Lesson #4: Basic Transformations with Parent Functions
Lesson #4:
Basic Transformations
with the Absolute Value
Function
The Absolute Value Function
Recall:
Basic
Parent
Function
y
Domain:
∞, ∞
Range:
,∞
Istheabsolute
valuefunction
one‐to‐one?
Explain.
No,becausethe
graphfailsthe
horizontallinetest.
[y‐valuesrepeat]
Algebra II with Trigonometry: Unit 1
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Lesson #4: Basic Transformations with Parent Functions
Basic Transformations using the
Absolute Value Function
1. If
, what happens to the
graph if 3 is added to the function?
The graph is shifted UP 3 units.
2. How does this transformation
change the range of the function?
The range becomes
or
,∞ .
3.
The equation becomes
Basic Transformations using the
Absolute Value Function
3. If
, what happens to the graph
if 3 is subtracted from the function?
The graph is shifted DOWN 3 units.
4. How does this transformation change
the range of the function?
The range becomes
The equation becomes
Algebra II with Trigonometry: Unit 1
3 or
,∞ .
3.
2
Lesson #4: Basic Transformations with Parent Functions
Basic Transformations using the
Absolute Value Function
5. If
, write the new equation if
the graph is moved 2 units to the right.
2
6. How does this transformation change
the range of the function?
0 or
The range remains
,∞ .
*Note:Horizontaltranslations‐
areopposite ofwhattheyappear tobe.
Basic Transformations using the
Absolute Value Function
7. If
, write the new equation if
the graph is moved 2 units to the left.
2
8. How does this transformation change
the range of the function?
The range remains
0 or
,∞ .
*Note:Horizontaltranslations‐
areopposite ofwhattheyappear tobe.
Algebra II with Trigonometry: Unit 1
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Lesson #4: Basic Transformations with Parent Functions
Transformations Using Function Notation
Vertical Translations
•If k is positive, the graph shifts up k units.
•If k is negative, the graph shifts down k units
Horizontal Translations
•If h is positive, the graph shifts to the right h units.
(Remember: if h “looks negative” h is positive)
•If h is negative, the graph shifts to the left h units.
(Remember: if h “looks positive” h is negative)
Basic Transformations using the
Absolute Value Function
9. If
, write the new equation if
the graph is reflected over the -axis.
*Recall:
Toreflectover
‐axis,negate
the ‐value.
10. How does this transformation
change the range of the function?
The range becomes
Algebra II with Trigonometry: Unit 1
0 or
∞,
.
4
Lesson #4: Basic Transformations with Parent Functions
Basic Transformations using the
Absolute Value Function
11. If
, write the new equation
if the graph is reflected over the -axis. *Recall:
12. How does this transformation
change the range of the function?
Toreflectover
‐axis,negate
the ‐value.
The graph remains
exactly the same!
Transformations Using Function Notation
Given:
Reflection over the -axis
Reflection over the -axis
•If the -value is
negated,
is
reflected over the -axis.
•If the -value is
negated,
is
reflected over the -axis.
Algebra II with Trigonometry: Unit 1
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Lesson #4: Basic Transformations with Parent Functions
Basic Transformations using the
Absolute Value Function
13. If
, what happens to the
graph if the function is multiplied by 3?
The graph is verticallystretchedby a
factor of 3.
14. How does this transformation
change the range of the function?
The range remains
0 or
,∞ .
3
The equation becomes
.
Basic Transformations using the
Absolute Value Function
15. If
, what happens to the
graph if the function is multiplied by ?
The graph is verticallyshrunkor
compressedby a factor of .
16. How does this transformation
change the range of the function?
The range remains
The equation becomes
Algebra II with Trigonometry: Unit 1
0 or
,∞ .
.
6
Lesson #4: Basic Transformations with Parent Functions
Transformations Using Function Notation
Given:
Vertically Stretching or Shrinking a Graph
∙
•If
1, multiply each -coordinate of
by ,
vertically stretching the graph of by the factor of .
•If 0
1, multiply each -coordinate of
by
, vertically shrinking the graph of by the factor of .
Basic Transformations using the
Absolute Value Function
17. If
, what happens to the graph
?
if the function changed to
The graph is horizontallyshrunkor
compressedby a factor of .
18. How does this transformation
change the range of the function?
The range remains
Algebra II with Trigonometry: Unit 1
0 or
,∞ .
7
Lesson #4: Basic Transformations with Parent Functions
Basic Transformations using the
Absolute Value Function
19. If
, what happens to the graph
?
if the function changed to
The graph is horizontallystretchedby a
factor of .
20. How does this transformation
change the range of the function?
The range remains
0 or
,∞ .
Transformations Using Function Notation
Given:
Horizontally Stretching or Shrinking a Graph
∙
•If
1, divide each -coordinate of
by ,
horizontally shrinking the graph of by the factor of .
•If 0
1, divide each -coordinate of
by ,
horizontally stretching the graph of by the factor of .
Algebra II with Trigonometry: Unit 1
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Lesson #4: Basic Transformations with Parent Functions
Transformations Using Function Notation
VerticalStretch&Shrink
HorizontalStretch&Shrink
Transformations Using Function Notation
CombinationsofTransformations:
A function involving more than one transformation can be
graphed by performing transformations in the following order:
1. Horizontal Shifting
2. Stretching or Shrinking
3. Reflecting
4. Vertical Shifting
Algebra II with Trigonometry: Unit 1
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Lesson #4: Basic Transformations with Parent Functions
Example 1: Writetheequationofeachgraphbelow.
Example 2: Use the graph of
to sketch the graph of
transformations in words.
and transformations
3
2. Also, describe the
H: Left 3
S: NONE
R: NONE
V: DOWN 2
Algebra II with Trigonometry: Unit 1
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Lesson #4: Basic Transformations with Parent Functions
Example 3: Use the graph of
to sketch the graph of
2
the transformations in words.
and transformations
1
1. Also, describe
H: Right 1
S:
Vertical Stretch
by factor of 2
*multiply ‐valuesby2*
R: NONE
V: UP 1
Using the Calculator
Inordertographanabsolutevalueequation:
Go to
**The quickest way to find the absolute value:
Second 0 and press enter.
abs( will appear or | | will appear.
This is the absolute value function.
Algebra II with Trigonometry: Unit 1
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