- One way of expressing a function is with a table. The following table defines the function
f
(
x
)
f(x).
Table 1
| x |
0 |
1 |
2 |
3 |
|
f
(
x
)
f(x) |
1 |
2 |
4 |
8 |
-
f
(
2
)
=
f(2)=
-
f
(
3
)
=
f(3)=
-
f
(
4
)
=
f(4)=
Now, I’m going to define a new function this way:
g
(
x
)
=
f
(
x
)
−2
g(x)=f(x)−2. Think of this as a set of instructions, as follows: Whatever number you are given, plug that number into
f
(
x
)
f(x), and then subtract two from the answer.
-
g
(
2
)
=
g(2)=
-
g
(
3
)
=
g(3)=
-
g
(
4
)
=
g(4)=
Now, I’m going to define yet another function:
h
(
x
)
=
f
(
x
−
2
)
h(x)=f(x−2). Think of this as a set of instructions, as follows: Whatever number you are given, subtract two. Then, plug that number into
f
(
x
)
f(x).
- h(2)=
- h(3)=
- h(4)=
- Graph all three functions below. Label them clearly so I can tell which is which!
- Standing at the edge of the Bottomless Pit of Despair, you kick a rock off the ledge and it falls into the pit. The height of the rock is given by the function
h
(
t
)
=
–16
t
2
h(t)=–16
t
2
, where t is the time since you dropped the rock, and
h
h is the height of the rock.
- Fill in the following table.
Table 2
| time (seconds) |
0 |
½ |
1 |
1½ |
2 |
2½ |
3 |
3½ |
|
height (feet)
|
|
|
|
|
|
|
|
|
-
h
(
0
)
=
0
h(0)=0. What does that tell us about the rock?
- All the other heights are negative: what does that tell us about the rock?
- Graph the function
h
(
t
)
h(t). Be sure to carefully label your axes!
- Another rock was dropped at the exact same time as the first rock; but instead of being kicked from the ground, it was dropped from your hand, 3 feet up. So, as they fall, the second rock is always three feet higher than the first rock.
- Fill in the following table for the second rock.
Table 3
| time (seconds) |
0 |
½ |
1 |
1½ |
2 |
2½ |
3 |
3½ |
|
height (feet)
|
|
|
|
|
|
|
|
|
- Graph the function
h
(
t
)
h(t) for the new rock. Be sure to carefully label your axes!
- How does this new function
h
(
t
)
h(t) compare to the old one? That is, if you put them side by side, what change would you see?
- The original function was
h
(
t
)
=
–16
t
2
h(t)=–16
t
2
. What is the new function?
h
(
t
)
=
h(t)=
(*make sure the function you write actually generates the points in your table!)
- Does this represent a horizontal permutation or a vertical permutation?
- Write a generalization based on this example, of the form: when you do such-and-such to a function, the graph changes in such-and-such a way.
- A third rock was dropped from the exact same place as the first rock (kicked off the ledge), but it was dropped 1½ seconds later, and began its fall (at
h
=
0
h=0) at that time.
- Fill in the following table for the third rock.
Table 4
| time (seconds) |
0 |
½ |
1 |
1½ |
2 |
2½ |
3 |
3½ |
4 |
4½ |
5 |
|
height (feet)
|
0 |
0 |
0 |
0 |
|
|
|
|
|
|
|
- Graph the function
h
(
t
)
h(t) for the new rock. Be sure to carefully label your axes!
- How does this new function
h
(
t
)
h(t) compare to the original one? That is, if you put them side by side, what change would you see?
- The original function was
h
(
t
)
=
–16
t
2
h(t)=–16
t
2
. What is the new function?
h
(
t
)
=
h(t)=
(*make sure the function you write actually generates the points in your table!)
- Does this represent a horizontal permutation or a vertical permutation?
Write a generalization based on this example, of the form: when you do such-and-such to a function, the graph changes in such-and-such a way.
"This is the "main" book in Kenny Felder's "Advanced Algebra II" series. This text was created with a focus on 'doing' and 'understanding' algebra concepts rather than simply hearing about them in […]"