Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.2
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TASK 1.4.2: PATTY PAPER REFLECTIONS
Solutions
Part I: Reflecting About the x-axis
Possible answers:
1. Graph y = x 2 on the grid. Using patty paper, sketch the reflection of y = x 2 about
the x-axis.
2. How could we algebraically describe the reflected graph?
y = !x 2 or ! f ( x )
3. Is the reflection a function? Why or why not?
Yes, the reflection is a function because for every x-value there is only one y-value.
4. Make a table that shows values for the original function and corresponding values for
the reflected function. Describe patterns.
For every x-value, the y-value of the reflection is the opposite of the y-value of the
function.
• How do you know if the reflection is a function or a relation? For the reflection
to be a function, it must pass the vertical line test. For every x-value, there is only
one y-value.
• How did you construct your table?
• Compare the domain and range of the original graph with its reflection. The
domains are the same. The ranges are exact opposites of each other.
5. Graph y = x3 on the grid. Using patty paper, sketch the reflection of y = x3 about the
x-axis.
6. How could we algebraically describe the reflected graph?
y = !x 3 or ! f ( x ) .
7. Is the reflection a function? Why or why not?
Yes, the reflection is a function because for every x-value there is only one y-value.
8. Make a table that shows values for the original function and corresponding values for
the reflected function. Describe patterns.
For every x-value, the y-value of the reflection is the opposite of the y-value of the
function.
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 4. Making Connections; Task 1.4.2
2
Part II: Reflecting About the y-axis
Possible answers:
1. Graph y = x 2 on the grid. Using patty paper, sketch the reflection of y = x 2 about
the y-axis.
2. How could we algebraically describe the reflected graph?
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y = ( !x ) or f ( !x )
3. Is the reflection a function? Why or why not?
Yes, the reflection is a function because for every x-value there is only one y-value.
4. Make a table that shows values for the original function and corresponding values for
the reflected function. Describe patterns.
For the opposite of every x-value, the y-value of the reflection is the same as the
y-value of the function.
• Does the reflected graph create a new graph? No, the reflection is the same as the
original function.
• Is y = x 2 an even function, an odd function, or neither? Why? y = x 2 = f ( x ) is
an even function because f ( !x ) = f ( x ) .
•
What are some other even functions? y = x 4 , y = x , y =
1
x2
5. Graph y = x 3 on the grid. Using patty paper, sketch the reflection of y = x 3 about the
y-axis.
6. How could we algebraically describe the reflected graph?
y = !x 3 or ! f ( x )
7. Is the reflection a function? Why or why not?
Yes, the reflection is a function because for every x-value there is only one y-value.
8. Make a table that shows values for the original function and corresponding values for
the reflected function. Describe patterns.
For the opposite of every x-value, the y-value of the reflection is the same as the yvalue of the function.
• Does the reflected graph create a new graph? Yes.
• Is y = x 3 an even function, an odd function, or neither? Why? y = x 3 = f ( x ) is
an odd function because f ( !x ) = ! f ( x ) .
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• What are some other odd functions? y = , y = 3 x
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 4. Making Connections; Task 1.4.2
•
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Compare the reflections of y = x3 about the x-axis and about the y-axis. What do
you find? Why? The reflections create the same graph.
Part III: Reflecting About the Line y = x
1. Graph y = x2 and y = x on the grid. Using patty paper, sketch the reflection of y = x2
about the line y = x.
2. Make a table that shows values for the original function and corresponding values for
the reflected function. Describe patterns.
The point (a, b) on the original graph becomes (b, a) on the inverse graph.
3. Is the reflection a function? Why or why not?
No, the reflection is not a function. Each input generates two outputs. For example,
there are two y-values when x = 4.
4. What could you do to make the inverse a function?
You can restrict the domain of the original function to a one-to-one function. Then its
reflection will be a function.
5. Graph y = x3 and y = x on the grid. Using patty paper, sketch the reflection of y = x3
about the line y = x.
6. Make a table that shows values for the original function and corresponding values for
the reflected function. Describe patterns.
The point (a, b) on the original graph becomes (b, a) on the inverse graph.
7. Is the reflection a function? Why or why not?
Yes, the reflection is a function. Each input generates one output.
Math notes
In this task, participants will now use algebraic representations of functions and relations in
addition to graphing to investigate reflections about the x-axis, y-axis, and line y = x.
Teaching notes
Discuss briefly how to use Patty Paper to sketch the reflections. Give transparencies of the task
to pairs of participants and ask them to do each part “live” on the overhead projector.
Participants are to complete Parts I-III along with the presenting pairs. See solutions above for
additional questions that may be asked.
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 4. Making Connections; Task 1.4.2
TASK 1.4.2: PATTY PAPER REFLECTIONS
Part I: Reflecting About the x-Axis
1. Graph y = x 2 on the grid. Using patty paper, sketch the reflection of
y = x 2 about the x-axis.
2. How could we algebraically describe the reflected graph?
3. Is the reflection a function? Why or why not?
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.
4
Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.2
4. Make a table that shows values for the original function and
corresponding values for the reflected function. Describe patterns.
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.
5
Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.2
5. Graph y = x 3 on the grid. Using patty paper, sketch the reflection of
y = x 3 about the x-axis.
6. How could we algebraically describe the reflected graph?
7. Is the reflection a function? Why or why not?
8. Make a table that shows values for the original function and
corresponding values for the reflected function. Describe patterns.
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 4. Making Connections; Task 1.4.2
Part II: Reflecting about the y-axis
1. Graph y = x 2 on the grid. Using patty paper, sketch the reflection of
y = x 2 about the y-axis.
2. How could we algebraically describe the reflected graph?
3. Is the reflection a function? Why or why not?
4. Make a table that shows values for the original function and
corresponding values for the reflected function. Describe patterns.
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 4. Making Connections; Task 1.4.2
5. Graph y = x 3 on the grid. Using patty paper, sketch the reflection of
y = x 3 about the y-axis.
6. How could we algebraically describe the reflected graph?
7. Is the reflection a function? Why or why not?
8. Make a table that shows values for the original function and
corresponding values for the reflected function. Describe patterns.
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.
8
Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.2
Part III: Reflecting About the Line y = x
1. Graph y = x 2 and y = x on the grid. Using patty paper, sketch the
reflection of y = x 2 about the line y = x.
2. Make a table that shows values for the original function and
corresponding values for the reflected function. Describe patterns.
3. Is the reflection a function? Why or why not?
4. What could you do to make the inverse a function?
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 4, Making Connections; Task 1.4.2
10
5. Graph y = x 3 and y = x on the grid. Using patty paper, sketch the
reflection of y = x 3 about the line y = x.
6. Make a table that shows values for the original function and
corresponding values for the reflected function. Describe patterns.
7. Is the reflection a function? Why or why not
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.