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REFLECTING GRAPHSAND
SYMMETRY
Section 4.3
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REFLECTING GRAPHS
x-axisy-axis
Line y = xAbsolute value of a graph
Around a point
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GRAPH USING CALCULATOR2
2
y x
y x
3 2
3 2
2
( 2 )
y x x
y x x
How are the graphs of y=f(x) and y = - f(x) related?
Reflected over the x-axis!!
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GRAPH
( 1)( 3)
( 1)( 3)
y x x x
y x x x
Any negative y values are reflected over the x-axis!!
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GRAPH USING CALCULATOR
y x
y x
How are the graphs of y = f(x)
and y = f(-x) related?
Reflected over the y-axis!!
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GRAPH2
2
y x
x y
How is the graph of an equation affected
when you interchange the variables in the equation?
The graph is reflected over the line y = x!!
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4
2
-2
-4
-5 5
Sketch the graphs of:( )
( )
( )
y f x
y f x
y f x
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Sketch the graphs of:( )
( )
( )
y f x
y f x
y f x
3
2
1
-1
-2
-3
-4 -2 2 4
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Sketch the graph and the reflection in the line y = x. Also, give the equation of the reflected graph.
2 4y x x 4
2
-2
-4
-5 5
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Bellwork
Given an isosceles right triangle shown below. Express the Area, A, of the triangle as a function of the hypotenuse, c.
ac
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Summary of Symmetry X-axis y = -f(x) (make y negative)
Y-axis y = f(-x) (make x negative)
Line y = x x = f(y) (switch x and y)
Origin y = -f(-x) (both x and y are negative)
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Is there any symmetry in the graph of the functions?
4 4 1x y 3 1x y
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Use symmetry to sketch the graph of
15
10
5
-5
-10
-15
-10 10
3 1x y
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Use symmetry to sketch the graph of 12xy
15
10
5
-5
-10
-15
-10 10
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Cubic Functions and Symmetry:
What type of symmetry is in a cubic?
What is the point of symmetry?
3
bx
a
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Given the polynomial below, what is the point of symmetry?
2 3( ) 9 6 2f x x x x
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The graph of a cubic function has a local max at (-5, 6) and a point of symmetry a (-1, 2). What is the local minimum?