Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.3
1
TASK 3.2.3: PARAMETER CHANGES FROM TABLES — LOOK AT THE MOVES
Solutions
Part I
1. Write a statement that explains the differences in the graphs for the function rules
y = x and y = x + 3.
(b)
(a)
The graph is shifted up three units. Adding three units to the function rule y = x shifts
the graph up three units.
Now look at a table of values for these two rules. Is your statement supported by what
you see in the tables? Do you need to revise your statement? Explain.
x
y1 = x
y2 = x + 3
2
5
8
12
26
2
5
8
12
26
5
8
11
15
29
Many students use the terms “shifted up” or “shifted down” without understanding the
connections to the y-values. The table should require the students to link the “shifted up
3 units” to the individual values on the table. Students need to realize that for every shift
up or down, every y-value for the domain is increased or decreased.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.3
2
2. Look at the function rules, graphs, and tables of y = x – 4 and y = x + 2. Explain what
you see.
(a)
(b)
Students may say that y = x - 4 is shifted up 6 units to make the graph y = x + 2.
x
3
5
8
12
27
y1 = x – 4
-1
1
4
8
23
y2 = x + 2
5
7
10
14
29
Does your statement for problem #1 work for these function rules, graphs, and tables?
Explain.
Here again, we want students to recognize that the “shift up” resulted in every y-value in
the graph of y = x – 4 being increased by 6 units.
3. Sometimes textbooks use the terms “shift up by ___” or a “shift down by ___” to
explain a translation in a graph. Explain how the terms “shift up” or “shift down” can be
seen in the graph, rule, and table in problem #2.
The terms “shifted up” and “shifted down” can be seen on the grid with the physical
movement of the graph. The shift is shown in the function rule by increasing or
decreasing the b in y = mx + b. The table displays the shift up or down with the increase
or decrease in the y values.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.3
3
4. Study each of the tables below and explain if the translations from y1 to y2 represent a
shift up or a shift down and by how much. Justify your answer.
x
3
5
8
12
27
y1
5
7
10
14
29
y2
-1
1
4
8
23
x
3
5
8
12
27
y1
6
8
11
15
30
y2
9
11
14
18
33
The table on the left shows a shift down by 6 units. Each of the y-values in y2 = is 6 less
than the corresponding y1 values.
The table on the right shows a shift up by three. Each of the y-values in y1 = have been
increased by 3 units.
5. Write a function rule and sketch the graph of the data from the tables in problem #4.
Make sure the translation is the same in the rule, graph, and table.
On the table on the left, y1 = x + 2 and y2 = x – 4. The b value in y1 is decreased by 6
units in the function y2. The graph of y1 is shifted down 6 units to model the function
y2 = x – 4.
y=x+2
y=x–4
In the table on the right y1 = x + 3 and y2 = x + 6. The b value in y1 has been increased 3
units thus creating a graphical shift up of 3 units.
y=x+6
y=x+3
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.3
6. Enter these functions into your graphing calculator: (use a standard window)
• y1 = x2
•
y2 = x2 + 3
•
y3 = x2 + 5
Explain what is happening to the graph of each function. Write a statement about
how the change in the rule correlates to the change in the graph.
Increasing the value of c in y = ax2 + c results in a shift up of the graph. The graph
of y = x2 + 3 is shifted up three units from the graph of y = x2.
7. Enter these functions into your graphing calculator:
• y1 = x2
• y2 = x2 – 2
• y3 = x2 – 6
Explain what is happening to the graph of each function. Write a statement about
how the change in the function rule correlates to the change in the graph.
Decreasing the value of c in y = ax2 + c results in a shift down of the graph. The
graph of y = x2 – 2 is shifted down two units from the graph of y = x2.
8. Look at the following graphs, function rules, and tables:
y1 = x2
x
1
2
3
4
6
8
y1 = x2
1
4
9
16
36
64
y2 = x2 + 4
y2 = x2+ 4
5
8
13
20
40
68
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
4
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.3
5
In questions 6 and 7 you wrote a statement comparing the changes in the function rule
to the changes in the graph. Does you statement reflect the changes in the table?
Explain.
Yes, the statements in question 6 and 7 explained that the change in c in y = ax2 + c
resulted in a shift of the graph up or down. This is seen in the table of values. Every
y-value in y2 has a value 4 more units than its corresponding y value in y1.
