# 1.9 Homework: horizontal and vertical permutations i

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This module provides practice problems designed to develop some concepts related to horizontal and vertical permutations of functions by graphing.
1. 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
1. $f\left(2\right)=$
2. $f\left(3\right)=$
3. $f\left(4\right)=$

Now, I’m going to define a new function this way: $g\left(x\right)=f\left(x\right)-2$ . 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.

1. $g\left(2\right)=$
2. $g\left(3\right)=$
3. $g\left(4\right)=$

Now, I’m going to define yet another function: $h\left(x\right)=f\left(x-2\right)$ . Think of this as a set of instructions, as follows: Whatever number you are given, subtract two. Then, plug that number into $f\left(x\right)$ .

1. h(2)=
2. h(3)=
3. h(4)=
4. Graph all three functions below. Label them clearly so I can tell which is which!
1. 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 $h\left(t\right)=–16{t}^{2}$ , where t is the time since you dropped the rock, and $h$ is the height of the rock.
1. Fill in the following table.
 time (seconds) 0 ½ 1 1½ 2 2½ 3 3½ height (feet)
1. $h\left(0\right)=0$ . What does that tell us about the rock?
2. All the other heights are negative: what does that tell us about the rock?
3. Graph the function $h\left(t\right)$ . Be sure to carefully label your axes!
1. 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.
1. Fill in the following table for the second rock.
 time (seconds) 0 ½ 1 1½ 2 2½ 3 3½ height (feet)
1. Graph the function $h\left(t\right)$ for the new rock. Be sure to carefully label your axes!
1. 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?
2. The original function was $h\left(t\right)=–16{t}^{2}$ . What is the new function?
$h\left(t\right)=$
(*make sure the function you write actually generates the points in your table!)
3. Does this represent a horizontal permutation or a vertical permutation?
4. 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.
1. 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.
1. 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
1. Graph the function $h\left(t\right)$ for the new rock. Be sure to carefully label your axes!
1. 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?
2. The original function was $h\left(t\right)=–16{t}^{2}$ . What is the new function?
$h\left(t\right)=$
(*make sure the function you write actually generates the points in your table!)
3. 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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