Transformations of Graphs 27/9/2013
Basic Transformations
Restructuring Graphs Vertical Translation
f(x) to f(x) + k
Horizontal Translation f(x) to f(x – h)
y
x
• •
•
x′
y′
x′ + h
y = f(x)
y = f(x) + k
y = f(x – h)y′ + k
Note: When x = x′ + h ,
y = f(x – h)= y′
= f(x′ + h – h)= f(x′)
Vertical move k units
Transformations of Graphs 37/9/2013
Basic Transformations Horizontal Translation
f(x) to f(x – h) Example
y
xx′
y′ • ••
• •
y = 2x
y = 2(x – 1)
x′ + 1
y = f(x) = 2xWhen x = x′ , y = 2x′
When x = x′ + 1,y = 2(x – 1) = 2((x′ + 1) – 1)
= 2x′
y = f(x – 1) = 2(x – 1)
= y′ = f(x′)
Transformations of Graphs 47/9/2013
Basic Transformations
y
x
• •
x
y
x′
y = f(x)y = f(x′)
y
x′
y = f(x′)
x′ = 0 when x = h So, x – h = 0 = x′
f(x′) = f(x – h)
h
x′
•
Horizontal Translation
Assume h > 0y = f(x – h)
Replacing x with
moves the graph h units to the right
x – h
Transformations of Graphs 57/9/2013
Basic Transformations
Restructuring Graphs Vertical Stretching
f(x) to af(x) , a > 1 stretch by factor of a
Note that when y = f(x) = 0 then y = af(x) = 0
y
x
y = f(x)
y = af(x)
•Stretch is away from x-axis
Note: •
Transformations of Graphs 67/9/2013
Basic Transformations
Restructuring Graphs Vertical Shrinking
f(x) to af(x) , 0 < a < 1 shrink by factor of a
Note that when y = f(x) = 0 then y = af(x) = 0
y
x
y = f(x)
y = af(x)
••Stretch is toward x-axis
Note:
Transformations of Graphs 77/9/2013
Basic Transformations
Restructuring Graphs Horizontal Stretching
f(x) to f(cx) , 0 < c < 1 stretch by factor of c
Note that when y
x
y = f(x)y = f(cx)
•Stretch is awayfrom y-axis
Note:
x = 0 then cx = 0so
y = f(x) = f(cx)
Transformations of Graphs 87/9/2013
Basic Transformations
Restructuring Graphs Horizontal Shrinking
f(x) to f(cx) , c > 1 shrink by factor of c Note that when y
x
y = f(x)y = f(cx)
•Shrink is toward y-axis
Note:
soy = f(x) = f(cx)
x = 0 then cx = 0
Transformations of Graphs 97/9/2013
Basic Transformations
Restructuring Graphs: Reflections Vertical reflection
through horizontal axis f(x) to -f(x)
y
x
y = f(x)
y = -f(x)
•
•
Each point on the graph of f(x) projects to acorresponding point on the graph of -f(x) reflected through the x-axis
Transformations of Graphs 107/9/2013
Basic Transformations
Restructuring Graphs: Reflections Horizontal reflection
through vertical axis f(x) to f(-x)
y
x
y = f(x)y = f(-x)
••Each point on the graph of f(x) projects to a corresponding point on the graph of f(-x) … reflected through the y-axis Note: When x = 0 ,
y = f(x) = f(-x)
Transformations of Graphs 117/9/2013
Translations
Vertical Translation y = f(x) Alters the value of y
Horizontal Translation y = f(x) Alters the value of x
x
yy = f(x) + k
k
y = f(x – h)
h
Transformations of Graphs 127/9/2013
Examples
Examples y = -x2
y = -x2
x
yy = -x2 + 4
y = -(x – 3)2
y = -x2 + 4
y = -(x – 3)2
Moves graph up 4 units
Moves graph right 3 units
Transformations of Graphs 137/9/2013
Examples
Vertical Stretching : y = f(x) Example:
Vertical Shrinking : y = f(x) Example:
x
y
y = af(x)
x
y
y = af(x)
y = 2(x – 4)2 – 6
a > 1
0 < a < 1
y = (x – 4)2 – 3
= 2y
y = (x – 4)2 – 3
y 12
32
–(x – 4)2 = 12= y
• •
••
f(x) = 0 = af(x)
f(x) = 0 = af(x)
Transformations of Graphs 147/9/2013
Consider y = f(x) = 2x + 1 y = f(x)
Horizontal Shrinking & Stretching
x
yg(x) = f(2x)
●
●●
●
●
f(x)
= 2
x +
1
f(2x
) = 4
x +
1
●
●●
f(x)f(2x)f(x/2)
x 0 1 2 1 3 51 5 9? ? ?
