Algebra 3 Investigation
Function Transformations – Investigation 1
Name:
Investigation 1, Part I: Investigating the Transformation y  T ( x )  k
In Winplot, select the menu “Window” and “2 dim” to create a two-dimensional graph. You will see a coordinate
plane appear with a viewing window that only includes values of x and y between -5 and 5. In order to change the
scale, select the “View” menu and “View” option. Click on “Set Corners,” and type in -10 and 10 for your left and
right values and -10 and 10 for your down and up values.
1. Select the “Equa” menu and “Explicit.” In the window that opens you can type functions just as you can on
your calculator.
2. Enter the function 𝑓(𝑥) = 𝑥 2 , and you should see the toolkit quadratic graph appear.
3. Select “Equa” and “Explicit” again. This time enter the function 𝑦 = 𝑥 2 + 𝑘. You will not see a new function
because Winplot defaults the value of “k” to zero. Write this new function as a transformation of 𝑓(𝑥) defined
in #2. Is the “+k” affecting the input (x’s) or output (y’s) of the toolkit function? How can you tell?
4. Select the “Anim” menu, and “Parameters A-W.” Make sure that you see the “K” in the “Current Value of”
window. You will see a slider bar that will enable you to slide your mouse to adjust the value of “k” and
observe what happens to the graph of the toolkit function with this type of transformation. To make sure that
we can observe what happens with positive and negative k-values, type 8 next to “K,” and click “Set L.”
Then, type 8 next to “K,” and click on the “Set R” button. This will enable our slider to slide between the
values -8 and 8.
5. Slowly move the slider between positive and negative values, and write down a few sentences to describe what
happens to the graph of the toolkit function when you change “k” to different values. Be specific. On the
coordinate plane provided, sketch the graph of the toolkit function and at least two transformed functions to
illustrate you statements. Label each graph, including a couple of points, and feel free to use colors to
distinguish between the graphs.
6. In the table provided below, find the corresponding y-values for the given x-values indicated below. One “k”
value is suggested. You should select three additional “k” values. Compare all y-values for the transformed
function with the y-value of the toolkit function for each x. What patterns do you see? How could you use this
information to find points on a transformed function without plotting points? The goal is to become more
efficient graphers!
k-value
function
0
yx
-1
y  x2 1
2
 1, y 
 1,1
 0, y 
 0, 0 
1, y 
1,1
(-1,
)
(0, )
(1, )
(-1,
(-1,
(-1,
)
)
)
(0, )
(0, )
(0, )
(1, )
(1, )
(1, )
7. Complete steps #1-6 with a different toolkit function, f ( x) 
x , and
y  x  k Do the same patterns that you noted with the previous
function still exist with this example?
y x
 0, y 
 0, 0 
1, y 
1,1
 4, y 
 4, 2 
y  x 1
(0, )
(1, )
(4, )
(0, )
(0, )
(0, )
(1, )
(1, )
(1, )
(4, )
(4, )
(4, )
k-value
function
0
-1
8. Complete steps #1-6 with toolkit function f ( x ) 
1
1
and y   k . This
x
x
time, pay special attention to the vertical and horizontal asymptotes. Do they
change? If so, how and why? Keep track of them in the table.
asymptotes
k-value
0
-1
function
y
y
1
x
1
1
x
 1, y  1, y 
 1, 1 1,1
 2, y 
vertical
horizontal
 1
 2, 
 2
(2, )
x0
y0
(-1,
)
(1, )
(-1,
)
(1, )
(2, )
(-1,
)
(1, )
(2, )
9. In your groups, identify the toolkit functions for the following transformed functions. Write a sentence to
describe how you think that the transformation will affect the graph of the toolkit function. Sketch a graph of
the toolkit function AND the transformed function by thinking through the transformations, not by plotting
points. Then label two points on transformed function. Also, label all asymptotes (note that they may move!).
Only use Winplot to check your work (Note: abs(x) is the Winplot notation for x ).
a.
y  x 4
b. y  2 x  5
c.
y  x3  4.2
d. y 
1
 2.5
x
10. Write a few sentences to summarize what happens to a toolkit function T(x) when you transform it to
y  T ( x)  k and 𝑦 = 𝑇(𝑥) − 𝑘 (𝑘 > 0). How can we use this information to graph functions more quickly
and efficiently?
Investigation I, Part 2: Investigating the Transformation y  T ( x  k )
1. You may wish to delete the functions in Winplot and start over. Select the “Equa” menu and “Explicit.” In the
window that opens you can type functions just as you can on your calculator.
2. Graph the function 𝑓(𝑥) = 𝑥 3 .
3. Select “Equa” and “Explicit” again. This time enter the function y   x  k  . You will not see a new
3
function because Winplot defaults the value of “k” to zero. Write this new function as a transformation of
𝑓(𝑥) defined in #2. Is the “+ k” affecting the input (x’s) or output (y’s) of the toolkit function? How can you
tell?
4.
Again, select the “Anim” menu, and “Parameters A-W.” Make sure that you see the “K” in the “Current Value
of” window. You will see a slider bar that will enable you to slide your mouse to adjust the value of “k” and
observe what happens to the graph of the toolkit function with this type of transformation. To make sure that
we can observe what happens with positive and negative k-values, type 8 next to “K,” and click “Set L.”
Then, type 8 next to “K,” and click on the “Set R” button. This will enable our slider to slide between the
values -8 and 8.
5. Slowly move the slider between positive and negative values, and write down a few sentences to describe what
happens to the graph of the toolkit function when you change “k” to different values. Be specific. On the
coordinate plane provided, sketch the graph of the toolkit function and at least two transformed functions to
illustrate you statements. Label each graph, including a couple of points, and feel free to use colors to
distinguish between the graphs.
6. Notice that the transformation in the second investigation affects the input or “x’s” of the toolkit function. To
understand why the graph changes the way it does, let’s complete another table analysis. Unlike investigation
1, we will hold the output, or y-values, constant. In the table provided below, find the corresponding x-values
that produce the y-values indicated below. One “k” value is suggested. You should select three additional “k”
values. Compare all y-values for the transformed function with the y-value of the toolkit function for each x.
What patterns do you see? How could you use this information to find points on a transformed function
without plotting points? Again, the goal is to become more efficient graphers!
k-value
function
0
y  x3
-1
y   x  1
3
 x, 1
 1, 1
 x, 0 
 0, 0 
 x,1
1,1
( ,-1 )
(
,0)
( ,1)
(
(
(
(
(
(
,0)
,0)
,0)
( ,1)
( ,1)
( ,1)
,-1 )
,-1 )
,-1 )
7. Complete steps #1-6 with a different toolkit function, f ( x) 
x and
y  x  k . Do the same patterns that you noted with the previous
function still exist with this example?
y x
 x, 0 
 0, 0 
 x,1
1,1
 x, 2 
 4, 2 
y  x 1
( , 0)
( ,1)
(
(
(
(
(
(
(
( , 2)
( , 2)
( , 2)
k-value
function
0
-1
, 0)
, 0)
, 0)
,1)
,1)
,1)
, 2)
1
1
and y 
. This
x
xk
8. Complete steps #1-6 with toolkit function f ( x ) 
time, pay special attention to the vertical and horizontal asymptotes. Do
they change? If so, how and why? Keep track of them in the table.
asymptotes
 x,1
k-value
function
 x, 1
0
1
x
1
y
x 1
 1, 1 1,1

