fill in the following table using your best guess at first. Then use your graphing calculator to
confirm your best guess.
Function
Degree
End Behavior
Sign on
Check
Term
A) h(x) = 3x4 + 2x3 - 4x
for positive x-values
for negative x-values
B) f(x) = -5x6 - 4x3 + 3x2 - 1
for positive x-values
for negative x-values
C) k(x) = x4 + 7x5 - 4x + 3
for positive x-values
for negative x-values
D) c(x) = x4 + 2x3 - 8x5
for positive x-values
for negative x-values
E) g(x) = 4x3 - 3x
for positive x-values
for negative x-values
F) b(x) = 3x6 + 4x3 - 5x2 + 2
for positive x-values
for negative x-values