UNIT ONE
TRANSFORMATIONS
MATH 621B
20 HOURS
Revised Apr 9, 02
18
SCO: By the end of grade
12, students will be
expected to:
C3 solve problems using
graphing technology
C23 express
transformations
algebraically and with
mapping rules
C47 investigate and
articulate how various
changes in the
parameters of an
equation affect the
graph
E3 use transformations to
draw graphs
Elaborations - Instructional Strategies/Suggestions
Translations (1.1)
See Teacher’s Resource p.6. Students should review graphing y = x, y =
x2, y = 1/x,
y=
x , y = x , y = x3 .
Invite student groups to read and discuss ex. 1-4 beginning on p.8.
Allow time for the groups to read and discuss the summary of vertical
and horizontal translations on p.9 & 10.
Vertical Translations
If y = f(x) is changed to y = f(x) + k, then the result is a vertical
translation. If k > 0 , then the translation is up.
If k < 0 , then the translation is down.
The transformation follows the mapping rule (x, y) Y (x, y + k).
Ex. If y = x2
is changed to y = x2 + 2
notice that k = 2 and the effect is a translation up 2 units.
Horizontal Translations
If y = f(x) is changed to y = f(x ! h), then the result is a horizontal
translation.
If h > 0 , then the translation is to the right.
If h < 0 , then the translation is to the left.
The transformation follows the mapping rule (x,y) Y (x ! h, y).
Ex. If y = x2
is changed to y = (x + 2)2
notice that h = ! 2 and the effect is a translation to the left 2 units.
Encourage students not to get flustered by all the different types of
function graphs shown in the text. It appears complicated but in fact,
upon closer inspection, the equations for the functions will be given and
the student is asked to place the translation coefficients in the correct
positions in the equation of the original function.
For a combined vertical and horizontal translation: y = f(x ! h) + k
The mapping rule is (x,y)
19
Y (x ! h, y + k).
Worthwhile Tasks for Instruction and/or Assessment
Translations (1.1)
Communication
Describe what translation results from changing y = f(x) to:
a) y = f(x) + 2
b) y = f(x) ! 3
c) y = f(x + 1)
d) y = f(x ! 4)
e) y = f(x ! 5) + 1
f) y = f(x + 2) ! 3
Activity
For the graph of the function y = f(x) shown below, sketch the
graph of each of the following transformations:
a) y = f(x) ! 2
b) y = f(x + 3)
c) y = f(x) + 1
d) y = f(x ! 2)
e) y = f(x + 2) ! 1
Suggested Resources
Translations (1.1)
Math Power 12 p.3 Mental Math #112
Math Power 12 p.10-12 #1-29odd,
31 a,c,e
32 a,c,e
33 a,c,e
34 a,c,e
35 a,c,e
36 a,c,e
39,43
Problem Solving Strategies
Math Power 12 p.15 #3-5,7
Group Activity/Presentation
The graph of the function drawn as a darker line is a vertical
translation of the function y = f(x) = x. Write the equation that
describes the darker line. Check your work on the TI-83.
20
SCO: By the end of grade
12, students will be
expected to:
C3 solve problems using
graphing technology
C23 express
transformations
algebraically and with
mapping rules
C47 investigate and
articulate how various
changes in the
parameters of an
equation affect the
graph
Elaborations - Instructional Strategies/Suggestions
Reflections (1.3)
In Unit 3, Math 521B, matrix multiplication was used to reflect a
geometric figure whose vertices were written in matrix form through the
x-axis, the y-axis and the y = x line. For details see Unit 3 in the Math
521B guide.
Reflection through the x-axis.
If y = f(x) is changed to y = ! f(x) , then the result is a reflection in
the x-axis. The mapping rule is (x,y) Y (x,!y).
Ex. If y = 2x + 1
is changed to y = ! 2x ! 1
notice the x values remain unchanged but the y values change sign,
resulting in a reflection in the x-axis.
Reflection through the y-axis
If y = f(x) is changed to y = f(!x), then the result is a reflection in
the y-axis. The mapping rule is (x,y) Y (!x,y).
Ex. If y = 2x + 1
is changed to y = 2(!x) + 1 = !2x + 1
E3 use transformations to
draw graphs
notice the y values remain unchanged but the x values change sign,
resulting in a reflection in the y-axis.
Reflection through the y = x line
If y = f(x) is changed to x = f(y), then the result is a reflection in the
y = x line. The mapping rule is: (x,y) Y (y,x).
