Transcript Graphical Transformations!!!

Graphical
Transformations!!!
Sec. 1.5a is amazing!!!
New Definitions
Transformations – functions that map real numbers
to real numbers
Rigid Transformations – leave the size and shape of
a graph unchanged (includes translations and
reflections)
Non-rigid Transformations – generally distort the
shape of a graph (includes stretches and shrinks)
New Definitions
Rigid Transformations
Vertical Translation – of the graph of y = f(x) is a
shift of the graph up or down in the coordinate plane
Horizontal Translation – a shift of the graph to the
left or the right
Translations
Let c be a positive real number. Then the following transformations result in translations of the graph of y = f(x):
Horizontal Translations
y = f(x – c)
a translation to the right by c units
y = f(x + c)
a translation to the left by c units
Vertical Translations
y = f(x) + c
a translation up by c units
y = f(x) – c
a translation down by c units
Each figure shows the graph of the original square root function,
along with a translation function. Write an equation for each
translation.
y  x5
y  x 4
y  x 1
New Definitions
Rigid Transformations
Points (x, y) and (x, –y) are reflections of each other across
the x-axis.
Points (x, y) and (–x, y) are reflections of each other across
the y-axis.
(–x, y)
(x, y)
(x, –y)
Reflections
The following transformations result in the reflections of
the graph of y = f(x):
Across the x-axis
y = – f(x)
Across the y-axis
y = f(–x)
Find an equation for the reflection of the given function across
each axis:
5x  9
f  x  2
x 3
Across the x-axis:
Across the y-axis:
5x  9 9  5x
y   f  x   2
 2
x 3 x 3
5x  9
5 x  9

y  f  x 
2
2
x  3 x  3
Let’s support our algebraic work graphically…
On to
vertical and horizontal stretches and
shrinks…
Stretches and Shrinks
Let c be a positive real number. Then the following transformations result in stretches or shrinks of the graph of
y = f(x):
Horizontal Stretches or Shrinks
x
y f 
c
a stretch by a factor of c
a shrink by a factor of c
if c > 1
if c < 1
Vertical Stretches or Shrinks
y  c f  x
a stretch by a factor of c
a shrink by a factor of c
if c > 1
if c < 1
3
Let C 1 be the curve defined by y 1 = f(x) = x – 16x. Find
equations for the following non-rigid transformations of C 1 :
1. C 2 is a vertical stretch of C 1 by a factor of 3
y2  3 f  x   3  x  16 x   3x3  48 x
3
2. C 3 is a horizontal shrink of C1 by a factor of 1/2
 x 
3
y3  f 
  f  2 x    2 x   16  2 x 
 1/ 2 
 8 x  32 x
3
Let’s verify our algebraic work graphically…
Whiteboard problems…
Describe how the graph of y  x
can be transformed to the given
equation.
3
y

x
Describe how the graph of
can be transformed to the given
equation.
y x
reflect across x-axis
y  2 x3
vertical stretch of 2
y  x5
shift right 5
y  (2 x)3
horiz. shrink of ½
y  x
reflect across y-axis
y  (0.2 x)3 horiz. stretch of 5
y  3 x
reflect across y-axis
shift right 3
y  0.3x
3
vertical shrink of 0.3
More whiteboard problems…
Describe how to transform the
graph of f into the graph of g.
f ( x)  x  2
g ( x)  ( x  3)2
g ( x)  2 x  4  1
right 6
g ( x)  x  4
f ( x)  ( x  1)2
Describe the translation of f ( x)  x
to
reflect across
x-axis, left 4
Reflected across x-axis
Vertical stretch factor of 2
Shift left 4
Shift up 1
Homework: p. 139-140 1-23 odd