precalculus 2015 1.4 transformation of functions objectives recognize graphs of common functions use...
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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
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The following basic graphs will be used extensively in this course. It is important to be able to sketch these from memory.
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The identity function f(x) = x
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The squaring function
2)( xxf
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xxf )(
The square root function
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xxf )(The absolute value function
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3)( xxf
The cubing function
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The cube root function 3( )f x x
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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.
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Graph Illustrating Vertical Shift.
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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.
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Graph Illustrating Horizontal Shift.
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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.
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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.
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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
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Use the basic graph to sketch the following:
( ) 3f x x
2( ) 5f x x
( ) 3f x x
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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
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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.
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Copyright © Houghton Mifflin Company. All rights reserved.
Digital Figures, 1–22
Graph of a Reflection across the x-axis.
What would f(x) look like if it were reflected across the y-axis?
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Graph of a Reflection across the y-axis.
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Use the basic graphs to sketch each of the following:
( )f x x
( )f x x 2( )f x x
( )f x x
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Vertical Stretching and Shrinking
The graph of af(x) can be obtained from the graph of f(x) by
stretching vertically for |a| > 1, orshrinking vertically for 0 < |a| < 1.
For a < 0, the graph is also reflected across the x-axis.
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VERTICAL STRETCH (SHRINK)y’s do what we
think they should: If you see 3(f(x)), all y’s are MULTIPLIED by 3 (it’s now 3 times as high or low!)
2( ) 3 4f x x
2( ) 4f x x
21( ) 4
2f x x
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Horizontal Stretching or Shrinking
The graph of y = f(cx) can be obtained from the graph of y = f(x) by
shrinking horizontally for |c| > 1, orstretching horizontally for 0 < |c| < 1.
For c < 0, the graph is also reflected across the y-axis.
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Horizontal stretch & shrink
Think of the coefficient on x as speed. It will either speed up or slow down the function.
( ) sin(2 )g x x
( ) sinf x x
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Transforming points
Use transformations to find 2 points on the graph of g(x)
Write the function g(x)
3( )
g(x) = 3 ( 2) 1
f x x
f x
3g(x)= 3( 2) 1x
0,0 2, 1
1,1 1,2
X values are shifted 2 units left. Y values (function values) are multiplied by 3 and shifted down 1 unit.
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Graph of Example3
3
( )
( ) 3 ( 2) 1 3( 2) 1
f x x
g x f x x
(0,0), (1,1)
(-2,-1), (-1,2)
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The point (-12, 4) is on the graph of y = f(x). Find a point on the graph of y = g(x).
g(x) = f(x-2)
g(x)= 4f(x)
g(x) = f(½x)
g(x) = -f(x)
(-10, 4)
(-12, 16)
(-24, 4)
(-12, -4)
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Discuss with your neighborCompare the graph of the function
below with the graph of . What transformations have taken place from the basic graph?
3( )f x x
32 1 3h x x
•Shift right by 1 unit•Vertical stretch by a factor of 2•Reflect across x-axis•Vertical shift up by 3
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Homework
Pg. 49: 3,9,11, 15-31 odd, 39-43 odd, 51, 59