1.9 Homework: horizontal and vertical permutations i

1. One way of expressing a function is with a table. The following table defines the function $f\left(x\right)$ .

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: $g\left(x\right)=f\left(x\right)\mathrm{-2}$ . 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.

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

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

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 $h\left(t\right)=\mathrm{–16}{t}^2$ , where t is the time since you dropped the rock, and $h$ is the height of the rock.

1. Fill in the following table.

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 $h\left(\mathrm{t}\right)$ . 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.

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 $h\left(t\right)=\mathrm{–16}{t}^2$ . 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.

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 $h\left(t\right)=\mathrm{–16}{t}^2$ . 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.