Chapter 3: Transformations of
Graphs and Data
Lesson 5: The Graph Scale-Change
Theorem
Mrs. Parziale
Vocabulary:
•
Vertical stretch: A scale change that makes the original graph taller or shorter
•
Horizontal stretch: a scale change that makes the original graph wider or
skinnier.
•
Scale change: a stretch or shrink applied to the graph vertically or horizontally
•
Vertical scale change: The value that changes the vertical values of the graph.
•
Horizontal scale change: The value that changes the horizontal values of the
graph.
•
Size change: When the same vertical and horizontal scale change occurs.
Example 1:
Consider the graph of y  x  4 x
(a) Complete the table and graph on the grid:
3
x y
-2
-1
0
1
2
y
5
4
3
2
1
–5
–4
–3 –2 –1
–1
–2
–3
–4
–5
1
22
33
44
55
xx
(b) Replace (y) with
y
3
.
y
9
1. Solve the new equation for y and
graph it on the same grid at right.
8
7
6
5
4
2. What happens to the ycoordinates?
3
2
1
–6
3. This is called a vertical stretch of
magnitude 3 .
–5
–4
–3
–2
–1
–1
–2
–3
–4
–5
–6
4. Under what scale change is the
new figure a vertical scale change
of the original?
–7
–8
–9
1
2
3
4
5
6
x
(c) Replace (x) with
x
y
.
9
2
1. Solve the new equation for
y and graph it.
2. What happens to the xcoordinates?
3. This is called a horizontal
stretch of magnitude 2 .
4. Under what scale change is
the new figure a horizontal
scale change of the
original?
8
7
6
5
4
3
2
1
–6
–5
–4
–3
–2
–1
–1
–2
–3
–4
–5
–6
–7
–8
–9
1
2
3
4
5
6
x
y
(d) Let f ( x )  x  4 x .
Find an equation for
g(x), the image of f(x)
under
9
3
8
7
6
5
4
3
2
1
–6
–5
–4
–3
–2
–1
–1
–2
S ( x , y )  (2 x , 3 y )
–3
–4
–5
–6
–7
–8
What is happening to
each part of the graph?
–9
1
2
3
4
5
6
x
S ( x , y )  (2 x , 3 y )
• How is the x changed?
• Change: horizontal
stretch two times wider.
y
9
8
7
6
5
4
3
2
1
–9 –8 –7 –6 –5 –4 –3 –2 –1
–1
–2
• How is the y changed?
• Change: vertical stretch
three times the original.
–3
–4
–5
–6
–7
–8
–9
1
2
3
4
5
6
7
8
9 x
Graph Scale-Change Theorem
In a relation described by a sentence in (x) and (y),
the following two processes yield the same
graph:
(1) replace (x) by
x
a
and (y) by
y
b
in the sentence
( x , y )  ( ax , by )
(2) apply the scale change __________________
to
the graph of the original relation.
size change
Note: If a = b, then you have performed a __________
If a = negative, the graph has been reflected (flipped) over the y-axis
If b = negative, the graph has been reflected over the x-axis
So, What’s the Equation?
(d) Find an equation for g(x), the image of f(x)
under S ( x , y )  (2 x , 3 y )
y
9
8
7
f ( x)  y  x  4 x
3
6
5
4
3
x
y
-2
0
x
y
2
1
–9 –8 –7
–6
–5
–4
–3
–2
–1
–1
–2
-1
3
0
0
1
-3
–3
–4
–5
–6
–7
–8
–9
2
0
1
2
3
4
5
6
7
8
9
xx
Example 2:
• Consider y  x . Find an equation for the
x
function under S ( x , y )   ,  2 y 
3

• Describe what happens to all of the x values:
• Describe what happens to all of the y values:
x

S ( x, y )   , 2 y 
3

• Find the equation for the transformed image
by
– Replace (x) with ______________
– Replace (y) with ______________
– Now make the new equation (remember to
simplify to y= form):
Graph It!
y
9
y  x
8
7
6
5
4
y  2 3x
3
2
1
–9
–8
–7
–6
–5
–4
–3
–2
–1
–1
–2
–3
–4
–5
–6
–7
–8
–9
1
2
3
4
5
6
7
8
9
x
Closure - Example 3:
The graph to the right is
x
y = f(x). Draw 3 y  f ( ) .
4
• What should happen to all
of the x values?
• What should happen to all
of the y values?