9. Look at the tables below and explain if they represent a “shift-up” or a “shift-down”
and by how much. Explain how you know. Write function rules and sketch the
graphs of the data from the tables below.
a.
x
y1
y2
1
1
-1
3
9
7
7
49
47
8
64
62
10
100
98
12
144
142
y1 = x2
y2 = x2 – 2 The change in
the value of c from 0 to -2 represents
a shift down in the graph. The table
demonstrates this change: for every yvalue in y1 its corresponding y-value
in y2 is 2 units less.
y = x2
y = x2 – 2
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.3
b.
x
y1
y2
3
9
4
5
25
20
7
49
44
8
64
59
11
121
116
13
169
164
y1 = x2
y2 = x2 – 5 The change in
the value of c from 0 to -5 represents a
shift down in the graph. The table
demonstrates this change: for every
y-value in y1 its corresponding y-value
in y2 is 5 units less.
y = x2
y = x2 – 5
c.
y = x2 + 7
x
y1
y2
2
4
11
3
9
16
5
25
32
6
36
43
13
169
176
15
225
231
y = x2
y1 = x2
y2 = x2 +7 The change in
the value of c from 0 to 7 represents a
shift up in the graph. The table
demonstrates this change: for every yvalue in y1 its corresponding y-value in
y2 is 7 units more.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
6
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.3
7
10. The first half of this assignment focused on translating linear functions and relating
the changes in the function rule to the changes in the graphs and tables. The second half
of the assignment focused on translations of quadratic functions and relating changes in
the function rule to changes in the graphs and tables. Write a paragraph explaining how
the translations are similar. Be sure to include all three representations – rules, tables,
and graphs.
Participant responses may differ, but all should note that the changes in b for linear, y =
mx + b and the changes in c for y = ax2 + c both result in the same shifts of the graphs.
If b is increased by 2 and if c is increased by 2, both graphs will be shifted up 2 units.
The tables for both would produce y-values decreased by 2 units. The idea is that a shift
in the graph means a increase or decrease for every point on the graph.
Math notes
Some participants may see y = x + 3 as a shift of y = x to the left 3 units (instead of a shift
up 3 units); however, the graph of y = x2 + 3 does not show a similar shift to the left. The
Algebra I TEKS only include the vertical translation y = x2 + c; Algebra II adds
horizontal translations of quadratics and other functions. Teachers may want to explore
this further.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.3
8
Part II
1. Write a statement that explains the differences in the graphs and function rules for
y = x and y = 3x.
Participants may say the graph got steeper.
Now look at a table of values for these two rules. Is your statement supported by the data
in the tables? Do you need to revise your statement? Explain.
x
2
5
7
9
12
y1 = x
2
5
7
9
12
y2 = 3x
6
15
21
27
36
The table does not show steepness. It shows the y-value in y2 increasing at three times
the rate of the y-value in y1.
2. Look at the function rules, graphs, and tables of y = 2x and y = 5x.
y = 5x
y = 2x
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.3
x
2
5
7
9
12
y1 = 2x
4
10
14
18
24
9
y2 = 5x
10
25
35
45
60
3. Does your statement for problem #1 work for the function rules, graphs, and tables I
problem #2? Explain.
Yes, even though the term “steeper” could describe the slope change in the graph, a
more accurate statement that would reflect both the change in the graph and the table
would be to discuss the rate at which y-values are increasing in y2 compared to the yvalues in y1. The y-values are increasing 2 ½ times as fast, resulting in a “steeper”
slope.
4. Some textbooks use the term “steeper” to explain the transformations in the graphs.
In earlier lessons the greater slope meant that y was getting bigger faster. Explain how
the term “steeper” and y getting “bigger faster” can be seen in the graph, function rule
and table.
If the y-values in y2 are increasing at a faster rate than the y-values in y1, it would mean
that for ever horizontal change the vertical change in y2 would be that much greater than
the vertical change for y1 .
5. Study each of the tables below and explain the transformation from y1 to y2 as
“steeper” or “less steep”. Include a reference to getting “bigger faster”.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.3
x
2
5
7
9
12
y1
2
5
7
9
12
y2
8
20
28
36
48
x
2
5
7
9
12
y1
6
15
21
27
36
10
y2
12
30
42
54
72
In the table on the left, the y-values in y2 are getting “bigger faster” than the y-values in
y1. The y-values in y2 are increasing at a rate 4 times as fast as the y-values in y1.
