Algebra II: Strand 1. Foundations of Functions; Topic 2. Transformations; Task 1.2.3
1
TASK 1.2.3: TRANSFORMATIONS WITH TECHNOLOGY
Solutions
(See worksheet for problems.)
I.
Vertical Translations
3. What type of transformation is created by adding a to the function f?
If a > 0 the function is shifted (slid) up by a units (vertically translated up). If a < 0
the function is shifted (slid) down by a units (vertically translated down).
II.
Horizontal Translations
3. What type of transformation occurs when you replace x with x-a when evaluating
the function f?
If a > 0 the function is shifted to the right (horizontal translation to the right). If a < 0
the function is shifted to the left (horizontal slide to the left).
III.
Vertical Stretch and Compressions (Dilations)
3. What happens to the graph of a function when you multiply the function by a?
When a > 1 the graph is stretched vertically (with any zeros of the function remaining
fixed). The new function appears thinner than the original parent function. The
function’s y-values get larger faster than the original parent function. When 0 < a
< 1 the graph is compressed vertically (with any zeros of the function remaining
fixed). The new function appears wider then the original parent function. This has the
effect of slowing down the rate at which the function values grow. For the following
cases, you may want to wait until part V to discuss them. When -1 < a < 0, the graph
is reflected first about the x-axis and then compressed as in the case when 0 < a < 1.
When a < -1, the graph is reflected first about the x-axis and then stretched as in the
case when a > 1.
IV.
Horizontal Stretches and Compressions (Dilations)
3. What type of transformation occurs when you replace x with ax when evaluating
the function f?
When a > 1 the graph of the function is stretched horizontally (with any y-intercepts
remaining fixed). When 0 < a < 1 the graph of the function is compressed
horizontally (with any y-intercepts remaining fixed). For the following cases, you
may want to wait until part V to discuss them. When -1 < a < 0, the graph of the
function is first reflected about the y-axis and then compressed as in the case when
0 < a < 1. When a < -1, the graph of the function is first reflected about the y-axis and
then stretched as in the case when a > 1. Note that if we begin with a power
function (a function of the form f(x) = bxn where b is a real number, b nonzero,
and n>0 is an integer), then a horizontal stretch by factor a is equivalent to a
vertical stretch by a factor an.
December 10, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 1. Foundations of Functions; Topic 2. Transformations; Task 1.2.3
V.
2
Reflections
2. What happens to the graph of a function when you multiply the function by -1?
The graph of the parent function is reflected about the x-axis.
4. What happens to the graph of a function when you replace x with -x?
The graph of the parent function is reflected about the y-axis. Keep in mind that this
effect is not noticeable for even functions.
VI.
Absolute Value
2. What happens when you graph the absolute value of a function?
The parts of the function for which f(x) > 0 remain unchanged. Those portions of the
function for which f(x) < 0 are reflected about the x-axis causing the appearance of
v-shaped vertices at the zeros of the function.
4. What happens to the graph of a function when you replace x with |x|?
The new graph is created by deleting the portion of the parent function that is in the
second and third quadrants and replacing this with the reflection about the y-axis of
the portion of the parent function that is in the first and fourth quadrants.
5. How do absolute value compositions compare with the reflections in part V?
The absolute value compositions can be thought of as partial reflections on the parent
function (see explanation in VI 2 & 4).
Math Notes
This set of exercises allows participants to develop prototypes for determining transformations
created by multiplying the function by a constant, adding a constant to the function, etc. The
following questions can be asked to emphasize certain global properties of transformations and
also their affects on important features or key points of the parent function.
I. Vertical Translations
• How does this transformation affect the domain of the original function?
It has no effect on the domain.
•
How does this transformation affect the range of the original function?
If the range of the original function is all real numbers, then it has no effect (i.e.
shifting (-∞,∞) to the right or left has no effect on the interval). If the range has
the form [c, ∞), (c, ∞), (∞, d) or (∞,d] then this will result in a change in the left
or right real number endpoint by a units to the right if a > 0 or a units to the left if
a < 0. If the range is a finite interval such as [c, d] or (c, d), then the range is
shifted either a units to the right if a > 0 or a units to the left if a < 0. If the range
is a collection of intervals or single points, the single points will be shifted by a
units and each interval will be affected in the same way as noted above.
•
Give an example of a function for which its range would not be affected.
y = x 3 , y = x , y = log x .
December 10, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 1. Foundations of Functions; Topic 2. Transformations; Task 1.2.3
•
•
3
Give an example of a function for which its range would be affected.
