Supplement 2 on Transformations of Functions
Page 1
Part I: Transformations on the Square Root Function
Given our toolkit functions, we shall explore the possible shifts and reflections for these
functions by examining the transformations on the square root function and applying what
we have learned about doing transformations on it to other functions.
f (x) =
x
y
x
0
0
1
1
4
2
9
3
3
1
4
9
x.
Now we shall find the domain and range of f (x) =
A) Domain: [0,  )
B) Range: [0,  )
Question: Why is it wrong to write the domain or range as [0,  ]?
Vertical Shifts
Now let us perform a vertical shift on f (x) by moving f (x) up 4 units. We shall call our new
function g (x). To do this shift we will add 4 to each of the y-values in the table for f (x).
x + 4.
The equation is g (x) =
f (x) =
x
x
y
0
0
1
1
4
2
9
3
g (x) =
x + 4
x
y
0
4
1
5
4
6
9
7
7
g (x) =
x + 4
4
1
Now we shall find the domain and range of g (x) =
A) Domain: [0,  )
4
9
x + 4
B) Range: [4,  )
Note that the domain remains the same as it was for f (x), but the range moves up 4 units.
Supplement 2 on Transformations of Functions
Page 2
Now compare f (x) and g (x) on the same graph, and write an equation for g (x) that includes
f (x).
g (x)
Equation: g (x) = f (x) + 4
f (x)
Exercise 1: Let f (x) =
x.
A) Write an equation that shifts f (x) down 2 units. Call your new function g (x), and graph
it.
B) Fill in the y-values for g (x).
x
y
0
1
4
9
C) Find the domain and range of g (x).
a) Domain:
b) Range
D) Write an equation for g (x) that includes f (x).
See answers in the margin to the right.
Rule for vertical shifts
Answer to Exercise 1:
Let f (x) be a function, and let k be some constant
number greater than zero.
A) g (x) =
x - 2
A) If g (x) = f (x) + k, then f (x) shifts up k units.
B) If g (x) = f (x) - k, then f (x) shifts down k units.
-2
B)
x
y
0
-2
1
-1
4
0
C) a) Domain: [0,  )
b) Range: [- 2,  )
D) g (x) = f (x) - 2
9
1
Supplement 2 on Transformations of Functions
Page 3
Horizontal Shifts
Now let us perform a horizontal shift on f (x) by moving f (x) to the left 4 units. We will call
our new function h (x). To do this shift we want to do something to the x-values of f (x) to
get the same y-values that we have in the f (x) function but at the same time will move f (x)
to the left 4 units. For this transformation we use the equation h (x) =
this equation h (x) =
x + 4 and not h (x) =
x - 4 ? If h (x) =
x + 4 . Why is
x + 4 , then we can
subtract 4 from each x-value in f (x) to get the x-values for h (x). Notice, however, that the
y-values for h (x) should be the same as the y-values for f (x). This has the effect of moving
f (x) 4 units to the left. For example if we subtract 4 from 0 (our first x-value in the table
-4 + 4 =
for f (x)), we get 0 - 4 = - 4. Thus h (- 4) =
0 = 0 which is the first y-value for
the first point in the table for f (x). Therefore, (- 4, 0) is one of the points needed to shift f (x)
to the left 4 units. So to move f (x) 4 units to the left we use the equation h (x) =
x + 4,
and to get the x-values for h (x) we subtract 4 from each of the x-values in the table of f (x)
but keep the same y-values for h (x) that we had for f (x).
f (x) =
x
x
y
h (x) =
0
0
1
1
4
2
9
3
h (x) =
x + 4
x
y
-4
0
-3
1
0
2
5
3
x + 4
3
-4
Now we shall find the domain and range of h (x) =
A) Domain: [- 4,  )
5
x + 4.
B) Range: [0,  )
Note that the range remains the same as it was for f (x), but the domain moves to the left 4
units.
Now compare f (x) and h (x) on the same graph, and write an equation for h (x) that
includes f (x).
h (x)
Equation: h (x) = f (x + 4)
-4
0
f (x)
Question: What is the difference between the shifts of g (x) = f (x) + 4 and
h (x) = f (x + 4) ?
Supplement 2 on Transformations of Functions
Exercise 2: Let f (x) =
Page 4
x.
A) Write an equation that shifts f (x) to the right 2 units. Call your new function h (x), and
graph it.
B) Fill in the x-values for h (x).
x
y
0
1
2
3
C) Find the domain and range of h (x).
a) Domain:
b) Range
D) Write an equation for h (x) that includes f (x).
See answers in the margin to the right.
Answers to Exercise 2:
Rule for horizontal shifts
A) h (x) =
x - 2
Let f (x) be a function, and let k be some constant
number greater than zero.
2
A) If g (x) = f (x + k), then f (x) shifts to the left k units.
B) If g (x) = f (x - k), then f (x) shifts to the right k
units.
B)
x
y
2
0
3
1
6
2
11
3
C) a) Domain: [2,  )
b) Range: [0,  )
D) h (x) = f (x - 2)
_______________________________
Answers to Exercise 3:
Question: What is the difference between the shifts of
g (x) = f (x) - 2 and h (x) = f (x - 2) ?
A) k (x) =
x + 3 - 5
(- 3, - 5)
B) a) Domain: [- 3,  )
b) Range: [- 5,  )
C) k (x) = f (x + 3) - 5
Supplement 2 on Transformations of Functions
Page 5
x.
Exercise 3: Let f (x) =
A) Write an equation that shifts f (x) horizontally to the left 3 units and then vertically
down 5 units. Call your new function k (x), and graph it.
B) Find the domain and range of k (x).
