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4.3 REFLECTING GRAPHS & SYMMETRY

4.3 REFLECTING GRAPHS & SYMMETRY. LEARNING OBJECTIVES Reflect Graphs Use Symmetry to Sketch Graphs Find Lines of Symmetry “How to use a line symmetry

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Page 1: 4.3 REFLECTING GRAPHS & SYMMETRY. LEARNING OBJECTIVES  Reflect Graphs  Use Symmetry to Sketch Graphs  Find Lines of Symmetry  “How to use a line symmetry

4.3 REFLECTING GRAPHS & SYMMETRY

Page 2: 4.3 REFLECTING GRAPHS & SYMMETRY. LEARNING OBJECTIVES  Reflect Graphs  Use Symmetry to Sketch Graphs  Find Lines of Symmetry  “How to use a line 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.”

Page 3: 4.3 REFLECTING GRAPHS & SYMMETRY. LEARNING OBJECTIVES  Reflect Graphs  Use Symmetry to Sketch Graphs  Find Lines of Symmetry  “How to use a line symmetry

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

Page 4: 4.3 REFLECTING GRAPHS & SYMMETRY. LEARNING OBJECTIVES  Reflect Graphs  Use Symmetry to Sketch Graphs  Find Lines of Symmetry  “How to use a line symmetry

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

Page 5: 4.3 REFLECTING GRAPHS & SYMMETRY. LEARNING OBJECTIVES  Reflect Graphs  Use Symmetry to Sketch Graphs  Find Lines of Symmetry  “How to use a line symmetry

EXAMPLESKETCH THE GRAPH OF

Page 6: 4.3 REFLECTING GRAPHS & SYMMETRY. LEARNING OBJECTIVES  Reflect Graphs  Use Symmetry to Sketch Graphs  Find Lines of Symmetry  “How to use a line symmetry

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

Page 7: 4.3 REFLECTING GRAPHS & SYMMETRY. LEARNING OBJECTIVES  Reflect Graphs  Use Symmetry to Sketch Graphs  Find Lines of Symmetry  “How to use a line symmetry

EXAMPLESKETCH THE GRAPH OF

Page 8: 4.3 REFLECTING GRAPHS & SYMMETRY. LEARNING OBJECTIVES  Reflect Graphs  Use Symmetry to Sketch Graphs  Find Lines of Symmetry  “How to use a line symmetry

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.

Page 9: 4.3 REFLECTING GRAPHS & SYMMETRY. LEARNING OBJECTIVES  Reflect Graphs  Use Symmetry to Sketch Graphs  Find Lines of Symmetry  “How to use a line symmetry

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)

Page 10: 4.3 REFLECTING GRAPHS & SYMMETRY. LEARNING OBJECTIVES  Reflect Graphs  Use Symmetry to Sketch Graphs  Find Lines of Symmetry  “How to use a line symmetry

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.

Page 11: 4.3 REFLECTING GRAPHS & SYMMETRY. LEARNING OBJECTIVES  Reflect Graphs  Use Symmetry to Sketch Graphs  Find Lines of Symmetry  “How to use a line symmetry

EXAMPLESKETCH THE GRAPH OF

Page 12: 4.3 REFLECTING GRAPHS & SYMMETRY. LEARNING OBJECTIVES  Reflect Graphs  Use Symmetry to Sketch Graphs  Find Lines of Symmetry  “How to use a line symmetry

IDENTIFY WHICH TYPE(S) OF REFLECTION CAN BE SEEN IN EACH

GRAPH.

Page 13: 4.3 REFLECTING GRAPHS & SYMMETRY. LEARNING OBJECTIVES  Reflect Graphs  Use Symmetry to Sketch Graphs  Find Lines of Symmetry  “How to use a line symmetry

ASSIGNMENT

Textbook pg. 136 #1-4

Page 14: 4.3 REFLECTING GRAPHS & SYMMETRY. LEARNING OBJECTIVES  Reflect Graphs  Use Symmetry to Sketch Graphs  Find Lines of Symmetry  “How to use a line symmetry

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

Page 15: 4.3 REFLECTING GRAPHS & SYMMETRY. LEARNING OBJECTIVES  Reflect Graphs  Use Symmetry to Sketch Graphs  Find Lines of Symmetry  “How to use a line symmetry

EQUATIONS FOR SYMMETRY

Page 16: 4.3 REFLECTING GRAPHS & SYMMETRY. LEARNING OBJECTIVES  Reflect Graphs  Use Symmetry to Sketch Graphs  Find Lines of Symmetry  “How to use a line 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)

Page 17: 4.3 REFLECTING GRAPHS & SYMMETRY. LEARNING OBJECTIVES  Reflect Graphs  Use Symmetry to Sketch Graphs  Find Lines of Symmetry  “How to use a line symmetry

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.

Page 18: 4.3 REFLECTING GRAPHS & SYMMETRY. LEARNING OBJECTIVES  Reflect Graphs  Use Symmetry to Sketch Graphs  Find Lines of Symmetry  “How to use a line symmetry

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.

Page 19: 4.3 REFLECTING GRAPHS & SYMMETRY. LEARNING OBJECTIVES  Reflect Graphs  Use Symmetry to Sketch Graphs  Find Lines of Symmetry  “How to use a line symmetry

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.

Page 20: 4.3 REFLECTING GRAPHS & SYMMETRY. LEARNING OBJECTIVES  Reflect Graphs  Use Symmetry to Sketch Graphs  Find Lines of Symmetry  “How to use a line symmetry

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.

Page 21: 4.3 REFLECTING GRAPHS & SYMMETRY. LEARNING OBJECTIVES  Reflect Graphs  Use Symmetry to Sketch Graphs  Find Lines of Symmetry  “How to use a line symmetry

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

Page 22: 4.3 REFLECTING GRAPHS & SYMMETRY. LEARNING OBJECTIVES  Reflect Graphs  Use Symmetry to Sketch Graphs  Find Lines of Symmetry  “How to use a line symmetry

ASSIGNMENT

Textbook pg. 135-137 #2, 7-8, 15-16