1
Algebra II: Strand 1. Foundations of Functions; Topic 2. Transformations; Task 1.2.1
TASK 1.2.1: EXPLORING TRANSFORMATIONS
Solutions
Given: Graph of f
Some values for f
f ( !2 ) = 1
f ( !1) = !1
f (0) = 2
f (1) = 1
f (2) = 3
f ( 3) = 0
f (4) = !1
Fill in Table for f
x
y = f (x)
-2
1
-1
-1
0
2
1
1
2
3
3
0
4
-1
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.
2
Algebra II: Strand 1. Foundations of Functions; Topic 2. Transformations; Task 1.2.1
Complete tables and sketch graphs of:
1. y = f ( x + 2 )
x
-4
-3
-2
-1
0
1
2
y
1
-1
2
1
3
0
-1
x
0
1
2
3
4
5
6
y
1
-1
2
1
3
0
-1
2. y = f ( x ! 2 )
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.
3
Algebra II: Strand 1. Foundations of Functions; Topic 2. Transformations; Task 1.2.1
3. y = f ( x ) + 2
x
-2
-1
0
1
2
3
4
y
3
1
4
3
5
2
1
x
-2
-1
0
1
2
3
4
y
-1
-3
0
-1
1
-2
-3
4. y = f ( x ) ! 2
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 2. Transformations; Task 1.2.1
5. y = f ( 2x )
x
-1
-1/2
0
1/2
1
3/2
2
y
1
-1
2
1
3
0
-1
x
-4
-2
0
2
4
6
8
y
1
-1
2
1
3
0
-1
!1 $
6. y = f # x &
"2 %
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 2. Transformations; Task 1.2.1
7. y = 2 f ( x )
8. y =
x
-2
-1
0
1
2
3
4
y
2
-2
4
2
6
0
-2
x
-2
-1
0
1
2
3
4
y
1
f ( x)
2
1
2
!1
2
1
1
2
3/2
0
!1
2
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 1.2.1
9. y = f ( !x )
x
-4
-3
-2
-1
0
1
2
y
-1
0
3
1
2
-1
1
x
-2
-1
0
1
2
3
4
y
-1
-3
0
-1
1
-2
-3
10. y = ! f ( 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.
7
Algebra II: Strand 1. Foundations of Functions; Topic 2. Transformations; Task 1.2.1
11. y = f ( x )
x
-2
-1
0
1
2
3
4
y
1
1
2
1
3
0
1
x
-4
-3
-2
-1
0
1
2
3
4
y
-1
0
3
1
2
1
3
0
-1
12. y = f ( x )
Math Notes
Notice that on each of the sketches to be completed by the participants, the parent function is
sketched lightly in the background. This was done to avoid any small differences in scale when
comparing various sketches. The simultaneous use of the tables and the graphs is purposefully
done to have participants compare among the sketches—what stays the same, what is different
and by how much. It is important to pay special attention to the effects of the absolute value on
the graph 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.
8
Algebra II: Strand 1. Foundations of Functions; Topic 2. Transformations; Task 1.2.1
Teaching Notes
Model the first exercise of the task and emphasize that when we work with transformations and
sketching we want to look for key features in the graph (“corners” or vertices, intercepts, linear
pieces, etc.) and decide where the transformation takes those “special” points or features. These
exercises already identify the special points and set them up in a table. As you model the first
exercise ask the following questions.
• What does f(1) refer to?
The y-value of the function f at x=1. That is, f(1)=1.
• When we fill in the table (for Exercise 1), the first x-value in the table is -4. Why?
We need to find out where the “important points” go and -4 corresponds to the old x-value 4+2=-2. That is f(-4+2)=f(-2)—this is the first important point for the parent function.
Make two sets of transparencies of the exercises. Divide these evenly among the different groups
with no group receiving the same exercise twice. Give each group a transparency pen. For
example, there will be two groups who were given a transparency of exercise 2—have a group
representative come forward and present their sketch to exercise 2. Then, have the second group
who had the same exercise come and lay their transparency over the one already presented to see
if they match (they should). Discuss what the differences were between the sketched
transformation of f and the original function f.
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 1.2.1
TASK 1.2.1: EXPLORING TRANSFORMATIONS
Given: Graph of f
Some values for f
f ( !2 ) = 1
f ( !1) = !1
f (0) = 2
f (1) = 1
f (2) = 3
f ( 3) = 0
f (4) = !1
Fill in Table for f
x
y = f (x)
1
-1
-1
2
1
2
3
-1
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 1.2.1
Complete tables and sketch graphs of:
1. y = f ( x + 2 )
x
-4
-3
-2
-1
0
1
2
y
x
0
1
2
3
4
5
6
y
2. y = f ( x ! 2 )
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.
1
0
Algebra II: Strand 1. Foundations of Functions; Topic 2. Transformations; Task 1.2.1
3. y = f ( x ) + 2
x
-2
-1
0
1
2
3
4
y
x
-2
-1
0
1
2
3
4
y
4. y = f ( x ) ! 2
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.
1
1
Algebra II: Strand 1. Foundations of Functions; Topic 2. Transformations; Task 1.2.1
5. y = f ( 2x )
x
-1
-1/2
0
1/2
1
3/2
2
y
x
-4
-2
0
2
4
6
8
y
!1 $
6. y = f # x &
"2 %
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.
1
2
Algebra II: Strand 1. Foundations of Functions; Topic 2. Transformations; Task 1.2.1
7. y = 2 f ( x )
8. y =
x
-2
-1
0
1
2
3
4
y
x
-2
-1
0
1
2
3
4
y
1
f ( x)
2
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.
1
3
Algebra II: Strand 1. Foundations of Functions; Topic 2. Transformations; Task 1.2.1
9. y = f ( !x )
x
-4
-3
-2
-1
0
1
2
y
x
-2
-1
0
1
2
3
4
y
10. y = ! f ( 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.
1
4
Algebra II: Strand 1. Foundations of Functions; Topic 2. Transformations; Task 1.2.1
11. y = f ( x )
x
-2
-1
0
1
2
3
4
y
x
-4
-3
-2
-1
0
1
2
3
4
y
12. y = f ( 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.
1
5