a) Domain:
b) Range
C) Write an equation for k (x) that includes f (x).
See answers on page 4 in the margin to the right.
Reflection of f (x) over the x-axis
Now we shall reflect f (x) over the x-axis. We shall call our new function g (x). To reflect
f (x) over the x-axis, we change the sign of every y-value in the table of f (x) from plus to
minus or from minus to plus. In our case every y-value for f (x) will be changed from a
positive number to a negative number. The equation for this reflection is g (x) = -
f (x) =
x
x
y
g (x) = -
0
0
1
1
4
2
9
3
g (x) = -
x
0
Now we shall find the domain and range of g (x) = A) Domain: [0,  )
x.
B) Range: (-  , 0]
Question: Why is it wrong to write the range as [0, -  )?
x
x
y
0
0
1
-1
4
-2
9
-3
x.
Supplement 2 on Transformations of Functions
Page 6
Now compare f (x) and g (x) on the same graph, and write an equation for g (x) that includes
f (x).
f (x)
Equation: g (x) = - f (x)
0
g (x)
Reflection of f (x) over the y-axis
Now we shall reflect f (x) over the y-axis. We shall call our new function h (x). To reflect
f (x) over the y-axis, we change the sign of every x-value in the table of f (x) from plus to
minus or from minus to plus. In our case every x-value for f (x) will be changed from a
positive number to a negative number. The equation is h (x) =
f (x) =
x
x
y
h (x) =
0
0
1
1
4
2
9
3
h (x) =
- x
- x
0
Now we shall find the domain and range of h (x) =
A) Domain: (-  , 0]
B) Range: [0,  )
Be careful not to write the domain as [0, -  ). Why?
- x.
x
y
- x.
0
0
-1
1
-4
2
-9
3
Supplement 2 on Transformations of Functions
Page 7
Now compare f (x) and h (x) on the same graph, and write an equation for h (x) that
includes f (x).
h (x)
f (x)
Equation: h (x) = f (- x)
0
Answers to Exercise 4:
Rule for reflection of f (x) over the x- and y- axes
A) g (x) = -
Let f (x) be a function.
- x
A) If g (x) = - f (x), then f (x) reflects over the x-axis.
B) If g (x) = f (- x), then f (x) reflects over the y-axis.
B) a) Domain: (-  , 0]
b) Range: (-  , 0]
Questions:
A) Now consider other toolkit functions. Let f (x) = x2,
g (x) =
x , and k (x) =
1
x
2
. Why don’t we
notice any change on the graph of these functions
when we reflect them over the y-axis?
B) Let us consider h (x) =
1
x
C) g (x) = - f (- x)
__________________________________________
Answers to Exercise 5:
A) h (x) =
=
-  x - 6 + 3
-x + 6 + 3
. Reflect h (x) over the
x-axis. Now reflect h (x) over the y-axis. Compare the
reflection of h (x) over the x-axis with the reflection of
h (x) over the y-axes. What do you notice in each
case? Prove your observations algebraically. Now
(6, 3)
consider f (x) = x3 and g (x) = 3 x . Reflect each
function over the x-axis. Again consider f (x) = x3
and g (x) = 3 x . Reflect each function over the y-
B) a) Domain: (-  , 6]
b) Range: [3,  )
axis. Compare the reflection of each function over
the x-axis with the reflection of each function over
the y-axes. What do you notice in each case? Prove
your observations algebraically.
C) h (x) = f (- (x - 6)) + 3
= f (- x + 6) + 3
Supplement 2 on Transformations of Functions
Page 8
x . Write an equation that reflect f (x) over the x-axis and then
Exercise 4: A) Let f (x) =
reflect the result over the y-axis. Call your new function g (x), and graph it.
B) Find the domain and range of g (x).
C) Write an equation for g (x) that includes f (x).
Exercise 5: Let f (x) =
x.
A) Write an equation that reflects f (x) over the y-axis, shifts the results to the
right 6 units, and then up 3 units. Call your new function h (x), and graph it.
B) Find the domain and range of h (x).
a) Domain:
b) Range
C) Write an equation for h (x) that includes f (x).
See answers on page 7 in the margin to the right.
The Quadratic Function
Given our toolkit functions, we shall explore the possible vertical stretches and
compressions for these functions by examining the quadratic function and applying what
we have learned about doing transformations on it to other functions.
f (x) = x2
x
y
-2
4
-1
1
Domain: (-  ,  )
0
0
1
1
2
4
Range: [0,  )
Supplement 2 on Transformations of Functions
Page 9
Vertical Stretch
Now let us perform a vertical stretch on f (x) by a factor of 4, and call the new function g (x).
To do this we will multiply every y-value in the table of f (x) by 4. The equation is
g (x) = 4 x2 .
f (x) = x2
x
y
-2
4
-1
1
0
0
1
1
g (x) = 4 x2
2
4
x
y
-2
16
-1
4
0
0
1
4
2
16
g (x) = 4 x2
Notice that g (x) seems taller or narrower because f (x) is being stretch upward by a factor
of 4.
Now compare f (x) and g (x) on the same graph, and write an equation for g (x) that includes
f (x).
g (x)
Equation: g (x) = 4 f (x)
f (x)
Supplement 2 on Transformations of Functions
Page 10
Vertical Compressions
Now let us perform a vertical compression on f (x) by a factor of
1
4
, and call the new
1
function h (x). To do this we will multiply every y-value in the table of f (x) by . The
4
equation is h (x) =
f (x) = x2
x
1

