Transcript Scaling - alleynmath

Transformations: Lesson 4
SCALING
Intro
 Remember when you were little and you
played with PlayDoh.....
You made a figure by either moulding it to
what you wanted or used a cookie cutter to
make a shape.....let’s say a star shape...
and because we were kids, we would play
around with that shape, either squishing it or
pulling it because we thought it was funny to
see the end result: A messed up looking star!!!
Intro
 The messed up looking star looked weird
because you were pulling it in either the xdirection only or the y-direction only.
 Pulling or squishing it distorts the picture and
makes it look funny.
 Let’s see an example.
Example
Take this
normal looking
picture or the
Eiffel Tower
If we pulled this picture in the
x-direction, we’d get
something that looks like this:
If we squished this picture in
the x-direction, we’d get
something that looks like this:
Pretty funny looking eh?
Notes
 In the previous slide, if we had placed the picture
of the Eiffel Tower on a Cartesian graph, then we’d
notice that the x-coordinates are being multiplied
by a certain number (factor) but the y-coordinates
are staying the same.
 The mapping of the points from the original figure
to the image is ( x, y )  (kx, y )
 If k  1 then your figure is getting horizontally
stretched.
 If 0  k  1 then your figure is getting horizontally
reduced
Horizontally Stretched
 Example
Suppose we had the following graph, and I
wanted to make it
bigger by a a factor of
A (-1, 6)
C’ (10, 4)
A’ (-2, 6)
2 in the x-direction.
C (5, 4)
The mapping is as
B (2,
B’ (4,
2) 2)
follows
The figure got distorted. Actually, the
figure was Horizontally Stretched.
But we could predict this because k > 1
( x, y )  ( 2 x, y )
A( 1,6)  A' ( 2,6)
B ( 2,2)  B ' ( 4,2)
C (5,4)  C ' (10,4)
Horizontally Reduced
 Example
Suppose we had the following graph, and I
wanted to make it
smaller by a a factor of
A (-1, 6)
A’ (-0.5, 6)
C’ (2.5, 4)
½ in the x-direction.
C (5, 4)
The mapping is as
B’
B (1,
(2, 2)
follows
The figure got distorted. Actually, the
figure was Horizontally Reduced.
But we could predict this because 0<k<1
1
x, y )
2
A( 1,6)  A' ( 0.5,6)
B ( 2,2)  B ' (1,2)
C (5,4)  C ' ( 2.5,4)
( x, y )  (
This also work in the y-direction
Take this
normal looking
picture or the
Eiffel Tower
If we pulled this picture in the
y-direction, we’d get
something that looks like this:
If we squished this picture in
the y-direction, we’d get
something that looks like this:
Now this is funny looking no?
Notes
 In the previous slide, if we had placed the picture
of the Eiffel Tower on a Cartesian graph, then we’d
notice that now the y-coordinates are being
multiplied by a certain number (factor) and it’s the
x-coordinates that are staying the same.
 The mapping of the points from the original figure
to the image is ( x, y)  ( x, ky)
 If k  1 then your figure is getting vertically
stretched.
 If 0  k  1 then your figure is getting vertically
reduced
Horizontally Stretched
 Example
Suppose we had the following graph, and I
A’ (-1, 12)
C’ (5, 8)
wanted to make it
bigger by a a factor of
A (-1, 6)
2 in the y-direction.
C (5, 4)
B’ (4, 2)
The mapping is as
B (2, 2)
follows
The figure got distorted. Actually, the
figure was Vertically Stretched.
But we could predict this because k > 1
( x, y )  ( x, 2 y )
A( 1,6)  A' ( 1,12)
B ( 2,2)  B ' ( 2,4)
C (5,4)  C ' (5,8)
Horizontally Reduced
 Example
Suppose we had the following graph, and I
A (-1, 6)
wanted to make it
smaller by a a factor of
½ in the y-direction.
C (5, 4)
A’ (1, 3)
The mapping is as
B (2, 2)C’ (5, 2)
follows
B’ (2, 1)
The figure got distorted. Actually, the
figure was Horizontally Reduced.
But we could predict this because 0<k<1
1
y)
2
A( 1,6)  A' ( 1,3)
B ( 2,2)  B ' ( 2,1)
C (5,4)  C ' (5,2)
( x , y )  ( x,
What if k is negative??
 The scale factor “k” is allowed to be negative.
 If your “k” value is negative, it doesn’t imply
that your figure will automatically get
smaller.
 The negative sign (-) means that it will be on
the opposite side of the axis you are moving
away from. Let me explain....
Example
Let’s look at point A(2,5)
A’
A’
A
If we applied a horizontal stretch of
( x, y )  (3 x, y )
we' d get A(2,5)  A' (6,5)
2
If we applied a horizontal stretch of
-6
6
( x, y )  (3x, y )
we' d get A(2,5)  A' (6,5)
Both of these are horizontal
stretches but take note of the
difference!!!
The same type of logic goes for horizontal reductions.
Example
A’
Let’s look at point A(2,5) again
If we applied a vertical stretch of
A
10
( x, y )  ( x, 2 y )
we' d get A(2,5)  A' (2,10)
5
If we applied a vertical stretch of
( x, y )  ( x, 2 y )
-10
A’
we' d get A(2,5)  A' (2,10)
Both of these are vertical stretches
but take note of the difference again!!!
The same type of logic goes for vertical reductions as well.
 That concludes this wonderful PowerPoint.
Hope you enjoyed the show!