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Translations and Reflections
Translations
Given an equation in x and y, the following changes in the equation result in the indicated
changes in the graph (where
):
Replacing Translation
x by x-h
h units to the right
x by x+h
h units to the left
y by y-h
h units up
y by y+h
h units down
Reflections
Given an equation in x and y, the following changes in the equation result in the indicated
changes in the graph:
Replacing
Reflection
x by -x
in the y-axis
y by -y
in the x-axis
x by -x and y by -y in the origin
x by y and y by x
in the line y=x
Ex 1 Find an equation of the curve which results by translating the graph of
units to the right and 3 units down.
Sol Replacing
by
and
by
Ex 2 Explain how to obtain the graph of
translations and reflections.
gives
5
or
from the graph of
.
using
Sol a First shift the graph of
in the y-axis to get
2 units to the left to get
.
Sol b First reflect the graph in the y-axis to get
to the right to get
, and then reflect
, and then shift the graph 2 units
.
Pr A Find an equation of the graph whic is obtained by reflecting the graph of
the line
in
.
Pr B Find an equation of the graph which results from reflecting the graph of
in the x-axis, and then shifting the resulting graph 3 units up.
Pr C Find an equation of the graph which is obtained by reflecting the graph of
the y-axis, and then shifting the resulting graph 4 units to the right.
in
Pr 1 Find an equation of the graph which results from translating the graph of
5 units to the left and 4 units up.
Pr 2 Find an equation of the graph which results from reflecting the graph of
the y-axis, and then shifting the graph 3 units to the left.
Pr 3 Find an equation of the graph which results from shifting the graph of
units to the left, and then reflecting in the y-axis.
Pr 4 Explain how to obtain the graph of
using translations and reflections.
from the graph of
Pr 5 Find an equation of the graph which is obtained by reflecting the graph of
in the line
, and then shifting the resulting graph 4 units down.
in
3
Pr 6 Find an equation of the graph which is obtained by shifting the graph of
4 units down, and then reflecting the resulting graph in the line
.
Pr 7 Find an equation of the graph which is obtained by reflecting the graph of
in the line
, then shifting the resulting graph 3 units to the right, and then shifting
the graph which results 4 units up.
Pr 8 Explain how to obtain the graph of
translations and reflections.
from the graph of
using
Translations and Reflections
Sol A Interchanging
Sol B Replacing
by
and
gives the equation
gives
gives
or
or
Sol C Replacing
by
or
by
Sol 2 Replacing
by
, and then replacing
.
gives
and
, and then replacing
by
by
gives
or
or
.
by
gives
.
Sol 1 Replacing
Sol 3 Replacing
or equivalently
gives
, and then replacing
.
by
gives
.
gives
.
, and then replacing
by
gives
by
Sol 4 Shift 5 units to the left to get
or
, and then reflect in the y-axis to get
. Then reflect in the x-axis to get
or
, and then shift 9 units up to get
or
. (This is not the only solution.)
Sol 5 Interchanging
and gives
, and then replacing
by
gives
.
Sol 6 Replacing
by
gives
and gives
or
, and then interchanging
.
Sol 7 Interchanging
gives
and gives the equation
or
. Next replacing
, and then replacing
by
by
gives the equation
.
Sol 8 Since
, we can shift 3 units to the right to get
reflect in the x-axis to get
or
or
, and then shift 1 unit up to get
. (This is not the only solution.)
,