**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 5 units to the right and 3 units down.

**Sol** Replacing by and by gives
or .

**Ex 2** Explain how to obtain the graph of
from the graph
of using translations and reflections.

**Sol a** First shift the graph of 2 units to the left to get
, and then reflect in the y-axis to get
.

**Sol b** First reflect the graph in the y-axis to get
, and
then shift the graph 2 units to the right to get
.

**Pr A** Find an equation of the graph whic is obtained by reflecting the
graph of in the line .

**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 in the y-axis, and then shifting the resulting graph 4 units to the
right.

**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
in 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
3 units to the left, and then reflecting in the y-axis.

**Pr 4** Explain how to obtain the graph of from the graph of
using translations and reflections.

**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.

**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
from the graph
of using translations and reflections.

Go to Solutions.

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