0-2: smart graphing objectives: identify symmetrical graphs identify odd/even functions sketch the...
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0-2: Smart Graphing
Objectives:•Identify symmetrical graphs•Identify odd/even functions•Sketch the graphs of functions using translations, reflections & dilations
© 2002 Roy L. Gover ([email protected]) Modified by Mike Efram 2004
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DefinitionPoint Symmetry: Two points, P & P’ are symmetric with respect to a point M if M is the midpoint of
'PP
P P’M
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...For a graph to have point symmetry with respect to a point M, M must be the midpoint of every set of points P & P’ on the graph. Examples...
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Example
2 2 2x y r Point SymmetryConsider:
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3( )f x x
M
Example
Point Symmetry:
M
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A graph that is symmetrical with the point (0,0) is symmetric with respect to the origin.
Definition
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Definition
A function f(x) is symmetric with respect to the origin if and only if
f(-x)=-f(x)
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Examplef(x)=x3 is symmetric with the origin because -30
-20
-10
0
10
20
30
1 2 3 4 5 6 7
f(-x)=-f(x). ie f(-2)=-8 & f(2)=8,therefore f(-2)=-f(2)
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Try This
Is f(x)=x2
symmetric with respect to the origin?
No
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Graphs that have line symmetry can be folded along the line of symmetry so that the two halves match exactly.
Important Idea
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Examples of Line Symmetry
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Symmetry with respect to x=0 ( y-axis ) exists if and only if:f(x)=f(-x)
Example: f(x)=x2-3
Definition
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Symmetry is useful in graphing functions. If you graph part of the function and understand the symmetry, the rest of the graph can be sketched.
Important Idea
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DefinitionEven Functions are functions symmetric with the y axis. They have exponents that are all even.
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Definition
Odd functions are functions symmetric with the origin. They have exponents that are all odd.
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Try ThisAre the following functions even, odd or neither:4 2 6y x x
3( )f x x x 5 3( ) 1g x x x
Even
Odd
Neither
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SummaryOdd functions:
f(-x) = -f(x)
Symmetry with origin (0, 0)
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SummaryEven functions:
f(x) = f(-x)
Symmetry with y-axis
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DefinitionReflections: the mirror image of a graph.
Example
f(x)=x2 f(x)=-x2
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Try This
Without using a graphing calculator, graph f(x)=-x3 using its parent graph as a starting point.
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Solution
3y x 3y x
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Definition
Translation: the sliding of a graph vertically or horizontally without changing its size or shape.
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Examples
f(x)=x2-3
f(x)=(x+3)2
f(x)=x2+3
f(x)=(x-3)2
VerticalTranslations
HorizontalTranslations
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Try ThisWrite the equation of this graph based on its parent graph.Hint: a vertical & horizontal translation is required.f x x( ) ( ) 3 32
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Try ThisWrite the equation of this graph based on its parent graph.Hint: a reflection & horizontal translation is required. f x x( ) ( ) 2 2
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Try ThisWithout using your calculator, sketch the graph of:
p x x( ) 2 2
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DefinitionDilation: changing a graph’s size. Making it either smaller or larger. Examples:
f x x( ) f x x( ) 1
4f x x( ) 4
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Example
The graph of f(x) is pictured at the right. Sketch a graph of:a) f(x+3)
b) f(x+3)-2
c) -f(x-3)-2
d) 2f(x+2)+3