( , -1) ( , 1)

,1

,1

,1
-1
y
(
(
, -1) (
, -1) (
, 1)
, 1)
 1
 x, 
 2
2, 1
2

2

2

2

vertical
horizontal
x0
y0
9. In your groups, identify the toolkit functions for the following transformed functions. Write a sentence to
describe how you think that the transformation will affect the graph of the toolkit function. Sketch a graph of
the toolkit function AND the transformed function by thinking through the transformations, not by plotting
points. Then label two points on transformed function. Also, label all asymptotes (note that they may move!).
Only use Winplot to check your work.
a.
y  x4
x 3
c. y  2
b. y   x  5 
d. y 
2
1
x  4.5
10. Write a few sentences to summarize what happens to a toolkit function T(x) when you transform it to 𝑇(𝑥 + 𝑘)
and 𝑇(𝑥 − 𝑘) (𝑘 > 0). Why do you think this happens?
Investigation 1, Part III: Wrapping It Up: Combining Shifts
For each of the following: i) For each function below, identify the appropriate toolkit function, and write a
sentence to describe the transformations to the toolkit function. ii) Sketch a graph of the toolkit function AND the
transformed function by thinking through the transformations, not by plotting points. iii) Label at least two points
on transformed function, and label all asymptotes (Remember: They may move!). iv) Calculate and label all
intercepts (x and y). v) State the domain and range of each function. Only use Winplot to check your work.
a. 𝑦 = √𝑥 + 2 − 3
b. 𝑦 = √𝑥 − 5 + 2
c. f ( x)   x  3  5
d. g ( x)   x  2.2   4.1
2
e. q ( x) 
1
3
x2
g. a( x)  4  x 10
3
x 6
f. h( x)  2  3
h. y   
1
x  2.72