Vertical Stretches about an Axis.
y = ⅔f(x)
The graph of y = f(x) is stretched vertically
about the x-axis by a factor of ⅔ .
Horizontal Stretches about Axis.
Graph has been stretched horizontally about the y-axis
y = f(bx)
y = f(3x)
1
by a factor of
1 

yf x
2 
3
.
The graph of y = f(x) has been stretched horizontally
by factor of 2.
Math 30-1
1
y  k  af (b( x  h))
** You must factor out the stretch if there is also a translation**
y + 4 = f(3x – 12)
1. Factor
2. Stretch
3. Translate
y + 4 = f(3(x – 4))
Math 30-1
2
The graph of the function y = f(x) is transformed to
produce the graph of the function y = g(x)
An equation for g(x)
in terms of f(x) is
A.
1
g ( x)  f (3x)
2
B. g ( x)  2 f (3x)
C.
g ( x) 
1
2
1 
f  x
3 
D. g ( x)  2 f  1 x 
3 
Math 30-1
3
1.2B Reflections
y  af (bx)
1.2B Exploring Reflections
Math 30-1
4
Reflect the Graph of y = f(x) in the x-axis
f(x) = | x |
(-6, 6)
(6, 6)
Family of Functions
f(x) = | x |
(-2, 2)
(2, 2)
(2, -2)
(-2, -2)
f(x) = -| x |
f(x) = -| x |
(-6, -6)
(6, -6)
Math 30-1
5
Reflections in the x-axis affect the y-coordinate.
y  af ( x)
where a  0
Notice (x, y) → (x, -y). The variable y is replaced
by (-y) in the ordered pair and in the function equation .
y = f(x) → -y = f(x) or y = -f(x)
The original equation y = |x| maps to
-y = |x| or y = -|x|
The invariant points are on the line of reflection, the x-axis.
The invariant points would be the x-intercepts
Math 30-1
6
Reflecting y = f(x)
Graph y = -f(x)
y = f(x)
y = -f(x)
Domain and Range
Math 30-1
7
Graphing a Reflection of y = f(x) in the y-axis
(-7, 7)
(-5, 5)
(-7, 3)
Notice domain and range.Math 30-1
(7, 7)
(5, 5)
(7, 3)
8
Reflections in the y-axis affect the x-coordinate.
y  f (bx)
where b  0
Notice (x, y) → (-x, y). The variable x is replaced by (-x).
y = f(x) → y = f(-x).
The original equation x - 5 = 0.5(y - 5)2 becomes
(-x) - 5 = 0.5(y - 5)2
The invariant points are on the line of reflection, the y-axis.
The invariant points would be the y-intercepts.
Math 30-1
9
Given the graph of f(x), graph the image function g(x) that would be the
graph of f(-x).
f(x)
g(x)
If the point (6, -1) is on the graph of f(x), what would be the
corresponding point on the graph of g(x).
(-6, -1)
Math 30-1
10
Which transformation is true regarding the two graphs
g(x) = f(-x)
y  f ( x)
g(x) = -f(x)
f(x) = g(-x)
y  g ( x)
f(x) = -g(x)
Math 30-1
11
Transformations of Functions
Given that the point (2, 6) is on the graph of f(x), state the
corresponding point after the following transformations of f(x).
a) f(x - 3) - 4
(5, 2)
b) -f(x + 2) - 1
(0, -7)
c) f(-x + 2) + 3
f(-(x - 2)) + 3
(0, 9)
Math 30-1
12
True or False
The graph of y = -f(-x - 2) – 3 is a horizontal
translation of the graph of y = f(x) 2 units right.
Multiple Choice
Which of the following transformations on y = f(x)
would have the y-intercepts as invariant points?
A. - y = f(x)
B. y = f(-x)
C. y = f(x + 2)
D. y = 2f(x)
Order of Transformations:
1. Factor
2. Reflections
3. Stretches
4. Translations
Math 30-1
13
Transformation Summary
Type of
Transformation
Replace what
in equation?
With What?
Resulting Effect
Vertical
Translation
y
y-k
Graph moves k
units up.
y-k = f(x)
Vertical
Translation
y
y+k
Graph moves k
units down.
y + k = f(x)
Horizontal
Translation
x
x-h
Graph moves h
units right.
y = f(x - h)
Horizontal
Translation
x
x+h
Graph moves h
units left.
y = f(x + h)
Vertical
Reflection
y
-y
Graph is reflected
in the x-axis
y = -f(x)
Horizontal
Reflection
x
-x
Graph is reflected
in the y-axis
y = f(-x)
Resulting
Equations
14
Page 28
1,3, 5c,d, 7b,d, 10, 15
Math 30-1
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