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Parent Functions and Transformations
Transformation of FunctionsRecognize graphs of common functions
Use shifts to graph functionsUse reflections to graph functionsGraph functions w/ sequence of transformations
The following basic graphs will be used extensively in this section. It is important to be able to sketch these from memory.
The identity function f(x) = x
The quadratic function
2)( xxf
xxf )(
The square root function
xxf )(The absolute value function
3)( xxf
The cubic function
The rational function1
( )f xx
We will now see how certain transformations (operations) of a function change its graph. This will give us a better idea of how to quickly sketch the graph of certain functions. The transformations are (1) translations, (2) reflections, and (3) stretching.
Vertical Translation
OUTSIDE IS TRUE!Vertical Translationthe graph of y = f(x) + d is the graph of y = f(x) shifted up d units;
the graph of y = f(x) d is the graph of y = f(x) shifted down d units.
2( )f x x 2( ) 3f x x
2( ) 2f x x
Horizontal Translation
INSIDE LIES!Horizontal Translationthe graph of y = f(x c) is the graph of y = f(x) shifted right c units;
the graph of y = f(x + c) is the graph of y = f(x) shifted left c units.
2( )f x x
22y x 2
2y x
The values that translate the graph of a function will occur as a number added or subtracted either inside or outside a function.
Numbers added or subtracted inside translate left or right, while numbers added or subtracted outside translate up or down.
( )y f x c d
Recognizing the shift from the equation, examples of shifting the function f(x) = Vertical shift of 3 units up
Horizontal shift of 3 units left (HINT: x’s go the opposite direction that you might believe.)
3)(,)( 22 xxhxxf
22 )3()(,)( xxgxxf
2x
Use the basic graph to sketch the following:
( ) 3f x x 2( ) 5f x x 3( ) ( 2)f x x ( ) 3f x x
Combining a vertical & horizontal shift
Example of function that is shifted down 4 units and right 6 units from the original function.
( ) 6
)
4
( ,
g x x
f x x
Use the basic graph to sketch the following:
( )f x x
( )f x x 2( )f x x
( )f x x
The big picture…
Example
Write the equation of the graph obtained when the parent graph is translated 4 units left and 7 units down.3y x
3( 4) 7y x
ExampleExplain the difference in the graphs
2( 3)y x 2 3y x
Horizontal Shift Left 3 Units
Vertical Shift Up 3 Units
Describe the differences between the graphs
Try graphing them…
2y x 24y x 21
4y x
A combinationIf the parent function is
Describe the graph of
2y x
2( 3) 6y x The parent would be horizontally shifted right 3 units and vertically shifted up 6 units
If the parent function is
What do we know about
3y x32 5y x
The graph would be vertically shifted down 5 units and vertically stretched two times as much.
What can we tell about this graph?
3(2 )y xIt would be a cubic function reflected across the x-axis and horizontally compressed by a factor of ½.