●
= 4x + 1
What changes ?
The graph shrinks toward the y-axis
Transformations of Graphs 157/9/2013
Consider y = f(x) = 2x + 1 y = f(x)
Horizontal Shrinking & Stretching
x
yh(x) = f((½)x)
●
●● ●
f(x)
= 2
x +
1
f(2x
) = 4
x +
1
f((½
)x) =
x +
1
●
f(x)f(2x)f(x/2)
x 0 1 2 1 3 51 5 91 2 3
●●●
= x + 1
What changes ?
The graph stretches away from the y-axis
Transformations of Graphs 167/9/2013
Consider y = f(x) = 2x + 1 y = f(x)
Horizontal Shrinking & Stretching
x
y
f(x)
= 2
x +
1
f(2x
) = 4
x +
1
f((½
)x) =
x +
1
f(x)f(2x)f(x/2)
x 0 4 8 16 1 17 331 33 651 9 17
For constant c, c ≠ 0, point (x, y) on the graph of y = f(x) corresponds to on the graph of f(cx)
xc( , y)
••
•
•
••
84
Let c = 2
xxc
16
(8,17)
•17
xc
9173
½
Transformations of Graphs 177/9/2013
How does it work ? y = f (x)
Horizontal Shrinking
x
y
y = g(x) = f(2x)
b
(b, f(b))●●●●
y = f(x)
x1
●
Note: g(x/2) = f(2(x/2))
For some x1 want g(x1) = f(b)
g(x1) = f(2x1) = f(b)
2x1 = b and x1 = b/2
g(b/2) = f(2(b/2)) = f(b)
Let
= f(x)f(b)g(b/2)
* *Note:We have assumed f is 1-1 on interval [x1, b]
Transformations of Graphs 187/9/2013
How does it work ? y = f (x)
Horizontal Shrinking - 2
x
y
y = g(x) = f(2x)
b
(b, f(b))●●●
y = f(x)
g(b/2) ●
Note: g(x/2) = f(2(x/2))
For some x2 want g(x2) = f(c)
g(x2) = f(2x2) = f(c)
2x2 = c and x2 = c/2
g(c/2) = f(2(c/2)) = f(c)
Let
= f(x)
●c
●● (c, f(c))
●x2
●
*
f(c)g(c/2)
*Note:We have assumed f is
1-1 on interval [x2, c]
Transformations of Graphs 197/9/2013
How does it work ? y = f (x)
Horizontal Shrinking - 3
x
y
y = g(x) = f(2x)
b
(b, f(b))●●●
y = f(x)
g(b/2) ●
Note: g(x/2) = f(2(x/2))
For some x3 want g(x3) = f(d)
g(x3) = f(2x3) = f(d)
2x3 = d and x3 = d/2
g(d/2) = f(2(d/2)) = f(d)
Let
= f(x)
●c
●● (c, f(c))●
*
f(c)g(c/2)
●
●●
d
(d, f(d))f(d)
x3 ●
●g(d/2)
*Note:We have assumed f is 1-1 on interval [b, d]
Transformations of Graphs 207/9/2013
How does it work ? y = f (x)
Horizontal Shrinking - 4
x
y
y = g(x) = f(2x)
b
(b, f(b))●
●
y = f(x)
g(b/2) ●
Note:g(x/2) = f(2(x/2))
Replacing x by 2x moves points (x,y) on the graph of f(x) to points (x/2, y) on the graph of f(2x)
Let
= f(x)●c
●● (c, f(c))●f(c)g(c/2)
●
●●
d
(d, f(d))f(d) ●g(d/2)
y = g(x) = f(2x)
●
This shrinks the graph toward the y-axis
Transformations of Graphs 217/9/2013
Horizontal Shrinking y = f1(x)
Note:
Example
Examples
x
y
