Graphical Transformations

1
Graphical Transformations

• Vertical and Horizontal Translations
• Vertical and Horizontal Stretches and Shrinks

2
Take the equation f(x) x2

• How do you modify the equation to translate the
  graph of this equation 5 units to the
  right?........ 5 units to the left?
• How do you modify the equation to translate the
  graph of this equation 3 units down?..............
  3 units up?
• What if you wanted to translate the graph of this
  equation 5 units to the left and 3 units down?

3
The parabola has been translated 5 units to the
right. How is the equation modified to cause this
translation?
4
Notice the change in the equation y x2 to
create the horizontal shift of 5 units to the
right.
f(x) x2
g(x) (x-5)2
5
The parabola is now translated 5 units to the
left. How is the equation modified to cause this
translation?
6
Notice the change in the graph of the equation
yx2 to create a horizontal shift of 5 units to
the left.
f(x)x2
h(x)(x5)2
7
The parabola has now been translated three units
down. How is the equation modified to cause this
translation?
8
Notice how the equation y x2 has changed to
make the Vertical translation of 3 units down.
f(x)x2
q(x)x2-3
9
The parabola has now been translated 3 units
up. How is the equation modified to cause this
translation?
10
Notice how the equation y x2 has been changed
to make the Vertical translation 3 units up.
f(x)x2
r(x)x23
11
Write what you think would be the equation for
translating the parabola 5 units to the left and
3 units up?
12
The equation would be
What would the graph would look like?
13
g(x) is the translation of f(x) 5 units to the
left and 3 units up.
f(x) x2 g(x) (x5)23
14
Vertical and horizontal stretches and shrinks

• How does the coefficient on the x2 term affect
  the graph of f(x) x2?
• What if we substitute an expression such as 2x
  into f(x)? How would that affect the graph of
  f(x) x2?

15
The parabola has been vertically stretched by a
factor of 2. Notice how the equation has been
modified to cause this stretch.
16
The parabola is vertically shrunk by a factor of
½. Notice how the equation has been modified to
cause this shrink.
17
By substituting an expression like 2x in for x in
f(x) x2 gives a different type of shrink. f(2x)
(2x)2. A horizontal shrink by a factor of ½.
18
Suppose we found g(1/2x). The equation would be
y (1/2x)2.. How would this affect the graph
of the function g(x) x2? It is a horizontal
stretch by a factor of 2.
19
If we were to write some rules for translations
of functions and stretches/shrinks of functions,
what would we write?
Horizontal translation Vertical
translation Vertical stretch Vertical
shrink Horizontal stretch Horizontal shrink