4
x2 .
-2
-1
0
1
2
h (x) =
y
h (x) =
4
1
4

1
0
1
4
1
4

x2
x
-2
y
1
-1
1
0
0
4
1
1
4
2
1
x2
Notice that h (x) seems shorter or fatter because f (x) is being compressed by a factor of
Now compare f (x) and h (x) on the same graph, and write an equation for h (x) that
includes f (x).
f (x)
Equation: h (x) =
h (x)
1
4

f (x)
1
4
.
Supplement 2 on Transformations of Functions
Page 11
Rule for vertical stretches and compressions
Let f (x) be a function, and let k be some constant number greater than zero.
A) If k > 1, then g (x) = k

f (x), and f (x) will stretch by a factor of k.
B) If 0 < k < 1, then g (x) = k

f (x), and f (x) will be compressed by a factor of k.
Example: Let f (x) = x2 .
A) Write an equation that will reflect f (x) over the x-axis, and then vertically stretches
the reflection by a factor of 2. We will call the new equation g (x).
B) Find the domain and range of g (x).
a) Domain:
b) Range
C) Write an equation for g (x) that includes f (x).
Solution
A) To reflect f (x) over the x-axis we write - x2 . Then to stretch this reflection by a factor
of 2, we write
g (x) = - 2 x2
B) a) Domain: (-  ,  )
b) Range: (-  , 0]
Note that the range is (-  , 0] and not [0, -  )
Why?
C) g (x) = - 2 f (x)
Supplement 2 on Transformations of Functions
Exercise 6: Let f (x) = x2 .
A) Write an equation that will reflect f (x) over the x-axis,
1
then compresses the reflection by a factor of . Call
5
the new equation g (x), and graph it.
Page 12
Answers to Exercise 6:
A) g (x) = -
1

