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Function Homework -- Horizontal and Vertical Permutations

Module by: Kenny M. Felder. E-mail the author

Summary: This module provides sample problems designed to develop some concepts related to horizontal and vertical permutations of functions by graphing.

Exercise 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(t)=16t2h(t)=16t2 size 12{h $$t$$ = - "16"t rSup { size 8{2} } } {}, where tt size 12{t} {} is the time since you dropped the rock, and h(t)=16t2h(t)=16t2 size 12{h $$t$$ = - "16"t rSup { size 8{2} } } {} is the height of the rock.

• a. Fill in the following table.
Table 1
time (seconds) 0 ½ 1 2 3
height (feet)
• b. h(0)=0h(0)=0 size 12{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)h(t) size 12{h $$t$$ } {}. Be sure to carefully label your axes!

Exercise 2

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.
 time (seconds) 0 ½ 1 1½ 2 2½ 3 3½ height (feet)
• b. Graph the function h(t)h(t) size 12{h $$t$$ } {} for the new rock. Be sure to carefully label your axes!
• c. How does this new function h(t)h(t) size 12{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)=16t2h(t)=16t2 size 12{h $$t$$ = - "16"t rSup { size 8{2} } } {}. What is the new function?
• h ( t ) = h ( t ) = size 12{h $$t$$ ={}} {}
• (*make sure the function you write actually generates the points in your table!)
• e. Does this represent a horizontal permutation or a verticalpermutation?
• 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.

Exercise 3

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=0h=0 size 12{h=0} {}) at that time.

• a. 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
• b. Graph the function h(t)h(t) size 12{h $$t$$ } {} for the new rock. Be sure to carefully label your axes!
• c. How does this new function h(t)h(t) size 12{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)=16t2h(t)=16t2 size 12{h $$t$$ = - "16"t rSup { size 8{2} } } {}. What is the new function?
• h ( t ) = h ( t ) = size 12{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.

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