1.8 Horizontal and vertical permutations

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

• A

  Fill in the following table.

• B

  $h(0)=0$ . What does that tell us about the rock?

• C

  All the other heights are negative: what does that tell us about the rock?

• D

  Graph the function $h(t)$ . 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.

• A

  Fill in the following table for the second rock.

• B

  Graph the function $h(t)$ for the new rock. Be sure to carefully label your axes!

• C

  How does this new function $h(t)$ compare to the old one? That is, if you put them side by side, what change would you see?

• D

  The original function was $h(t)=-\text{16}{t}^2$ . What is the new function?
• $h(t)=$
• (*make sure the function you write actually generates the points in your table!)

• E

  Does this represent a horizontal permutation or a vertical permutation ?

• F

  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 1½seconds later , and began its fall (at $h=0$ ) at that time.

• A

  Fill in the following table for the third rock.

• B

  Graph the function $h(t)$ for the new rock. Be sure to carefully label your axes!

• C

  How does this new function $h(t)$ compare to the original one? That is, if you put them side by side, what change would you see?

• D

  The original function was $h(t)=-\text{16}{t}^2$ . What is the new function?
• $h(t)=$
• (*make sure the function you write actually generates the points in your table!)

• E

  Does this represent a horizontal permutation or a vertical permutation ?

• F

  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.