5
x2
B) Find the domain and range of g (x).
a) Domain:
b) Range
C) Write an equation for g (x) that includes f (x).
B) a) Domain: (-  ,  )
b) Range:
(-  , 0]
C) g (x) = -
Exercise 7: Let f (x) = x2 .
A) Write an equation that stretches f (x) by a factor of 3,
shifts the result to the left 2 units, then shifts it down
4 units. Call the new equation h (x), and graph it.
1
f (x)
5
__________________________
Answers to Exercise 7:
2
A) h (x) = 3 (x + 2) - 4
B) Find the domain and range of h (x).
a) Domain:
b) Range
C) Write an equation for h (x) that includes f (x).
Exercise 8: Let f (x) = x2 .
A) Write an equation that stretches f (x) by a factor of 7,
1
then compresses it by a factor of . Call the new
4
equation k (x), and graph it.
B) a) Domain: (-  ,  )
b) Range: [- 4,  )
C) h (x) = 3 f (x + 2) - 4
_____________________________
Answers to Exercise 8:
A) k (x) = (7
1

4
) x2 =
7
4

x2
B) Find the domain and range of k (x).
a) Domain:
b) Range
C) Write an equation for k (x) that includes f (x).
B) a) Domain: (-  ,  )
b) Range: [0,  )
1
7
C) k (x) = (7  ) x2 =
f (x)
4
4
Supplement 2 on Transformations of Functions
Page 13
Transformations on a Non-Toolkit Function
We shall consider a non-toolkit function and perform the transformations that we have
learned in supplement 2 on this function.
Let f (x) be defined by the graph below. The end points of f (x) are (- 3, - 2), (- 1, 2), (1, 0),
(4, 4), and (6, 4). We shall change the endpoints of f (x) in such a way as to perform the
transformations that we want.
The graph of f (x)
The endpoints of f (x)
Now we shall do the shifts, reflections, and vertical stretches and compressions that we did
on various toolkit functions on our new function.
Our basic function has the endpoints
x
y
-3
-2
-1
2
1
0
4
4
6
4
Vertical Shifts
Suppose we wanted to shift f (x) 3 unis down. To do this transformation, we must subtract 3
from each of the y-value endpoints of f (x). We shall call our new function g (x). The
equation is g (x) = f (x) - 3.
f (x)
x
y
-3
-2
-1
2
1
0
4
4
6
4
g (x) = f (x) - 3
x
y
-3
-5
-1
-1
1
-3
4
1
6
1
Supplement 2 on Transformations of Functions
Page 14
Exercise 9:
A) Write an equation that shifts f (x) up 4 units. Call your new function g (x).
B) Fill in the correct endpoints that will perform the above transformation.
f (x)
x
y
-3
-2
-1
2
1
0
4
4
6
4
g (x) =
x
y
-3
-1
1
4
6
C) Use the graph paper here, and graph g (x).
See answers in the margin to the right on page 15.
Horizontal Shifts
Now we want to shift f (x) to the right 5 units. We will call the new equation h (x). To do
this shift we write the equation h (x) = f (x - 5). We will add 5 units to each of the
x-values in f (x) to get the x-values for h (x) but at the same time keep the y-values for f (x)
and h (x) the same. The result will be to move f (x) to the right 5 units. For example if we
add 5 to - 3 (our first x-value in the table for f (x)) we get 5 + - 3 = 2. Thus h (2) = f (2 - 5)
= f (- 3) = - 2, and (2, - 2) is one of the points needed to shift f (x) to the right 5 units. So to
move f (x) 5 units to the right we add 5 to each of the x-values in the table of f (x) but keep
the same y-values for h (x) that we had for f (x).
f (x)
x
y
-3
-2
-1
2
1
0
4
4
6
4
h (x) = f (x - 5)