In the table on the left, the y-values in y2 are getting “bigger faster” than the y-values in
y1. The y-values in y2 are increasing at a rate 2 times as fast as the y-values in y1
6. Write a function rule and sketch the graph of the data from the tables in problem #5.
Make sure the transformation is the same in the rule, graph, and table.
y = 6x
y = 4x
y=x
y = 3x
7. Enter these three function rules into your graphing calculator:
• y=x2
•
y = 3x2
•
y = 6x2
Explain what is happening to the graph of each function rule. Write a statement
about how the change in the rule correlates to the change in the graph.
Participants may say the graphs get skinnier as the coefficient of x2 increases.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.3
11
8. Look at the following graphs, function rules, and tables:
y = 4x2
y = x2
x
y1 = x2
y2 = 4x2
−2
4
16
−1
0
1
0
4
0
1
1
4
2
4
16
In question #7 you wrote a statement comparing the changes in the function rule to the
changes in the graph. Is your statement supported by the table, graphs, and rules shown
here? Explain.
The table shows for every y-value in y1 the corresponding y-value in y2 is 4 times greater.
The graph is not just getting skinnier, the y-value is increasing faster causing a vertical
stretch in the graph.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.3
12
9. Look at the tables below and explain in which, y1 or y2, the y value is getting “bigger
faster”. Write a function rule and sketch the graphs of the tables below.
a.
x
−3
−2
−1
0
1
2
3
y1
9
4
1
0
1
4
9
y2
18
8
2
0
2
8
18
y1 = x2 y2 = 2x2 The y-values in y2 are
“getting bigger” faster than the y-values
in y1. The y-values are increasing twice
as fast in y2 as in y1.
y=2x2
y = x2
b.
y1 = 2x2 y2 = 5x2 The y-values in y2 are
“getting bigger” faster than the y-values in
y1. The y-values are increasing 2.5 times as
fast in y2 as in y1.
x
−3
−2
−1
0
1
2
3
y1
18
8
2
0
2
8
18
y2
45
20
5
0
5
20
45
y = 5x2
y = 2x2
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.3
c.
x
−3
−2
−1
0
1
2
3
y1
27
12
3
0
3
12
27
y2
54
24
6
0
6
24
54
13
y1 =3x2 y2 = 6x2 The y-values in y2
are “getting bigger” faster than the yvalues in y1. The y-values are
increasing two times as fast in y2 as in
y1.
y = 6x2
y = 3x2
10. In this activity you observe the effects of changing the m in y = mx and the a in y =
ax2. Write a paragraph comparing and explaining the effects of increasing m and a.
Be sure to include all three representations – function rules, tables, and graphs.
Student answers will vary, but each should include references that relate the increasing
value of a to the resulting increase in the y-values. As the a increases the y-values grows
at a faster rate resulting in a “skinnier” graph. This is similar to the affect of increasing
the value of m in y = mx + b.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.3
14
TASK 3.2.3: PARAMETER CHANGES FROM TABLES — LOOK AT THE MOVES
Part I
1. Write a statement that explains the differences in the graphs for the function rules
y = x and y = x + 3.
(b)
(a)
Now look at a table of values for these two rules. Is your statement supported by what
you see in the tables? Do you need to revise your statement? Explain.
x
y1 = x
y2 = x + 3
2
5
8
12
26
2
5
8
12
26
5
8
11
15
29
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.3
15
2. Look at the function rules, graphs, and tables of y = x – 4 and y = x + 2. Explain what
you see.
(a)
(b)
x
3
5
8
12
27
y1 = x – 4
-1
1
4
8
23
y2 = x + 2
5
7
10
14
29
Does your statement for problem #1 work for these function rules, graphs, and tables?
Explain.
3. Sometimes textbooks use the terms “shift up by ___” or a “shift down by ___” to
explain a translation in a graph. Explain how the terms “shift up” or “shift down” can be
seen in the graph, rule, and table in problem #2.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.3
16
4. Study each of the tables below and explain if the translations from y1 to y2 represent a
shift up or a shift down and by how much. Justify your answer.
x
3
5
8
12
27
y1
5
7
10
14
29
y2
-1
1
4
8
23
x
3
5
8
12
27
y1
6
8
11
15
30
y2
9
11
14
18
33
5.
Write a function rule and sketch the graph of the data from the tables in problem #4.
Make sure the translation is the same in the rule, graph, and table.
6.
Enter these functions into your graphing calculator: (use a standard window)
• y1 = x2
•
y2 = x2 + 3
•
y3 = x2 + 5
Explain what is happening to the graph of each function. Write a statement about
how the change in the rule correlates to the change in the graph.
