Math 140
1.7: Graphing Techniques  Transformations
Trigonometry
Here is a summary of the transformations of y = f(x). See Summary of Graphing Techniques on page 74.
1. Vertical translation (or shift): y  f ( x)  k
For k > 0,
 The graph of y  f ( x)  k is the parent graph, y  f (x) , shifted up k units.
 The graph of y  f ( x)  k is the parent graph, y  f (x) , shifted down k units;
2. Horizontal translation (or shift): y  f ( x  h)
For h > 0,
 The graph of y  f ( x  h) is the parent graph, y  f (x) , shifted left h units.
 The graph of y  f ( x  h) is the parent graph, y  f (x) , shifted right h units;
3. Reflections:
a.)
Reflection about the x-axis:
 The graph of y   f (x) is the reflection of the parent graph, y  f (x) , about the x–axis.
b.)
Reflection about the y-axis:
 The graph of y  f ( x) is the reflection of the parent graph, y  f (x) , about the y–axis.
4. Vertical Stretch or Compression: y  af ( x)
The graph of y  af ( x) is the graph of the parent graph, y  f (x) , …

Stretched vertically by a multiplication factor of a for a  1 , or

Compressed vertically by a multiplication factor of a for 0  a  1 .
For a < 0, the graph is also reflected across the x–axis.
5. Horizontal Stretch or Compression: y  f (ax)
The graph of y  f (ax) is the graph of the parent graph, y  f (x) , …

Compressed horizontally by a multiplication factor of

Stretched horizontally by a multiplication factor of
1
a
1
a
for a  1 , or
for 0  a  1 .
For a < 0, the graph is also reflected across the y–axis.
Order of Operations for
Transformations
1. Stretch/Compress
2. Reflect
3. Translate
Hint: When reflecting in the
y–axis AND translating
horizontally, factor out a –1.
See p. 75, Example 6, also #65, #66
from homework.
Vertical and Horizontal
Translations
Vertical Stretching
and Compressing
Horizontal Stretching
and Compressing
y  f (x) - bold
y  f ( x)  3 - thin
y  f ( x  4) - dotted
y  f (x) - bold
y  2 f ( x) - thin
y  12 f ( x) - dotted
y  f (x) - bold
y  f (2 x) - thin
y  f ( 12 x) - dotted
For problems 39 – 67: Graph each function using the techniques of shifting, compressing, stretching, and/or
reflecting.
 Start with the graph of the parent function and show all stages.
 Be sure to show at least three key points on the final graph.
 Find the domain and range of each function.
 Verify your results with a graphing utility.
Remember: (1) Stretch/Compress; (2) Reflect; (3) Translate
(40) f  x   x 2  4
(44) h x   x  1
(48) g  x  
(50) h  x   3 2 x
1
x
2
(60) g  x   3 x  1  3
(46) f  x    x  2  3
3
(56) h x  
(like 63) f  x   x  2  3
2
1
2
x
(64) f  x   4 x  1
Hole at (1, 0)
(66) g  x   4 2  x
Finding the Cube Root Key
MATH, then 4:
Hole at (2, 0)
3
(
Finding the Absolute Value Key
MATH, then arrow key to the right
to get the NUM menu.
Then 1: abs(
OR
2nd CATALOG, then the first item
on the list will be abs( , once it is
highlighted hit ENTER.
Question:
What about f(x) = (2x)3 ? Do I think of the x as getting cut in half, or rewrite the function as
f(x) = 8x3, and think of the y–values as getting multiplied by 8.
Answer:
Regardless of how you conceptualize f(x) the net effect on the graph is the same.