x
y
2
-2
4
2
6
0
9
4
11
4
Supplement 2 on Transformations of Functions
Page 15
Answers to Exercise 9:
The graph for f (x) shifts to the right 5 units.
A) g (x) = f (x) + 4
B)
x
y
-3
2
-1
6
1
4
4
8
6
8
C)
Exercise 10:
A) Write an equation that will shift f (x) to the left 6 units. Call your new function h (x).
B) Fill in the correct endpoints that will perform the above transformation.
f (x)
x
y
-3
-2
-1
2
1
0
4
4
6
4
h (x) =
C) Use the graph paper here, and graph h (x).
See answers in the margin to the right on page 16
x
y
-2
2
0
4
4
Supplement 2 on Transformations of Functions
Page 16
Reflection over the x-axis
Answers to Exercise 10:
We will perform the transformation of reflecting f (x)
over the x-axis. Call the new function k (x). We change
the signs of all y-values in the table of f (x) from plus to
minus or from minus to plus.
A) h (x) = f (x + 6)
f (x)
x
y
-3
-2
-1
2
1
0
4
4
6
4
k (x) = - f (x)
x
y
-3
2
-1
-2
1
0
4
-4
6
-4
B)
x
y
-9
-2
-7
2
-5
0
-2
4
C)
Exercise 11: Reflecting f (x) over the y-axis
A) Write an equation that will reflect f (x) over the y-axis. Call your new function m (x).
B) Fill in the correct endpoints that will perform the above transformation.
f (x)
x
y
-3
-2
-1
2
1
0
4
4
6
4
m (x) =
x
y
C) Use the graph paper here, and graph m (x). Answers, see margin page 17.
0
4
Supplement 2 on Transformations of Functions
Page 17
Vertical Compression
Answers to Exercise 11:
We will perform a vertical compression on f (x) by a factor
1
of . For this compression to happen every y-value in the
2
A) m (x) = f ( - x)
1
table of f (x) should be multiplied by . Call the new
2
function n (x).
f (x)
n (x) =
1
2
f
(x)
x
y
-3
-2
-1
2
1
0
4
4
6
4
x
y
-3
-1
-1
1
1
0
4
2
6
2
B)
x
y
3
-2
1
2
-1
0
-4
4
-6
4
C)
_______________________________
Answers to exercise 12:
A) p (x) = 2 f (x)
B)
x
y
C)
-3
-4
-1
4
1
0
4
8
6
8
Supplement 2 on Transformations of Functions
Page 18
Exercise 12: Vertical Stretch
A) Write an equation that vertically stretches f (x) by a factor of 2. Call your new function
p (x).
B) Fill in the correct endpoints that will perform the above transformation.
f (x)
x
y
-3
-2
-1
2
1
0
4
4
6
4
p (x) =
x
y
-3
-1
1
4
6
C) Use the graph paper here, and graph p (x). Answers, see margin page 17.
Exercise 13: Combing transformations
A) Write an equation that reflects f (x) over the y-axis, vertically stretches it
by a factor of 3, then shifts it down 4 units. Call your new function r (x).
B) Fill in the correct endpoints that will perform the above transformation.
f (x)
x
y
-3
-2
-1
2
1
0
4
4
6
4
r (x) =
x
y
C) Use the graph paper provide here, and graph g(x). Answers, see margin page 19.
Supplement 2 on Transformations of Functions
Answers to exercise 13:
1) Basic Function
f (x)
x
y
-3
-2
-1
2
1
0
4
4
6
4
1
2
-1
0
-4
4
3
-6
1
6
-1
0
-4 ;-6
12 12
3
- 10
1
2
2) Reflection over y-axis
Reflection
over y-axis
x
y
3
-2
-6
4
3) Stretch by a factor of 3
x
y
Stretch by 3
4) Shift down 4 units
Shift down
by 4
x
y
-1
-4
-4
8
-6
8
5) Final Results
A) r (x) = 3 f (- x) - 4
B) r (x) =
C)
x
y
3
- 10
1
2
-1
-4






































-4
8














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
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

