Ex. If
y = 2x + 1
is changed to
x = 2y + 1 or
y =
1
1
x −
2
2
Notice that the
x and y values
are interchanged resulting in a reflection in the y = x line. Invite
students to use the vertical line test on the inverse relation or the
horizontal line test on the original to determine if the inverse is an
inverse function f!1(x).
21
Worthwhile Tasks for Instruction and/or Assessment
Reflections (1.3)
Performance
Shown below is the graph of y =f(x). On the same graph draw
the graph of y = ! f(x). Use the Reflect-View to check your
answer.
Suggested Resources
Reflections (1.3)
Math Power 12 p.25 #1-21,27
Applications p.27 #34,36,37,39,43
Group Activity
The thick graph is a reflection in the y = x line of the thinner
line. The equation of the thinner line is y = ! 2x + 2. Write the
equation of the thick graph.
p.29 Exploring Even and Odd
Functions Green #1,2
Journal
Write a concise explanation of how a function can be reflected
through the x-axis.
22
SCO: By the end of grade
12, students will be
expected to:
C3 solve problems using
graphing technology
C23 express
transformations
algebraically and with
mapping rules
C47 investigate and
articulate how various
changes in the
parameters of an
equation affect the
graph
E3 use transformations to
draw graphs
Elaborations - Instructional Strategies/Suggestions
Stretches of Functions (1.4)
In Unit 3, Math 521B, scalar multiplication was used to create dilations
of a geometric figure. Here students will get a chance to stretch
functions. Stretches can be broken into two types:
a) Vertical Stretches
If y = f(x) is changed to y = af(x), then the result is a vertical
stretch of the function. If a > 1, then the stretch is an expansion.
If 0 < a < 1, then the stretch is a compression.
Y
The transformation follows the mapping rule: (x,y)
(x, ay).
y = 2(x ! 1)(x + 1)(x + 3)
Ex. y = (x ! 1)(x + 1)(x + 3)
notice that a > 1 and thus an expansion takes place. In the two tables for
the same x values the y values are twice as large, hence the graph is
expanded (rises more quickly than normal). The zeros remain the same.
b) Horizontal Stretches
If y = f(x) is changed to y = f(kx), then the result is a horizontal stretch
of the function. If k > 1, then the stretch is a compression.
If 0 < k < 1, then the stretch is an expansion.
The transformation follows the mapping rule: (x,y)
YF
Gx , yIJ.
Hk K
Ex. y = (x ! 1)(x + 1)(x + 3)
y =(
Note: A vertical expansion
leaves the zeros unchanged
while a horizontal
expansion changes the
zeros by that factor
1
1
1
x − 1)( x + 1)( x + 3)
2
2
2
notice that k < 1 and thus an expansion takes place. In the tables for the
same y values the x values are 2 times what they were originally
(expanded). The zeros were multiplied by a factor of 2.
23
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Stretches of Functions (1.4)
Stretches of Functions (1.4)
Journal
Describe the effect the following transformations have on y =
f(x):
a) y = 2 f ( x )
Math Power 12 p.38 #1-6,13-16,
19-21,24-27,
29,31,33-37,
41,42
b) y = f ( 3 x )
1
f ( x)
4
1
d) y = f
x
2
e) y = 3 f ( 4 x )
c) y =
Problem Solving Strategies
F
IJ
G
HK
f) y = −2 f
Math Power 12 p.43 #1,2,5,8
1 I
F
G
H3 xJ
K
g) y = f ( −2 x )
Group Presentation
Pick a function of your own choice and show the graphs of the
original function and the above transformations of it.
Communication/Technology
The lighter graph is a stretch of the darker graph. The
equation of the darker graph is given. Write an equation for
the lighter graph. Check your work on the TI-83.
b)
y = sin x
a) y = x2
c)
y = sin x
24
Note: The best functions to work with
are cubic and sinusoidal to
demonstrate expansions or
compressions. Quadratic functions
can have a vertical expansion look
the same as a horizontal
compression.
SCO: By the end of grade
12, students will be
expected to:
C3 solve problems using
graphing technology
Elaborations - Instructional Strategies/Suggestions
Combinations of Transformations (1.6)
In this section students will explore combinations of transformations and
develop a standard order in which the transformations are to be
applied(unless otherwise specified).