7. Enter these functions into your graphing calculator:
• y1 = x2
• y2 = x2 – 2
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.3
•
17
y3 = x2 – 6
Explain what is happening to the graph of each function. Write a statement about
how the change in the function rule correlates to the change in the graph.
8. Look at the following graphs, function rules, and tables:
y1 = x2
x
1
2
3
4
6
8
y1 = x2
1
4
9
16
36
64
y2 = x2 + 4
y2 = x2+ 4
5
8
13
20
40
68
In questions 6 and 7 you wrote a statement comparing the changes in the function rule
to the changes in the graph. Does you statement reflect the changes in the table?
Explain.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.3
18
9. Look at the tables below and explain if they represent a “shift-up” or a “shift-down”
and by how much. Explain how you know. Write function rules and sketch the
graphs of the data from the tables below.
a.
x
y1
y2
1
1
-1
3
9
7
7
49
47
8
64
62
10
100
98
12
144
142
x
y1
y2
3
9
4
5
25
20
7
49
44
8
64
59
11
121
116
13
169
164
x
y1
y2
2
4
11
3
9
16
5
25
32
6
36
43
13
169
176
15
225
231
b.
c.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.3
19
10. The first half of this assignment focused on translating linear functions and relating
the changes in the function rule to the changes in the graphs and tables. The second half
of the assignment focused on translations of quadratic functions and relating changes in
the function rule to changes in the graphs and tables. Write a paragraph explaining how
the translations are similar. Be sure to include all three representations – rules, tables,
and graphs.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.3
20
Part II
1.
Write a statement that explains the differences in the graphs and function rules
for y = x and y = 3x.
Now look at a table of values for these two rules. Is your statement supported by the data
in the tables? Do you need to revise your statement? Explain.
x
2
5
7
9
12
y1 = x
2
5
7
9
12
y2 = 3x
6
15
21
27
36
2. Look at the function rules, graphs, and tables of y = 2x and y = 5x.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.3
y = 5x
x
2
5
7
9
12
y1 = 2x
4
10
14
18
24
21
y = 2x
y2 = 5x
10
25
35
45
60
3. Does your statement for problem #1 work for the function rules, graphs, and
tables I problem #2? Explain.
4.
Some textbooks use the term “steeper” to explain the transformations in the
graphs. In earlier lessons the greater slope meant that y was getting bigger faster.
Explain how the term “steeper” and y getting “bigger faster” can be seen in the
graph, function rule and table.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.3
22
5. Study each of the tables below and explain the transformation from y1 to y2 as
“steeper” or “less steep”. Include a reference to getting “bigger faster”.
x
2
5
7
9
12
y1
2
5
7
9
12
y2
8
20
28
36
48
x
2
5
7
9
12
y1
6
15
21
27
36
y2
12
30
42
54
72
6. Write a function rule and sketch the graph of the data from the tables in problem
#5. Make sure the transformation is the same in the rule, graph, and table.
7. Enter these three function rules into your graphing calculator:
• y=x2
•
y = 3x2
•
y = 6x2
Explain what is happening to the graph of each function rule. Write a statement
about how the change in the rule correlates to the change in the graph.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.3
23
8. Look at the following graphs, function rules, and tables:
y = 4x2
y = x2
x
y1 = x2
y2 = 4x2
−2
4
16
−1
0
1
0
4
0
1
1
4
2
4
16
In question #7 you wrote a statement comparing the changes in the function rule to the
changes in the graph. Is your statement supported by the table, graphs, and rules shown
here? Explain.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.3
24
9. Look at the tables below and explain in which, y1 or y2, the y value is getting
“bigger faster”. Write a function rule and sketch the graphs of the tables below.
a.
x
−3
−2
−1
0
1
2
3
y1
9
4
1
0
1
4
9
y2
18
8
2
0
2
8
18
x
−3
−2
−1
0
1
2
3
y1
18
8
2
0
2
8
18
y2
45
20
5
0
5
20
45
b.
c.
x
−3
−2
−1
0
1
2
3
y1
27
12
3
0
3
12
27
y2
54
24
6
0
6
24
54
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.3
25
10. In this activity you observe the effects of changing the m in y = mx and the a in y
= ax2. Write a paragraph comparing and explaining the effects of increasing m
and a. Be sure to include all three representations – function rules, tables, and
graphs.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.