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-6
8
Page 19
Exercises for Part I of Supplement 2 on Transformations of Functions
Page 20
S.2 Exercises for the Supplement 2 on Transformations of Functions
Let f (x) be a toolkit function. For exercises 1-10 do A) - C) below.
A) Write an equation that describes the transformation on f (x), and call the new function
g (x). Graph g (x).
B) Find the domain and range of g (x).
C) Write an equation for g (x) that includes f (x).
1) f (x) =
1
is shifted 5 units to the left.
x
2) f (x) = x2 is shifted down 6 units.
9) f (x) =
x is shifted to the left 1 unit, then
shifted up 4 units.
10) f (x) =
x is reflected over the y-axis,
shifted to the right 5 units, then
shifted down 3 units.
3) f (x) = x3 is shifted to the right 4 unit,
then shifted up 2 units.
4) f (x) =
1
x2
is reflected over the x-axis.
5) f (x) = 3 x is reflected over the x-axis,
then shifted down 4 units.
6) f (x) =
x is reflected over the x-axis,
then shifted to the left 3 units.
x is reflected over the x-axis,
7) f (x) =
then reflected over the y-axis, then
shifted up 5 units.
8) f (x) =
1
x2
is shifted to right 2 units,
then shifted down 3 units.
Supplement 2 on Transformations of Functions
Page 21
S.2 Exercises for the Supplement 2 on Transformations of Functions
For problems 11 to 22 write the equation for each graph below that describes its
transformation from the basic toolkit function (supplement 1 page 3), and find the domain
and the range of the transformation. Each tic mark represents one unit on both the x- and
y-axes.
11)
12)
13)
Equation:___________________
Equation:___________________
Equation:___________________
Domain: ___________________
Domain: ___________________
Domain: ___________________
Range:
Range:
Range:
___________________
___________________
___________________
14)
15)
16)
Equation:___________________
Equation:___________________
Equation:___________________
Domain: ___________________
Domain: ___________________
Domain: ___________________
Range:
Range:
Range:
___________________
___________________
___________________
Supplement 2 on Transformations of Functions
Page 22
S.2 Exercises for the Supplement 2 on Transformations of Functions
For problems 11 to 22 write the equation for each graph below that describes its
transformation from the basic toolkit function (supplement 1 page 3), and find the domain
and the range of the transformation. Each tic mark represents one unit on both the x- and
y-axes.
17)
18)
19)
Equation:___________________
Equation:___________________
Equation:___________________
Domain: ___________________
Domain: ___________________
Domain: ___________________
Range:
Range:
Range:
___________________
___________________
___________________
20)
21)
22)
Equation:___________________
Equation:___________________
Equation:___________________
Domain: ___________________
Domain: ___________________
Domain: ___________________
Range:
Range:
Range:
___________________
___________________
___________________
Supplement 2 on Transformations of Functions
Page 23
S.2 Exercises for the Supplement 2 on Transformations of Functions
Let f (x) be a toolkit function. For exercises 23-32 do A) - C) below.
A) Write an equation that describes the transformation on f (x), and call the new function
g (x). Graph g (x).
B) Find the domain and range of g (x).
C) Write an equation for g (x) that includes f (x).
1
23) f (x) =
x
29) f (x) = x3 is vertically compressed by
1
a factor of , shifted to the left 5 units,
3
then shifted down 2 units
is stretched vertically by a
factor of 15.
24) f (x) = x2 is compressed vertically by a
1
factor of .
6
25) f (x) =
then vertically stretched by a factor
of 5, then shifted to the right 3 units.
x is reflected over the x-axis,
vertically compressed by a factor of
3
5
then shifted down 4 units.
26) f (x) =
x is reflected over the x-axis,
then vertically stretched by a factor of
3, then shifted up 8 units.
27) f (x) =
1
x
a factor of
is vertically compressed by
1
2
, then shifted to the left
4 units.
28) f (x) = 3 x is vertically stretched by
a factor of 2, then shifted to the right
4 units.
x is reflected over the y-axis,
30) f (x) =
,
31) f (x) =
1
x2
is reflected over the x-axis,
then vertically compressed by a factor
1
of
, then shifted to the left 2 units,
4
then shifted up 6 units.
32) f (x) = x2 is reflected over the x-axis,
then vertically stretched by a factor of
2, then vertically compressed by a
1
factor of
.
7
Supplement 2 on Transformations of Functions
Page 24
S.2 Exercises for the Supplement 2 on Transformations of Functions
Let f (x) is the function pictured below.
List the key endpoints for the function f (x) showed in the graph above. List the key
endpoints for each transformation below, and sketch it.
Exercise
Old End Points
33) g (x) = f (x) - 3
34) g (x) = f (x + 2)
35) g (x) = - f (x - 3)
New End Points
x
y
-6 -3
0 -3
0
5
2
0
5
2
x
y
x
y
-6 -3
0 -3
0
5
2
0
5
2
x
y
x
y
-6 -3
0 -3
0
5
2
0
5
2
x
y
x
y
36) g (x) = 2 f (x)
x
y
-6 -3
0 -3
0
5
2
0
5
2
x
y
37) g (x) = f (- x) + 1
x
y
-6 -3
0 -3
0
5
2
0
5
2
x
y
x
y
38) g (x) =
1
2
f (x)
x
y
-6 -3
0 -3
0
5
2
0
5
2
x
y
Supplement 2 on Transformations of Functions
Page 25
Graph paper for problems 33-38.
33)
34)
g (x) = f (x) - 3
35)
g (x) = f (x + 2)
36)
g (x) = - f (x - 3)
37)
g (x) = 2 f (x)
38)
g (x) = f (- x) + 1
g (x) =
1
2
f (x)
Supplement 2 on Transformations of Functions
Page 26
Answers for the exercises of Supplement 2 on Transformation of Functions
Each tic mark represents one unit on both the x- and y-axes.
1)
A) Equation:
2)
3)
A) Equation: g (x) = x2 - 6
A) Equation:
1
g (x) =
 x + 5
B) Domain:
(-  , - 5)