For
y = af(b(x ! h)) + k the standard order is:
a) expansions and compressions (Multiplication by a constant)
C23 express
transformations
algebraically and with
mapping rules
C47 investigate and
articulate how various
changes in the
parameters of an
equation affect the
graph
E3 use transformations to
draw graphs
b) reflections (Multiplication of x or y by !1 or interchanging x and y)
c) translations ( adding or subtracting a constant)
Ex: Compare y = x2 to
The mapping rule is
 1

y = 2
x − 3
 4

2
−1
x
( x, y ) ⇒ ( + h, ay + k ) .
b
So for our specific example: (x,y)
Y (4x + 3, 2y ! 1).
Notice that some of the ordered pairs in the right table are the
transformed ordered pairs from the left table.
2
1

y = 2  ( x − 3)  − 1
4

25
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Combinations of Transformations (1.6)
Combinations of Transformations
Activity
The screens below show the graph of y = (x + 3)2 + 2 and its
image after a reflection through a) the y-axis; b) the x-axis;
c) the y = x line; d) and finally the x-axis then the y-axis:
Write the equation of each of the image reflections:
Math Power 12 p.47 #1-21 odd, 22
25,27,29,44,45
a)
c)
b)
d)
Pencil/Paper
Given f ( x ) = x , sketch the graph of each of the
following:
a) y = f ( x + 1) − 5
b) y = 2 f ( x )
c) y = f
d) y = f ( x + 3) + 1
e) y =
1 I
F
G
H3 xJ
K
1
f (3x )
2
26
Applications
p.48 #49,55
SCO: By the end of grade
12, students will be
expected to:
C47 investigate and
articulate how various
changes in the
parameters of an
equation affect the
graph
Elaborations - Instructional Strategies/Suggestions
Reciprocal and Absolute Value Functions (1.7)
Note to Teachers: This section may be placed previous to section 4.6 in
Unit 4. The students may need a change from all the transformation
work at this time.
Reciprocal Functions
Invite student groups to read p.50; allow time to do the explore exercise
on p.50 and the 7 inquire questions on p.50-51. Using a graphing
calculator may help the students visualize the situations discussed here.
Have them read and discuss the general rules for obtaining from a
function y = f(x) The reciprocal function y =
C90 demonstrate an
understanding for
asymptotic behaviour
in rational functions
1
on p.51.
f ( x)
Challenge student groups to read and discuss on p.51 the rules for
obtaining a reciprocal function. Allow time for students to examine ex.
1-3 & 6 in the text.
Absolute Value Functions
Challenge students to read and discuss ex.4,5 & 7 on p.54,55. Note the
generalization after ex.4:
< when f(x) $ 0, then the graph of y =
f ( x ) is the same as
the graph of y = f(x).
< when f(x) < 0, then the graph of y =
f ( x ) is the graph of
y = f(x) reflected through the x-axis.
An example of interest to students is shown below. Even though
students have not yet graphed trig functions it might be useful to show
them the y = sin x graph using the TI-83.
This is the profile of AC current. The term AC
refers to the fact that the current reverses
direction 60 times/second. AC current can be
rectified to DC current by using a pair of
semi-conductor diodes which allows current
to flow in only one direction. Now the signal looks
like this:
y = sin x
Most electronic devices require DC current.
The DC current that we now have is still not
satisfactory because the device(your
calculator for instance) would be turned off
and on 120 times/second. What must be
added to the circuit is a capacitor which will smooth
out the current output and still have the current as DC.
27
Worthwhile Tasks for Instruction and/or Assessment
Reciprocal and Absolute Value
Functions
Reciprocal and Absolute Value Functions (1.7)
Pencil/Paper/Technology
Draw the graph of y = f(x) and y =
1
f ( x)
where f(x) = x ! 2. Check your work on the TI-83.
Pencil/Paper
Graphs of y = f(x) are shown. Sketch the graph of
y =
1
for each of the following:
f ( x)
Pencil/Paper/Technology
Sketch the graph of y = f(x) and y =
Suggested Resources
f ( x ) where:
a) f(x) = x ! 2
b) f(x) = x2 ! 3
Check your work on the TI-83.
Pencil/Paper
The graph of y = f(x) is shown. Sketch the graph of
y = f ( x) .
Journal
Explain the effect of putting absolute value symbols around a
function.
28
Math Power 12 p.56 #1-19 odd,
20-22,
23-33 odd,
34,35,37