Precalculus 2015 1.4 Transformation of Functions Objectives Recognize graphs of common functions Use shifts to graph functions Use reflections to graph functions Use stretching & shrinking to graph functions Graph functions w/ sequence of transformations
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Precalculus 2015 1.4 Transformation of Functions Objectives Recognize graphs of common functions Use shifts to graph functions Use reflections to graph.
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Precalculus 20151.4 Transformation of Functions Objectives
Recognize graphs of common functions
Use shifts to graph functions
Use reflections to graph functions
Use stretching & shrinking to graph functions
Graph functions w/ sequence of transformations
The following basic graphs will be used extensively in this course. It is important to be able to sketch these from memory.
The identity function f(x) = x
The squaring function
2)( xxf
xxf )(
The square root function
xxf )(The absolute value function
3)( xxf
The cubing function
The cube root function 3( )f x x
Vertical and Horizontal Shifts
Let c be a positive real number. Vertical and horizontal shifts in the graph of y = f(c) are represented as follows:
1.Vertical shift c units upward:
2.Vertical shift c units downward:
3.Horizontal shift c units to the right:
4.Horizontal shift c units to the left:
h x f x c
h x f x c
h x f x c
h x f x c
Numbers added or subtracted outside translate up or down, while numbers added or subtracted inside translate left or right.
Graph Illustrating Vertical Shift.
Vertical Translation
Vertical TranslationFor c > 0, the graph of y = f(x) + c is the graph of y = f(x) shifted up c units;
the graph of y = f(x) c is the graph of y = f(x) shifted down c units.
Graph Illustrating Horizontal Shift.
Horizontal TranslationFor c > 0, the 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.
Why translations work the way they do
Upward Vertical TranslationConsider the function f(x) = x2 . If we add, say 4 units, to f(x) then the function becomes g(x) = f(x) + 4. The graph of g(x) is an upward translation of the graph of f(x) shifted vertically by 4 units. The reason why the graph shifted upward is because 4 units have been added to every y-coordinate of the graph of f(x), and the y-coordinate of f(x) happens to be f(x) itself or x2.
Thus, adding 4 to x2 causes the y-coordinate of every ordered pair of f(x) to increase by 4.
Why translations work the way they do
Horizontal TranslationConsider the function f(x) = x2 . In order for a function to have its graph shifted n units to the right, then all we have to do is add n units to every x-coordinate of the function.
The x-coordinate of a graph of a function can be found by solving for x. So if our function is y = x2, then solving for x:
If we want the function y = x2 to have its graph shifted to the right, say 3 units, then we add 3 to the right side of the equation above as follows:
All the x-coordinates of f(x) have now been shifted 3 units to the right; and if we solve for y: 2
3y x
x y
3x y
3x y
Use the basic graph to sketch the following:
( ) 3f x x
2( ) 5f 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.
What is the equation of the translated function?
( ) , f x x
( ) 6 4 f x x
Reflections
The graph of f(x) is the reflection of the graph of f(x) across the x-axis.
The graph of f(x) is the reflection of the graph of f(x) across the y-axis.