Transformations Activity

Transformations Activity

 

In this activity you will discover various ways in which functions can be transformed (i.e., stretched or compressed vertically or horizontally, shifted up or down or left or right).

 

Start by clicking on the link below or by typing the link into a web browser.

https://www.desmos.com/calculator/fneiikwfiy

 

Notice that there are two functions defined.

And

 

 

The function  is defined in terms of . The parameters , , and  are modifiers of the function . The initial values of the parameters are . With these initial conditions,  is identical to , i.e., the parameters have no effect on the original function. But, what happens if we change the values of  and/or ?

 

Note: Since  is defined in terms of , we call  the parent function.

 

 

1. Exploring

Let’s discover the effects of parameter  first. As you follow through the next set of instructions, notice how the graph of  changes. Also notice how the values in the table change.

 

1. Set and

 

1. Change the value of by adjusting the slider so that d is greater than 0 (d > 0), describe how the graph of g changes as d increases. How does the graph of g compare to f ? Why?

 

 

 

 

 

 

 

 

 

1. Change the value of d by adjusting the slider so that d is less than 0 (d < 0), describe how the graph of g changes as d decreases. How does the graph of g compare to f ? Why?

 

 

 

 

 

 

 

 

 

2. Set d = 2.
3. Describe specifically how compares to . Remember to look at the table as well as the graph to help you describe specifically how  compares to . Explain why the relationship that you are describing occurs.

 

 

 

 

 

 

 

1. Write the formula for the function in terms of .

(I am providing the answer this time so that you can see what I mean by the statement “ in terms of ”.)

 

 

iii.  Given that , write the formula for the function  in terms of .

(Again, I am providing the answer this time so that you can see what I mean by the statement “ in terms of ”.)

 

 

5. Set d = –5.
6. Describe specifically how compares to . Remember to look at the table as well as the graph to help you describe specifically how  compares to . Explain why the relationship that you are describing occurs.

 

 

 

 

 

 

 

1. Write the formula for the function in terms of .

 

 

 

 

iii.  Given that , write the formula for the function  in terms of .

 

 

 

 

1. How would we describe the transformation effect that a non-zero value for has on the graph of a function? Be specific.

 

 

 

 

2. Exploring

Now let’s discover the effects of parameter . As you follow through the next set of instructions, notice how the graph of  changes. Also notice how the values in the table change.

1. Set and

 

1. Change the value of a by adjusting the slider so that a is greater than 1 (a > 1), describe how the graph of g changes as a increases. How does the graph of g compare to f ? Why?

 

 

 

 

 

 

 

1. Change the value of a by adjusting the slider so that a is between 0 and 1 (0 < a < 1), describe how the graph of g changes for these values of a. How does the graph of g compare to f ? Why?

 

 

 

 

 

 

 

1. Change the value of a by adjusting the slider so that a is less than 0 (a < 0), describe how the graph of g changes as a decreases. How does the graph of g compare to f ? Why?

 

 

 

 

 

 

 

1. Set a = 0.5.
2. Describe specifically how compares to . Remember to look at the table as well as the graph to help you describe specifically how  compares to . Explain why the relationship that you are describing occurs.

 

 

 

 

 

 

1. Write the formula for the function in terms of .

 

 

 

iii.  Given that , write the formula for the function  in terms of .

 

5. Set a = 5.
6. Describe specifically how compares to . Remember to look at the table as well as the graph to help you describe specifically how  compares to . Explain why the relationship that you are describing occurs.

 

 

 

 

 

 

 

 

 

1. Write the formula for the function in terms of .

 

 

 

 

iii.  Given that , write the formula for the function  in terms of .

 

 

 

 

 

1. Set a = –1.
2. Describe specifically how compares to . Remember to look at the table as well as the graph to help you describe specifically how  compares to . Explain why the relationship that you are describing occurs.

 

 

 

 

 

 

 

 

 

 

1. Write the formula for the function in terms of .

 

 

 

 

iii.  Given that , write the formula for the function  in terms of .

 

 

 

 

 

 

 

2. Set a = –2.
3. Describe specifically how compares to . Remember to look at the table as well as the graph to help you describe specifically how  compares to . Explain why the relationship that you are describing occurs.

 

 

 

 

 

 

 

 

 

1. Write the formula for the function in terms of .

 

 

 

 

iii.  Given that , write the formula for the function  in terms of .

