Math 30
1.
2.
Transformation Review
bg
Sketch the graph of the following functions, y  f x , then analyze them by
stating the domain, range, intercepts and asymptotes:
a.
absolute value, y  x
c.
cubic, y  x 3
d.
square root, y  x
e.
exponential, y  a x
f.
reciprocal, y 
b.
quadratic, y  x 2
1
x
bg b g
If the functions above were translated to the function g x  f x  3  2 , state
which ones would have a different:
a.
domain
b.
range
c.
x-, y-intercepts
page 2
3.
Math 30
Transformation Review
bg
If f x  x 2  3 , state the effect on the domain and range if the transformation is:
Domain
Range
f  x
bg
x
bg
f x
f
1
bg
a f x , a 1
f  bx  , b  1
4.
Given f  x   2x as shown to the right.
Sketch the graph of y   f  x  , y  f   x  , f 1  x  ,  f   x  . Two on each.
page 3
5.
Math 30
Transformations Review
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The graph of y  f x is shown below. Sketch the graphs of:
bg
a.
y2  f x
c.
y  f x
bg
b g
b.
y  f x 1  3
d.
y  f 1  x 
page 4
Math 30
e.
6.
7.
Transformations Review
bg
1
y  f 2x
2
f.
y
b g
1
f 2x  6  2
2
bg
Describe the series of transformations that have resulted to the graph of y  f x if :
a.
x is replaced by 2x and y by 2y.
b.
x is replaced by 3x , then y with
c.
x is replaced
1
y, then y with y  4
2
1
1
x and y with y , then x with x  2 and y with y  1
2
3
Given the following series of transformations, write the equation of the
transformed function:
a.
b.
y  x 2 is stretched vertically about the x-axis by a factor of 2 then
translated 3 units to the left, and translated 4 units up.
f  x   x is stretched horizontally about the y-axis by a factor of
1
, then
2
reflected in the x-axis, and translated 4 units to the left.
3
is reflected in the x-axis and translated 3 units to the right.
x 4
c.
y
d.
y  4 x 2  3 is stretched horizontally about the y-axis by a factor of 2
e.
x  y 2 is reflected in the line y  x , horizontally stretched by a factor of 3
and then vertically translated 2 units down.
2
page 5
8.
Math 30
Transformation Review
The graph of a function, f  x  and its transformed function, g  x  are shown
below. Find the equation of g  x  . Presume b = 1
a.
b.
g  x
g  x
f  x  x
f  x  x
c.
f  x   x2
g  x
9.
b g
b g
10.
bg
The point P 6,2 lies on the graph of y  f x . Find the image of P on the
transformed graph:
1
y  f 3 x  1
a.
b.
y   f 2x  6  1
2
c b gh
Describe the series of replacements required to transform the graph of the first
function to the graph of the second function:
1
1
2
y   x 2  1 to y    x  3  5 b.
y  x  3 to y   2  x  1  1
a.
2
2
c.
y  4  x  1  6 to y  2  x 1  3
page 6
11.
Math 30
Transformation Review
The graph of f  x   x undergoes a series of transformations. What is the
resulting equation if the point  x, y  is changed to  x  3, 2 y  5 ?
12.
Draw the reciprocal for the graph given below, labeling all the critical points and
asymptotes.
13.
The graph of y  f x is shown below. Sketch the graph of y 
bg
1
.
f  x
page 7
14.
Math 30
Transformation Review
b g
bg
The point P 4,2 lies on the graph of y  f x . Find the point on the following
transformations.:
1
a.
b.
y
x  f  y
f x
bg
15.
Explain what happens to the x-intercepts of f  x  when the graph of
16.
The graph of f  x  is shown below:
a.
1
is sketched.
f  x
g  x  is the formed when f  x 
is vertically stretched about the
line y  1 by a factor of 2.
Determine the coordinates of the
vertex of g  x  .
b.
Determine the y  intercept of g  x  from  0, 1.25 , the y-intercept on f  x  .
c.
Give the invariant points for g  x  .
d.
Determine the equation of g  x  and write the equation in
the form of y  a  x  h   k .
2
page 8
17.
Math 30
Transformation Review
The point P  5, 4  is on the graph of f  x  . The graph is stretched horizontally about the line
x  2 by a factor of 2. The x-coordinate of the translated point P is ________.
18.
A transformed graph, g of a graph f is shown below. Describe two different ways of
transforming the graph f to graph g.
f
g
19.
If the graph of f is transformed into g  x   f  2 x  , complete the following table of
values for g  x  .
x
0
1
y
2
4
f
page 9
Math 30
Transformation Review
ANSWER KEY
1.
2.
a.
x  R, y  0, x, y  intercept  0,0
c.
x, y  R; x, y  intercept
e.
x  R, y  0, y  intercept of  0,1
y  0 is the horizontal asymptote
a.
d, f
b.
 0,0
a, b, d, e, f
 0,0
b.
x  R, y 0, x, y  intercept
d.
x0, y0, x, y  intercept
f.
x  0, x  R; y  0, y  R; no intercepts
 0,0
x  0, y  0 are the asymptotes
c.
all
page 10
Math 30
3.
4.
5.
domain
Transformations Review
range
domain
f x
no effect
no effect
 f  x
no effect
y3
f 1  x 
a.
b.
c.
d.
Endpoints are:
a.
 4, 2 ,  4, 4 ,  2,0
b.
c.
e.
 4, 2 ,  4,0 ,  2, 2
 2,0 ,  2, 4 , 1, 4
d.
f.
range
x  3
af  x 
no effect
f  bx 
same
 3, 3 , 5, 1 , 3, 5
 2, 2 ,  2, 4 ,  0, 4
 1, 3 ,  5, 2 ,  2, 1
yR
y   3a
same
page 11
Math 30
6.
a.
Transformations Review
stretched vertically about the x-axis by a factor of
about the y-axis by a factor of
b.
1
, and stretched horizontally
2
1
2
reflected horizontally, stretched horizontally about the y-axis by a factor of
1
,
3
then stretched vertically about x-axis by a factor of 2, and then vertically
translated 4 units up.
c.
horizontally stretched about the y-axis by a factor of 2 then stretched vertically
about the x-axis by a factor of 3 then horizontally translated by 2 units left, and
vertically translated 1 unit down.
a.
y  2  x  3  4
c
y
e.
y
a.
g  x    x  3  2
c
g  x   2  x  2   3
9.
a.
 6,3
10.
a.
x is replaced by x  3, y is replaced by y  4
b.
x is replaced by 2 x, x is replaced by x  1
y is replaced by  y, y is replaced by y  1
c.
x is replaced by x  2, y is replaced by 2 y
7.
8.
11.
2
3
or
  x  3  4
2
3
 x  3
2
4
b.
y   2  x  4
d.
y  x2  3
b.
g  x  2
b.
 3, 4
1 2
x 2
9
2
f  x  2
 x  3  5
 x  4  2
page 12
12.
Math 30
Transformations Review
Asymptotes are at x  2, 1, 3
13.
a.
b.
1

 4,  
2

b.
 2, 4
14.
a.
15.
The x-intercepts of f  x  become the asymptotes of
16.
a.
1, 5
d.
y  1.5  x  1  5
b.
 0, 3.5
c.
1
.
f  x
 1,1 and 3,1
2
x  2 x, x   x or x  2 x, x  x  12
17.
x 8
19.
 0, 2 , 1,0 ,  2, 2 ,  4,6 
18.
y
1
1
x, x   x or y  x, x  x  12
4
4