1
, y= x.
y = x2 , y =
1 + x2
How does it affect the shape of the graph?
It does not change the shape of the graph. It only slides the graph up or down.
II. Explore Horizontal Translations
• How does this transformation affect the domain?
Similar to discussion on how the range is affected by a vertical translation.
• How does this transformation affect the range?
It has no effect.
• How does it affect the shape of the graph?
It does not change the shape of the graph. It only slides the graph left or right.
III. Explore Vertical Stretches and Compressions (Dilations)
Have participants trace to any point on y1. Use the down arrow key to move the
cursor to y2. Notice that for a given x-coordinate, the y-coordinate of y2 is three
times the value of the y-coordinate of y1.Also examine the effect a has on the
y-values by looking at the table on the graphing calculator. Notice again that for a
given x-coordinate, the y-coordinate of y2 is three times the value of the ycoordinate of y1.
•
•
•
Do these transformations affect the domain?
No.
Do these transformations affect the range?
Sometimes.
When?
When the leading coefficient is negative, the range of the following functions
1
changes from y ≥ 0 to y ≤ 0: y = x 2 , y = x , y = x 2 , y = 2 , y = e x , y = x .
x
IV. Explore Horizontal Stretches and Compressions (Dilations)
• Does this transformation affect the domain?
Horizontal transformations may affect the domain of a function.
• Does this transformation affect the range?
No.
• Do they affect the shape of the graph?
Yes.
• How is the shape of the graph affected?
If a >0 then the function is horizontally stretched. If a <0 then the function is
compressed horizontally.
December 10, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 1. Foundations of Functions; Topic 2. Transformations; Task 1.2.3
4
V. Explore Reflections
• Does the transformation -f(x) affect the domain of the function?
No.
• Does it affect the range of the function?
Yes, the range is affected for some of the parent functions. Some parent
functions whose ranges are affected are as follows: y = x 2 , y = x ,
1
y = x2 , y = 2 , y = 2x , y = x .
x
• Does the transformation f(-x) affect the domain of the function?
Yes, the domain is affected for some of the parent functions. Some parent
functions whose domains are affected are as follows: y = x , y = ln ( x ) .
• Does it affect the range of the function?
No.
• For which functions does the graph of f(x) look the same as the graph of
f ( !x ) ?
1
y = x 2 , y = x 2 , y = 2 , y = x , and y = cos ( x ) .
x
• What common feature do these graphs share?
They all are symmetric with respect to the y-axis. They are all “even”
functions.
VI. Explore Absolute Value
• Note that y = x is not a transformation of y = x , but if f ( x ) = x , then
h ( x ) = x is the composition of g ( x ) = x and f ( x ) = x , g o f .
• Does this composition affect the domain of the function?
No.
• Does this composition affect the range of the function?
In some cases it does. Whenever the range of the parent function includes
values less than 0, then it changes to include only positive values.
• Identify a parent function such that y1 = | y1|.
Any function whose range is y ≥ 0 fits this criteria, e.g., y = x 2 , y = x ,
1
y= 2, y= x .
x
• Does the composition f o g affect the domain of the function?
In some cases it does. For example when the domain of the parent function is
x ≥ 0, the domain of the transformed function becomes all real numbers.
•
Does this composition affect the range of the function?
In some cases it does. For example the range of y = x3 is all real numbers.
The range of y = | x| 3is y ≥ 0.
•
Does this transformation affect the shape of the graph?
December 10, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 1. Foundations of Functions; Topic 2. Transformations; Task 1.2.3
5
If the graph of the parent function is symmetric with respect to the y-axis, then
the shape of the transformed function is not affected. If the graph of the
parent function is not symmetric with respect to the y-axis, then the shape of
the transformed function is affected.
Teaching Notes
Explain to participants that they are going to use composition of functions and their graphing
calculators as tools to explore transformations. They will enter parent functions one at a time
into y1 . They will enter y2 as a transformation of y1 . This composition approach gives
participants a sort of transformation exploring machine. Illustrate by doing Exercise 1a in
Activity 1 with participants as follows. Enter y1 = x and y2 = y1 ( x ! 1) . Explain that this is
analogous to writing f ( x ) = x and g ( x ) = f ( x ! 1) . Show that y2 = y1 ( x ! 1) = x ! 1 , so the
graph of y2 should be a transformation of y1 .
Instruct participants to work in pairs to sketch the given functions and respective transformations
in Task1.2.3 I. Suggest to participants that they use different graph styles to help distinguish the
graph of the parent function from the graph of the transformed function. Also instruct
participants to graph a, b, and c of each set and then guess the graph of d before checking with
the calculator. Give transparency pens and a transparency of the activity sheets for the different
types of transformations I – IV to some of the groups. Ask the groups to prepare a presentation
of their findings.