Range:
(-  , 0)
Domain: (-  ,  )
Range:
(- 5,  )

g (x) =
(0,  )
[- 6,  )
(-  ,  )
C) g (x) = f (x - 4) + 2
C) g (x) = f (x + 5)
4)
A) Equation: g (x) =
B) Domain:
(-  , 0)
Range:

x2
(0,  )
(-  , 0)
C) g (x) = - f (x)
-1
5)
6)
A) Equation:
A) Equation:
g (x) = - 3 x - 4
B) Domain: (-  ,  )
Range:
(-  ,  )
C) g (x) = - f ( x) - 4
+ 2
B) Domain: (-  ,  )
Range:
C) g (x) = f (x) - 6
3
x - 4
g (x) = -
x+3
B) Domain: (-  ,  )
Range:
(-  , 0]
C) g (x) = - f (x + 3)
Supplement 2 on Transformations of Functions
Page 27
Answers for the exercises of Supplement 2 on Transformation of Functions
Each tic mark represents one unit on both the x- and y-axes.
7)
8)
A) Equation:
A) Equation:
g (x) -
-x + 5
9)
A) Equation:
g (x) =
1
x
2
- 2
- 3
B) Domain: (-  , 0]
Range:
B) Domain:
(-  , 2)
(-  , 5]
Range:

(2,  )
(- 3,  )
C) g (x) = f (x - 2) - 3
10)
A) Equation:
=
-  x - 5 - 3
- x + 5
- 3
B) Domain: (-  , 5]
Range:
(-  , - 3]
C) g (x) = f (- (x - 5 )
= f (- x + 5)
x+1 + 4
B) Domain: (-  ,  )
C) g (x) = - f ( – x) + 5
g (x) =
g (x) =
- 3
- 3
Range:
[4,  )
C) g (x) = f (x + 1) + 4
Supplement 2 on Transformations of Functions
Page 28
Answers for the exercises of Supplement 2 on Transformation of Functions
Each tic mark represents one unit on both the x- and y-axes.
11)
Equation:
g (x) = -
x+3
12)
13)
Equation:
Equation: g (x) = -
g (x) =
Domain: [- 3,  )
Range:
Domain:
(-  , - 4)
Range:
(-  , 3)
(-  , 0]
14)
Equation:_
g (x) =
Domain:
Range:
x
1
+ 3
x + 4
2
+ 1 - 3
(-  ,  )
[- 3,  )

(- 4,  )

(3,  )
(-  , 4]
Range:
16)
Equation: g (x) = x3 + 2
Equation:
(-  ,  )
Range:
(-  ,  )
+ 4
Domain: (-  ,  )
15)
Domain:
x
g (x) =
1
x
2
- 4
+ 3
Domain:
(-  , 4)
Range:

