Transcript 1.4 - School District 67 Okanagan Skaha

Chapter 1 Transformations
1.4
MATHPOWERTM 12, WESTERN EDITION 1.4.1
Vertical Stretches of Functions
f(x) = 2 | x |
f(x) = | x |
f(x) = 3 | x |
x
y
x
y
x
y
0
0
0
0
0
0
1
1
1
2
1
3
2
2
2
4
2
6
3
3
3
6
3
9
x
1
x
3
y
0
0
0
1
1
2
1
2
1
2
3
3
2
3
x
1
x
2
y
0
f (x) 
f (x) 
1
3
2
3
1
1.4.2
Vertical Stretches of Functions [cont’d]
f(x) = 3| x |
f(x) = 2| x |
f(x) = | x |
1
x
2
1
f (x)  x
3
f (x) 
A stretch can be
an expansion.
A stretch can be
compression.
1.4.3
Graphing y = af(x)
y = 3| x |
y=|x|
Given the graph of y = f(x), there is a vertical expansion
when a > 1 for the transformation of y = af(x).
1.4.4
Graphing y = af(x)
compression
y=|x|
y=
1
|x|
2
Given the graph of y = f(x), there is a vertical compression
when 0 < a < 1 for the transformation of y = af(x).
1.4.5
Vertical Stretching and Reflecting of y = f(x)
In general, for any function y = f(x), the graph of a function
y = af(x) is obtained by multiplying the y-value of each point
on the graph of y = f(x) by a.
That is, the point (x, y) on the graph of y = f(x) is transformed
into the point (x, ay) on the graph of y = af(x).
• If a > 1, the graph y = f(x) expands vertically by a factor of a.
• If 0 < a < 1, the graph y = f(x) compresses vertically by a
factor of a.
• If a < 0, the graph is also reflected in the x-axis.
1.4.6
Vertical Stretching and Reflecting of y = f(x)
For y = af(x), there is a vertical stretch.
y y==f(x)
f(x)
y = f(x)
1
yy=2f(x)
f (x)
2
-2f(x)
y y==f(x)
When a < 0, there is a
reflection in the x-axis.
1.4.7
Horizontal Stretching of y = f(x)
f(x) = x2
f(x) = (2x)2
y
x
y
0
0
0
0
1
0.5
2
1
2
4
1
1
4
4
4
3
9
1.5
9
6
9
x
y
0
0
1
x
f(x) = (0.5x)2
Each point on the graph of y = (2x)2 is half as far from the
y-axis as the related point on the graph of y = x2.
The graph of y = f(2x) is a horizontal compression of the
graph of y = f(x) by a factor of 0.5.
Each point on the graph of y = (0.5x)2 is twice as far from the
y-axis as the related point on the graph of y = x2.
The graph of y = f(0.5x) is a horizontal expansion of the
graph of y = f(x) by a factor of 2.
1.4.8
Horizontal Compression of y = f(kx) when k > 1
y = (2x)2
y = x2
(-1, 1)
(-0.5, 1)
(0.5, 1)
(1, 1)
For y = f(kx), there is a
horizontal compression
when k > 1.
1.4.9
Horizontal Expansion of y = f(kx) when 0 < k < 1
y = x2
(-2, 1)
(-1, 1)
(2, 1)
(1, 1)
2
 1  
y    x 
2
For y = f(kx), there is a
horizontal expansion,
when 0 < k < 1.
1.4.10
Comparing y = f(x) With y = f(kx)
In general, for any function y = f(x), the graph of the function
y = f(kx) is obtained by dividing the x-value at each point
on the graph of y = f(x) by k.
That is, the point (x, y) on the graph of the function y = f(x) is
x
transformed into the point ( , y) on the graph of y = f(kx).
k
• If k > 1, the graph of y = f(x) is compressed horizontally by
1
a factor of .
k
• If 0 < k < 1, the graph of y = f(x) is expanded horizontally by
1
a factor of .
k
• If k < 0, there is also a reflection in the y-axis.
1.4.11
Graphing y = f(kx) and its Reflection
y = f(x) y = f(x)
y=
1f(x)
y=
f(x)
y  f ( x)
2
Graph y = f(2x).
Graph y = f(1/2x).
y = f(1/2x)
When k < 0, thereyis= f(2x)
Graph y = f(-1/2x).
a reflection in the y-axis
as well as a compression
or expansion.
For
1 
Forthe
thegraph
graphofofy =
y =f(2x),
f(1/2x),
  there
y  f   by
x 
is there
a horizontal
compression
is a horizontal expansion
2
a of
factor
of ½. of y = f(x) by
the graph
a factor of 2.
1.4.12
Describing the Horizontal or Vertical Stretch of a Function
Describe what happens to the graph of a function y = f(x).
a) y = f(3x)
b) 3y = f(x)
Vertical compression by a
Horizontal compression by a
1
factor of
3
1
c) y = f( x)
2
factor of
d) -2y = f(x)
Vertical compression by a
Horizontal expansion by a
factor of 2
e) 2y = f(2x)
Horizontal compression by a
1
factor of and vertical compression
2
1
by a factor of
2
1
3
factor of
f)
1
and reflected in the x-axis
2
1
y = f(-3x)
4
Horizontal compression by a
1
factor of , reflection in the
3
y-axis, and vertical expansion by a
factor of 4
1.4.13
Stating the Equation of y = af(kx)
The graph of the function y = f(x) is transformed as described.
Write the new equation in the form y = af(kx).
a) Horizontal compression factor of one-third, and a vertical
y = 2f(3x)
expansion factor of two
b) Horizontal expansion factor of two, a vertical compression
by a factor of one-third, and a reflection in the x-axis
y 
1 1
f ( x)
3 2
c) Horizontal compression factor of one-fourth, a vertical
expansion factor of three, and a reflection in the y-axis
y = 3f(-4x)
d) Horizontal expansion factor of three, vertical compression
factor of one-half, and a reflection in both axes
y 
1
1
f ( x)
2
3
1.4.14
Sketching
y =af(kx)
( x  4 )2,
Given the graph
of y = 16
2
y

2
16

(x

4)
sketch the graphs with the following transformations.
a) Expand horizontally by a factor of 2.
y =
2
1
16   x  4
2
y  16 (x  4)2
b) Expand vertically by a factor of 2.
1
2
y = 
16  ( 2x  4 )
2
c) Reflect in the x-axis, with a vertical compression
factor of 1 and a horizontal compression factor of 2.
2
1.4.15
Graphing a Polynomial and its Transformations
State the zeros of this polynomial, and a possible equation of P(x).
y  (x  2)( x  1)( x  2)
x = -2, 1, 2
P(x) = (x + 2)(x - 1)(x - 2)
y  (x  2)( x  1)( x  2)
State the zeros after the following transformations of P(x).
a) y = -2P(x)
Because this is a vertical
stretch, the zeros will
remain the same, even
with the reflection.
1
b) y = P(- x)
2
For this transformation,
there is a reflection in the
y-axis and a horizontal
expansion of 2. Therefore,
the zeros are -4, -2, and 4.
1
1
1
y  ( x  2)( x  1)( x  2)
2
2
2
y  2(x  2)( x  1)( x  2)
1.4.16
Suggested Questions:
Pages 38-40
1-26, 27-41,
45-51
1.4.17