2.6
TRANSCRIPT
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2.6 FAMILIES OF FUNCTIONS
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FAMILIES OF FUNCTIONS
There are sets of functions, called families that share certain characteristics. A parent function is the simplest form in a set
of functions that form a family. Each function in the family is a transformation
of the parent function
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FAMILIES OF FUNCTIONS
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TRANSLATIONS
One type of transformation is a translation. A translation shifts the graph of the parent
function horizontally, vertically, or both without changing shape or orientation.
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EXAMPLEHow can you represent each translation of y = |x| graphically?
1.
2.
2g x x
1h x x
Shift the parent graph
down 2 units
Shift the parent graph
left 1 unit
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EXAMPLE
How can you represent each translation of y = |x| graphically?
1.
2.
3 1g x x
2 3h x x
Shift the parent graph right 3 units and up 1 units
Shift the parent graph left 2 units and down 3 units
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REFLECTIONS
A reflection flips the graph over a line (such as the x – or y – axis) Each point on the graph of the reflected function
is the same distance from the line of reflection as its corresponding point on the graph of the original function.
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REFLECTIONSWhen you reflect a graph in the y-axis, the x values change signs and the y-values stay the same.
When you reflect a graph in the x-axis, the y-values change signs and the x-values stay the same.
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REFLECTING A FUNCTION ALGEBRAICALLY
Let and be the reflection in the x-axis. What is a function rule for
?
3 3f x x g x g x
g x f x
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REFLECTING A FUNCTION ALGEBRAICALLY
Let and be the reflection in the y-axis. What is a function rule for ?
3 3f x x g x g x
g x f x
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2.6 CONTINUED
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VERTICAL STRETCH AND VERTICAL COMPRESSION A vertical stretch multiplies all y-values of
a function by the same factor greater than 1. A vertical compression reduces all
y-values of a function by the same factor between 0 and 1.
Why do you think the value being multiplied is always positive?
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EXAMPLEThe table represents the function f(x). Complete the table to find the vertical stretch and vertical compression. Then graph the functions.
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EXAMPLE: COMBINING TRANSFORMATIONS
The graph of g(x) is the graph of f(x) = 4x compressed vertically by the factor ½ and then reflected in the y-axis. What is the function rule for g(x)?
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EXAMPLE: COMBINING TRANSFORMATIONS
The graph of g(x) is the graph of f(x) = x stretched vertically by the factor 2 and then translated down 3 units. What is the function rule for g(x)?
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EXAMPLE: COMBINING TRANSFORMATIONS
What transformations change the graph of f(x) to the graph of g(x)?
2 22 6 1f x x g x x