(4,  )
(3,  )
Supplement 2 on Transformations of Functions
Page 29
Answers for the exercises of Supplement 2 on Transformation of Functions
Each tic mark represents one unit on both the x- and y-axes.
17)
18)
19)
Equation: g (x) = 3 x + 3
Equation: g (x) =
-x - 4
Equation: g (x) =
Domain:
( - , )
Domain:
( -  , 0]
Domain: ( -  ,  )
Range:
( - , )
Range:
[- 4 ,  )
Range:
[0,  )
20)
21)
22)
Equation:
Equation:
Equation: -
g (x) = -
x
+ 4
Domain: ( -  ,  )
Range:
3
( - , )
+ 2
-1
g (x) =
- 2
 x - 3
Domain:
(-  , 3)

Range:
(-  , - 2)
(3  )

(- 2,  )
x - 4
x
2
- 4
Domain: ( -  ,  )
Range:
( -  , 5]
+ 5
Supplement 2 on Transformations of Functions
Page 30
Answers for the exercises of Supplement 2 on Transformation of Functions
Each tic mark represents one unit on both the x- and y-axes.
23)
24)
A) Equation: g (x) =
B) Domain:
(-  , 0)
Range:
(-  , 0)
15
x
25)
A) Equation: g (x) =
6
(-  ,  )
B) Domain:

(0,  )

(0,  )
x2
[ 0,  )
Range:
C) g (x) =
1
1
6
f (x)
A) Equation:
g (x) =
(-  , - 4]
Range:
C) g (x) =
-3
5
26)
27)
28)
A) Equation:
A) Equation:
1
1
g (x) = 
2
(x + 4)
B) Domain:
(-  , - 4)  (- 4,  )
A) Equation:
x + 8
B) Domain: [0,  )
Range:
(-  , 8]
C) g (x) = - 3 f (x) + 8
Range:
(-  , 0)
C) g (x) =
1
2

(0,  )
f ( x + 4)
x - 4
5
B) Domain: (-  ,  )
C) g (x) = 15 f (x)
g (x) = - 3
-3
f (x) - 4
g (x) = 2 3 x - 4
B) Domain:
(-  ,  )
Range:
(-  ,  )
C) g (x) = 2 f (x - 4)
Supplement 2 on Transformations of Functions
Page 31
Answers for the exercises of Supplement 2 on Transformation of Functions
Each tic mark represents one unit on both the x- and y-axes.
29)
30)
31)
A) Equation:
1
g (x) =

3
A) Equation:
A) Equation:
- 1
g (x) =

4
x
3
+ 5
- 2
= 5
B) Domain: (-  ,  )
C) g (x) =
Range:
1
f (x + 5) - 2
3
32)
A) Equation:
g (x) = ( - 2
1

7
)=
-2
7
B) Domain: (-  ,  )
(-  , 0]
Range:
C) g (x) = ( - 2
=
-2
7
1

7
) x2
f (x )
-  x - 3
-x + 3
B) Domain: (-  , 3]
(-  ,  )
Range:
g (x) = 5
x2
[ 0,  )
C) g (x) = 5 f ( - ( x - 3))
= 5 f (- x + 3)
B) Domain:
(-  , - 2)
x

2
+ 2
+ 6
(- 2,  )
(-  , 6)
Range:
C) g (x) =
1
- 1
4
f (x + 2) + 6
Supplement 2 on Transformations of Functions
Page 32
Answers for the exercises of Supplement 2 on Transformation of Functions
Basic Function p 26
x
y
-6 -3
0 -3
0
5
2
0
5
2
33)
34)
x
y
-6
-3
-3
-6
0
2
2
-3
5
-1
x
y
g (x) = f (x) - 3
-8 -5
0 -3
-2
5
0
0
3
2
g (x) = f (x + 2)
35)
36)
reflection
x-axis
x
y
-6
0
-3
3
0
-5
2
0
5
-2
shift right
3 units
x
y
-3
0
0
3
3
-5
5
0
8
-2
g (x) = - f (x - 3)
x
y
-6 -3
0 -6
0
10
g (x) = 2 f (x)
2
0
5
4
Supplement 2 on Transformations of Functions
Page 33
Answers for the exercises of Supplement 2 on Transformation of Functions
37)
38)
reflection x
y-axis
y
6
0
3
-3
0
5
-2
0
-5
2
x
shift up 1
unit
6
1
3
-2
0
6
-2
1
-5
3
y
x
y
g (x) = f (- x) + 1
-6
0
-3
0
-3
5
2
2
g (x) =
1
2
2
5
0
1
f (x)