This module provides practice problems designed to develop some concepts related to horizontal and vertical permutations of functions by graphing.
One way of expressing a function is with a
table. The following table defines the function
$f\left(x\right)$ .
x
0
1
2
3
$f\left(x\right)$
1
2
4
8
$f\left(2\right)=$
$f\left(3\right)=$
$f\left(4\right)=$
Now, I’m going to define a new function this way:
$g\left(x\right)=f\left(x\right)\mathrm{-2}$ . Think of this as a set of instructions, as follows:
Whatever number you are given, plug that number into$f\left(x\right)$ ,
and then subtract two from the answer.
$g\left(2\right)=$
$g\left(3\right)=$
$g\left(4\right)=$
Now, I’m going to define yet another function:
$h\left(x\right)=f(x-2)$ . Think of this as a set of instructions, as follows:
Whatever number you are given, subtract two. Then, plugthatnumber into$f\left(x\right)$ .
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
$h\left(t\right)=\mathrm{\u201316}{t}^{2}$ , where t is the time since you dropped the rock, and
$h$ is the height of the rock.
Fill in the following table.
time (seconds)
0
½
1
1½
2
2½
3
3½
height (feet)
$h\left(0\right)=0$ . 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
$h\left(\mathrm{t}\right)$ . 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.
time (seconds)
0
½
1
1½
2
2½
3
3½
height (feet)
Graph the function
$h\left(t\right)$ for the new rock. Be sure to carefully label your axes!
How does this new function
$h\left(t\right)$ compare to the old one? That is, if you put them side by side, what change would you see?
The original function was
$h\left(t\right)=\mathrm{\u201316}{t}^{2}$ . What is the new function?
$h\left(t\right)=$ (*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
$h=0$ ) at that time.
Fill in the following table for the
third rock.
time (seconds)
0
½
1
1½
2
2½
3
3½
4
4½
5
height (feet)
0
0
0
0
Graph the function
$h\left(t\right)$ for the new rock. Be sure to carefully label your axes!
How does this new function
$h\left(t\right)$ compare to the
original one? That is, if you put them side by side, what change would you see?
The original function was
$h\left(t\right)=\mathrm{\u201316}{t}^{2}$ . What is the new function?
$h\left(t\right)=$ (*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.
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