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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.) ,