y = f2(cx)
y = | 2x |
, c > 1
x
(x, y)●●●
●●
= f1(x)
y
f2(c( )) = f1(x) = ycx
for x1 = cx
cxx1 =
f1(x) f2(cx)
f2(cx1) = f2(c( ))cx
y = | x |
Transformations of Graphs 227/9/2013
x
y Horizontal Stretching
y = f1(x)
Examples
y = f2(cx)for 0 < c < 1
y = x 12
= f1(x)
for x1 = cx
f2(cx1) = f2(c( ))cx
Note: x
(x, y)●●●
●●
y
cxx1 =
f1(x)f2(cx) f2(c( )) cxf1(x) =
Example
y = | x |
Transformations of Graphs 237/9/2013
Examples Vertical Reflection
y = f(x)
Example
y = |x|
Horizontal Reflection y = f(x)
Example
y = x + 1
x
y
y = –f(x)
y = f(-x)
y = –|x|
y = -x + 1
y
x
f(x)
–f(x)
f(x)
f(-x)
Transformations of Graphs 247/9/2013
Exercise What Transformations Are These ?
x
yf1(x) – 2b = (ax + b) – 2b
f1(x) = ax + b
●(0, b)
●(-b/a, 0)
f2(x) = ax – b
●(0, -b)
●(b/a, 0)
f1(x) f2(x)?
= ax – b = f2(x)
f1(x – 2b/a) = a(x – 2b/a) + b = ax – 2b + b
= f2(x)
OR
Vertical Translation
Horizontal Translation
●
Transformations of Graphs 257/9/2013
Exercise What Transformations Are These ?
x
y
f1(x) = ax + b
●
(0, b)
●(-b/a, 0) ●(b/a, 0)
f2(x) = -ax + b
f1(x) f2(x)?
f1(-x) = a(-x) + b = -ax + b
= f2(x) Horizontal
Reflection
Transformations of Graphs 267/9/2013
Exercise What Transformations Are These ?
x
y
x
y
f1(x) = ax + b
● (0, b)●
(-b/a, 0)
● (0, -b)
f2(x) = -ax – b
f1(x) f2(x)?
-f1(x) = -(ax + b) = -ax – b
= f2(x) Vertical Reflection
f1(x) = ax + b
●(0, b)●
(-b/a, 0)●
f2(x) = 2ax + b
(-b/(2a), 0)
f1(x) f2(x)?
f1(2x) = a(2x) + b = 2ax + b
= f2(x) Horizontal Shrink
Transformations of Graphs 277/9/2013
Exercise What Transformations Are These ?
x
y
x
y
f1(x) = ax + b
●(0, b)
●(-b/a, 0)
●(0, -2b/a)
f1(x) f2(x)?
= f2(x) Horizontal Stretch
f2(x) = ax + b 12 f1(x) = ax + b
●●
(0, b)(-b/a, 0)
f2(x) = 2ax + 2b
●(0, 2b)
f1(x) f2(x)?
= f2(x) Vertical Stretch
2f1(x) = 2(ax + b) = 2ax + 2b ax + b 1
2=
a( x) + b
12f1( x) 1
2 =
Transformations of Graphs 287/9/2013
Exercise What Transformation Is This ?
x
y
f1(x) = b●
(0, b)
f1(x) f2(x)?
= f2(x)
Horizontal Stretchf2( x) = b 1
2
... or Shrink
... or Reflection
f1(2x) = f2(x) = b
f1(-x) = f2(x) = b f1( x) = b
12
Transformations of Graphs 297/9/2013
Exercise What Transformation Is This ?
x
y
f1(x) = b
●
(0, b)
f2(x) = -2b (0, -2b)
f1(x) f2(x)?
= f2(x)
Vertical Stretch and Reflection
-2f1(x) = -2b
●
... or Vertical Translation
f1(x) – 3b = b – 3b = –2b
= f2(x)