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4.3 REFLECTING GRAPHS & SYMMETRY
LEARNING OBJECTIVES
Reflect GraphsUse Symmetry to Sketch GraphsFind Lines of Symmetry
“How to use a line symmetry or point symmetry to sketch a graph.”
Line of Reflection acts like a mirror located halfway between a point and its reflection
Reflecting Across:1. x-axis2. y-axis3. line y = x
REFLECTING GRAPHS
X-AXIS REFLECTION
If the red line represents f(x) and the purple line represents –f(x), describe the rule for reflecting across the x axis.
To Graph or a reflecting across the x-axis
EXAMPLESKETCH THE GRAPH OF
Y-AXIS REFLECTION
The red line represents f(x) and the blue line represents f(-x). Describe the rule for reflecting across the y axis.
To Graph or a reflecting across the y-axis
EXAMPLESKETCH THE GRAPH OF
The graph of f(x) = x2 + 2 is shown.
Y = X REFLECTION
To graph the reflection over the line y = x, swap x and y.
If a reflection over y = x is found by swapping x and y, what is the notation for this reflection?
Find the equation for the reflection of f(x) over y = x line.
EXAMPLESKETCH GRAPH ACROSS Y = X
a) Sketch the graph of b) Give an equation for the reflected graph
across y = xc) Sketch the graph reflected across y = x
b)
To graph find plot all values of above the x axis.Note: is identical to when y > 0
ABSOLUTE VALUE REFLECTION
The absolute value reflection of f(x) = x2 – 5 is shown. Describe a rule for absolute value reflections.
EXAMPLESKETCH THE GRAPH OF
IDENTIFY WHICH TYPE(S) OF REFLECTION CAN BE SEEN IN EACH
GRAPH.
ASSIGNMENT
Textbook pg. 136 #1-4
A line l is called an axis of symmetry of a graph if it is possible to pair the points of the graph in such a way that l is the perpendicular bisector of the segment joining each pair.
A point O is called point of symmetry of a graph if it is possible to pair the points of the graph in such a way that O is the midpoint of the segment joining each pair.A point of symmetry signifies a 180 rotation.
l is the axis of symmetry O is the point of symmetry
SYMMETRY
EQUATIONS FOR SYMMETRY
EXAMPLEFIND A LINE OF SYMMETRY
a) Graph the fb) Find the equation for the axis of symmetry.b) Graph the axis of symmetry.
b)
2
Type of Symmetry Example
1
:
Symmetry in the
-axis y x
Meani g
x
n
2
, is on the graph whenever , is. 1
:
In the equation, leave alone and
substitute for . Does an equivalent
equivalent
x y x y y x
Testing an equation of a graph
x
y y
equation result?
Type of Symmetry Example
SPECIAL TESTS FOR THE SYMMETRY OF A GRAPH
Both points (x, y) and (x, –y) on the graph
You only need to graph one branch. The other branch can be obtained by reflecting the graphed branch about the x-axis.
2
Type of Symmetry Example
:
Symmetry in
the -ax
is y x
Meaning
y
2
, is on the graph whenever , is.
:
In the equation, substitute for and
leave alone. Does an equivalent
eq
equivalent
x y x y y x
Testing an equation of a graph
x x
y
uation result?
SPECIAL TESTS FOR THE SYMMETRY OF A GRAPH
Type of Symmetry Example
p. 134
Both points (x, y) and (–x, y) on the graph
You only need to graph one branch. The other branch can be obtained by reflecting the graphed branch about the y-axis.
3 3 1
: equivale
Symmetry in the lin
, is on the graph whenever , is.
e
t
n
y x x y
Meaning
y x x y
3 3 1
:
In the equation, interchange and .
Does an equivalent equation result?
y x
Testing an equation of a graph
x y
Type of Symmetry Example
p. 134
SPECIAL TESTS FOR THE SYMMETRY OF A GRAPH
Both points (x, y) and (y, x) on the graph
You only need to graph one branch. The other branch can be obtained by reflecting the graphed branch about the line y = x.
3
: e
Symmetry in the or
quivalent
, is
i
o
gi
n the graph when r
n
eve
y x
Meaning
x y
3 , is.
:
In the equation, substitute for and
for . Does an equivalent equation
result?
x y y x
Testing an equation of a graph
x x
y y
Type of Symmetry Example
p. 134
SPECIAL TESTS FOR THE SYMMETRY OF A GRAPH
Both points (x, y) and (–x, –y) on the graph
You only need to graph one branch. The other branch can be obtained by reflecting the graphed branch about the origin.
Reflect x-axisReflect y-axisAbove x-axis
c whole number – stretchc fraction – shrink
(vertically)
c whole number – shrinkc fraction – stretch
(horizontally)
Shift up or down
Shift right or left
TRANSLATIONS – SUMMARY
ASSIGNMENT
Textbook pg. 135-137 #2, 7-8, 15-16