 

 

 

 

 

 

1. How would we describe the transformation effect that a non-zero value for a has on the graph of a function? Be specific.

 

 

 

 

 

 

 

 

 

3. Using parameters and  to transform functions

Suppose you are given a function m(x). Note,  has not been defined as any particular function. It is not important what the function  is, it is just a generic function.

1. Suppose you want to create a new function which is a vertical compression of the parent function  by a factor of 0.43.
2. Write the formula for the function in terms of m.

 

 

 

1. How does this transformation affect all the outputs of the original function?

 

 

 

 

1. Suppose you want to create a new function which adds 10 to all the outputs of the parent function .
2. Write the formula for the function in terms of m.

 

 

 

1. What kind of transformation is this?

 

 

 

 

 

2. Suppose you want to create a new function which stretches m vertically by a factor of 2.5 and shifts  down 7 units.
3. Write the formula for the function in terms of m.

 

 

 

 

1. Does it matter in which order the transformations are completed?

 

 

 

 

 

 

4. Exploring

Now let’s discover the effects of parameter . As you follow through the next set of instructions, notice how the graph of  changes. Also notice how the values in the table change.

1. Set and

 

1. Change the value of c by adjusting the slider so that c is greater than 0 (c > 0), describe how the graph of g changes as c increases. How does the graph of g compare to f ? Why?

 

 

 

 

 

 

 

 

1. Change the value of c by adjusting the slider so that c is less than 0 (c < 0), describe how the graph of g changes as c decreases. How does the graph of g compare to f ?  Why?

 

 

 

 

 

 

 

 

2. Set c = 2.
3. Describe specifically how compares to . Remember to look at the table as well as the graph to help you describe specifically how  compares to . Explain why the relationship that you are describing occurs.

 

 

 

 

 

 

 

 

 

1. Write the formula for the function in terms of .

 

 

 

iii.  Given that , write the formula for the function g in terms of x.

 

 

 

 

5. If c = –5.
6. Describe specifically how compares to . Remember to look at the table as well as the graph to help you describe specifically how  compares to . Explain why the relationship that you are describing occurs.

 

 

 

 

 

 

 

 

 

1. Write the formula for the function in terms of .

 

 

 

 

iii.  Given that , write the formula for the function g in terms of x.

 

 

 

 

 

1. How would you describe the transformation effect that a non-zero value for c has on the graph of a function? Be specific.

 

 

Given a parent function f and a transformed function

 

 

the transformations caused by parameters  are as follows:

 

Parameter  causes a vertical shift. If , then there is a shift up. If  then there is a shift down.

 

Parameter  causes a vertical stretch or compression. If , then there is a vertical stretch.                If  , then there is a vertical compression.

 

Parameter  causes a vertical reflection if  is negative.

 

Parameter  causes a horizontal shift. If , then the shift is to the right. If , then the shift is to the left.

 

 

 

5. Using parameters and  to transform functions

Suppose you are given a function . Note,  has not been defined as any particular function. It is not important what the function  is, it is just a generic function.

 

For the following problems, create a new function , by transforming the parent function  using the transformations stated. Write the formula for the function  in terms of .

 

1. Vertical shift up 5, Horizontal shift left 2

 

 

 

 

 

10. Vertical stretch by a factor of 10.5 and vertical reflection

 

 

 

 

 

1. Vertical shift down 3, Horizontal shift right 19

 

 

 

 

 

 

 

For Questions #6 – 9, assume the parent function is , and complete the following

1. Graph the parent function and the given function in the same window on your graphing device.
2. Describe the transformation(s) of the given function as compared to the parent function (e.g. translation to the left/right/up/down, vertical stretch/shrink, vertical reflection)
3. Write the new function in terms of .
4. State the vertex (point where the maximum or minimum occurs) of the function.

 

6. g(x) = 2f (x) + 3 7.    h(x) = f (x + 2) – 1

 

Vertical stretch by a factor of 2                                                  Horizontal shift left 2

Vertical shift up 3                                                                     Vertical shift down 1

 

 

 

Vertex:                                                                            Vertex:

 

 

 

 

 

 

 

8. k(x) = –0.5f (x – 4)                                                             9.   m(x) = 5f (x – 2) – 10