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 $h>0$):

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 $y=x^2$ 5 units to the right and 3 units down.

Sol Replacing $x$ by $x-5$ and $y$ by $y+3$ gives $y+3=(x-5)^2$ or $y=(x-5)^2-3$.

Ex 2 Explain how to obtain the graph of $y=\frac{1}{2-x}$ from the graph of $y=1/x$ using translations and reflections.

Sol a First shift the graph of $y=1/x$ 2 units to the left to get $y=\frac{1}{x+2}$, and then reflect in the y-axis to get $y=\frac{1}{2-x}$.

Sol b First reflect the graph in the y-axis to get $y=\frac{1}{-x}$, and then shift the graph 2 units to the right to get $y=\frac{1}{-(x-2)}=\frac{1}{2-x}$.

Pr A Find an equation of the graph whic is obtained by reflecting the graph of $y=x^3$ in the line $y=x$.

Pr B Find an equation of the graph which results from reflecting the graph of $y=x^2+5$ 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 $y=e^x$ 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 $\frac{x^2}{9}+\frac{y^2}{16}=1$ 5 units to the left and 4 units up.

Pr 2 Find an equation of the graph which results from reflecting the graph of $y=\sqrt{x}$ 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 $y=\sqrt{x}$ 3 units to the left, and then reflecting in the y-axis.

Pr 4 Explain how to obtain the graph of $y=9-\ln(5-x)$ from the graph of $y=\ln x$ using translations and reflections.

Pr 5 Find an equation of the graph which is obtained by reflecting the graph of $y=x^2-5$ in the line $y=x$, 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 $y=x^2-5$ 4 units down, and then reflecting the resulting graph in the line $y=x$.

Pr 7 Find an equation of the graph which is obtained by reflecting the graph of $y=\sqrt{x}$ in the line $y=x$, 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 $y=\frac{x-4}{x-3}$ from the graph of $y=1/x$ using translations and reflections.

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