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Transcript of Symmetry & Transformations. Transformation of Functions Recognize graphs of common functions Use...
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Symmetry & Transformations
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Transformation of Functions
Recognize graphs of common functions Use vertical shifts to graph functions Use horizontal shifts to graph functions Use reflections to graph functions Use vertical stretching & shrinking to graph
functions Use horizontal stretching & shrinking to graph
functions Graph functions w/ sequence of
transformations
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Basic Functions
You should memorize the following basic functions. They are vital to understanding upper level math
courses. See the back section of your book,
Algebra’s Common Functions, for all of the basic functions you should
memorize.
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Types of Transformations Vertical translation: shifts basic graph up or
down; up if y = f(x) + b or down if y = f(x) – b Horizontal translation: shifts basic graph left or
right; right if y = f(x - d) or left if y = f(x + d) Reflection: crosses an axis;
across x axis if y = -f(x) & across y axis if y = f(-x) Vertical stretching and shrinking: y = af(x)
stretch if |a| > 1; shrink if 0<|a|<1 Horizontal stretching and shrinking: y = f(cx)
stretch if 0<|c|<1; shrink if |c|>1
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Vertical shifts Moves the graph up or
down Impacts only the “y”
values of the function No changes are made
to the “x” values Horizontal shifts
Moves the graph left or right
Impacts only the “x” values of the function
No changes are made to the “y” values
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Recognizing the shift from the equation, look at examples of shifting the function f(x)=x2.
Vertical shift of 3 units up
Horizontal shift of 3 units left (HINT: x’s go the opposite direction that you might think.)
3)(,)( 22 xxhxxf
22 )3()(,)( xxgxxf
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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.
46)(,)( xxgxxf
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Types of SymmetrySymmetry with respect to the y-axis (x, y) & (-x, y) are
reflections across the y-axis Origin (x, y) & (-x, -y) are
reflections across the origin x-axis (x, y) & (x, -y) are
reflections across the x-axis
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Tests of Symmetry
f(x) = f(-x) symmetric to y-axis (same) even function
- f(x) = f(-x)symmetric to origin(opposite) odd
function
f(x) = - f(x) symmetric to x-axis neither even nor odd
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Example – Determine any/all symmetry of 6x + 7y = 0.
a) y-axis
b) origin
c) x-axis
Is the function even, odd, or neither?
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Example – Determine any/all symmetry of 2( ) 1f x x
d) y-axis
e) origin
f) x-axis
Is the function even, odd, or neither?
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Horizontal stretch & shrink We’re MULTIPLYING by
an integer (not 1 or 0). x’s do the opposite of
what we think they should. (If you see 3x in the equation where it used to be an x, you DIVIDE all x’s by 3, thus it’s compressed or shrunk horizontally.)
4)3()( 2 xxg
f(x) = |x2 – 4|
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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!)
43)( 2 xxh
f(x) = |x2 – 4|
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Sequence of transformations
Follow order of operations. Select two points (or more) from the original
function and move that point one step at a time.
1)2(31)2(3
)(3
3
xxf
xxf
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Graph of Example
1)2(31)2(3)(
)(
3
3
xxfxg
xxf
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Transformations withthe Squaring Function
Function Transformation g(x) = x2 + 4 ________________________ h(x) = x2 – 5 _______________________ j(x) = (x – 3)2 ________________________ k(x) = (x + 1)2 ________________________ q(x) = -x2 ________________________ r(x) = 2x2 ________________________ s(x) = ¼x2 ________________________ t(x) = (5x)2 ________________________
2( )f x x
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Your turn. Describe these transformations with the Absolute Value Function.
Function Transformation g(x) = -|x| ________________________ h(x) = |2x| ________________________ j(x) = 3|x| ________________________ k(x) = |x + 4| ________________________ q(x) = |x - 5| ________________________ r(x) = |x| - 1 ________________________ s(x) = -½|x| ________________________ t(x) = |x| + 2 ________________________
( ) | |f x x
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Summary of Transformations
See instructor webpage.