Henley Task – Day 2 Name ______

The next few items depart from Henley Chocolates to explore graph transformations that will give us insight about the function A for the area of the bottom of a mini truffle box. We will return to the function A in item #1 on day 3.

  1. Graph each of the following functions on the same axes with the graph of the function f defined by f (x) = x2. Use a new set of axes for each function listed below, but repeat the graph of f each time. For each function listed, describe a rigid transformation of the graph of f that results in the graph of the given function. Make a conjecture about the graph of , where h is any real number.
  2. Conjecture

x / (x - 3)2 / y
0
2
3
4
6

x / (x - 6)2 / y
3
5
6
7
9


x / (x - 4)2 / y
1
3
4
5
7
  1. Graph each of the following functions on the same axes with the graph of the function f defined by f (x) = x2. Use a new set of axes for each function listed below, but repeat the graph of f each time. For each function listed, describe a rigid transformation of the graph of f that results in the graph of the given function.

a. 

x / (x + 2)2 / y
-5
-3
-2
-1
1

b. 

x / (x + 5)2 / y
-8
-6
-5
-4
-2

c. 

x / (x + 6)2 / y
-9
-7
-6
-5
-3
  1. We can view the exercises in item 1 as taking a function, in this case the function, f (x) = x2, and replacing the “x” in the formula with “x – h”. We can view the exercises in item 2 as replacing the “x” in the formula with “x + h”, but we can also view these exercises as replacing the “x” in the formula with “x – h”.

a.  How can this be done?

b.  Does your conjecture from item 1 agree with the transformations you described for item 2? If so, explain how it works. If not, adjust the statement of your conjecture to include these examples also.

c.  What do you think the will happen if we replace the “x” in the formula with “x – h” for other functions in our basic family of functions? Have you seen any examples of such replacements before?

  1. For each pair of functions below, predict how you think the graphs will be related and then graph the two functions on the same axes and check your prediction.

x / / y / x / / y
-3 / -2
-2 / -1
0 / 0
2 / 1
3 / 2

x / / y / x / / y
-2 / 2
-1 / 3
0 / 4
1 / 5
2 / 6

x / / y / x / / y
-4 / -4
-1 / -2
0 / 0
1 / 2
4 / 4

x / / y / x / / y
-4 / -4
-2 / -2
0 / 0
2 / 2
4 / 4
x / / y / x / / y
-2 / 2
-1 / 3
0 / 5
1 / 7
2 / 8