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2-6 Families of Functions
Algebra 2
Prentice Hall, 2007
Think… “The Function Family”
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Definitions A Family of Functions is made up of
functions with certain characteristics. A parent function is the simplest form of a
function. Examples:
(line with slope 1 passing through origin)
(a V-graph opening up with vertex at origin)
(a U-graph opening up with vertex at origin)
y x
y x
y x 2
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Definitions Offspring of parent functions are:
Translations (or shifts of the parent) Stretches (or skinnier versions of the parent) Shrinks (or fatter versions of the parent) Reflections (or flipped versions of the parent)
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Translations
Translations are shifts up/down, left/right, or a combination of the 2.
The graph is the same size & shape as the parent, but moved to a new location.
Ex: Parent Offspring
y x
y x 26
4
2
-2
-4
-6
-5 5
6
4
2
-2
-4
-6
-5 5
y x 26
4
2
-2
-4
-6
-5 5
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Translation Functions If the parent function is , then translation
offspring would be: shift up k units shift down k units shift right h units shift left h units
What kind of translation is ? Shift up k and right h
NOTE: shift left/right according to “k” only when the coefficient of x is 1!!!
y x
y x k
y x k
y x h
y x h
y x h k
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Examples
Ex. 1 Graph
Remember what the parent looks like and go from there…
…the parent shifted right 3 units!
y x 3
6
4
2
-2
-4
-6
-5 5
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Examples Ex. 2 Graph
The parent shifted down 3 units.
What would look like?
y x 3
6
4
2
-2
-4
-6
-5 5
y x 2 1
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Stretches
A stretch multiples all the y-values by the
same factor greater than 1, thereby
the graph vertically
(making it skinnier than the parent!) If the parent function is , then
stretched offspring would be …
… provided .
y x
y a x
a 1
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Shrinks
A shrink reduces all the y-values by the
same factor less than 1, thereby
the graph
vertically (making it fatter than the parent!) If the parent function is , then
shrunken offspring would be …
… provided .
y x
y a x
0 a 1
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Examples Graph the parent function: Graph the offspring:
Stretch:• y’s are 3 times bigger
Shrink:• y’s are half as big
y 3 x
y 1
2x
y x
6
4
2
-2
-4
-6
-5 5
6
4
2
-2
-4
-6
-5 5
stretch
parent
6
4
2
-2
-4
-6
-5 5
shrink
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Reflections
A reflection over the x-axis changes the y-values to their opposites…(i.e. the parent flips!)
If the parent function is , then a reflection offspring would be: reflection over the x-axis ( )
y x
y x
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Example Graph the parent function: Graph the reflection offspring:
y x
6
4
2
-2
-4
-6
-5 5
y x
6
4
2
-2
-4
-6
-5 5
reflection
parent
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Just for kicks…
Describe this “offspring”:
y 4 x 2 56
4
2
-2
-4
-6
-5 5
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Homework
2-6 PRWS 2-6 Families of Functions