December 10, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 1. Foundations of Functions; Topic 2. Transformations; Task 1.2.3
TASK 1.2.3: TRANSFORMATIONS WITH TECHNOLOGY
I.
Explore Vertical Translations
1. Enter into y2 : y2 = y1 + 1
Enter the following functions, one at a time, into y1 .
Use a friendly window.
Sketch the graph of y1 and y2 .
a. y1 = x
b. y1 = log x
c. y1 = 3
d. y1 = !x + 2
c. y1 = 2 x
d. y1 = ! x
2. Now enter y2 = y1 ! 2
a. y1 = 0.5x
b. y1 = x 2
3. Generalize: What happens to the graph of a function when you add a constant to the
function rule?
December 10, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
6
Algebra II: Strand 1. Foundations of Functions; Topic 2. Transformations; Task 1.2.3
II. Explore Horizontal Shifts
1. Enter into y2 : y2 = y1 ( x ! 1)
Enter the following functions, one at a time, into y1 .
Use a friendly window.
Sketch the graphs of y1 and y2 .
a. y1 = x
b.
y1 = x 2
c. y1 =
1
x
d. y1 = x + 1
2. Now enter y2 = y1 ( x + 2 )
a. y1 = 9 ! x 2
b. y1 = log x
c. y1 = x 3
d. y1 = x
3. Generalize: What happens to the graph of a function when you replace x with x ! a ?
December 10, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
7
Algebra II: Strand 1. Foundations of Functions; Topic 2. Transformations; Task 1.2.3
8
III. Explore Vertical Stretches and Compressions
1. Enter into y2 : y2 = 3y1
Enter the following functions, one at a time, into y1 .
Use a friendly window.
Sketch the graphs of y1 and y2 .
a. y1 = 10 x
2. Now enter y2 =
a. y1 = x
b. y1 = x 3
c. y1 = x 3 ! x
d. y1 = x
c. y1 = 2 x
d. y1 = x ! 1
1
y1 .
3
b. y1 = x 4 ! 5x 2 + 4
3. Generalize: What happens to the graph of a function when you multiply the function rule
by a?
December 10, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 1. Foundations of Functions; Topic 2. Transformations; Task 1.2.3
IV. Explore Horizontal Stretches and Compressions
1. Enter into y2 : y2 = y1 (2x) .
Enter the following functions, one at a time, into y1 .
Use a friendly window.
Sketch the graphs of y1 and y2 .
a. y1 = x 3 ! x
b. y1 = x 2
c. y1 = x
d. y1 = 9 ! x 2
c. y1 = ! x
d. y1 = 9 ! x 2
!1 $
2. Now enter y2 = y1 # x &
"2 %
a. y1 = x 4 ! 5x 2 + 4
b. y1 = x 3
3. Generalize: What happens to the graph of a function when you replace x with ax ?
December 10, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
9
Algebra II: Strand 1. Foundations of Functions; Topic 2. Transformations; Task 1.2.3
10
V. Explore Reflections
1. Enter into y2 : y2 = !y1
Enter the following functions, one at a time, into y1 .
Use a friendly window.
Sketch the graphs of y1 and y2 .
a. y1 = x
b. y1 = log x
c. y1 = x 3 ! x
d. y1 = x 2
2. Generalize: What happens to the graph of a function when you multiply the function by
!1 ?
3. Now enter y2 = y1 ( !x )
a. y1 = x
b. y1 = x 2
c. y1 = 2 x
d. y1 = x 3
4. Generalize: What happens to the graph of a function when you replace x with !x ?
December 10, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 1. Foundations of Functions; Topic 2. Transformations; Task 1.2.3
VI.
11
Explore Absolute Value
1. Enter into y2 : y2 = y1
Enter the following functions, one at a time, into y1 .
Use a friendly window.
Sketch the graphs of y1 and y2 .
a. y1 = x 2 ! 2
b. y1 = log x
c. y1 = x 4 ! 5x 2 + 4
d. y1 = x
2. Generalize: What happens when you graph the absolute value of a function?
3. Now enter y2 = y1 ( x )
a. y1 = log x
b. y1 = x 2 ! x ! 6
c. y1 = 2 x
d. y1 = x 2
4. Generalize: What happens to the graph of a function when you replace x with x ?
5. How do absolute value compositions compare with the reflections in part V?
December 10, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.