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# Graphs of Reflections

## Graph images given preimage and line of reflection

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Reflections: Graphs and Rules

### Review

Transformation
A transformation is an operation that moves, ____, or changes a shape to create a new shape.
Reflection
A reflection is an example of a transformation that flips _____ of a shape over the same line.
Line of Reflection
The line of reflection is the line that a shape _____ (flips) across when undergoing a reflection.

Each point on the preimage will be the same distance from the line of reflection as it's corresponding point in the image.



• reflections across the x\begin{align*}x\end{align*} -axis: y\begin{align*}y\end{align*} values are multiplied by ___.
• reflections across the y\begin{align*}y\end{align*} -axis: __ values are multiplied by -1.
• reflections across the line y=x\begin{align*}y=x\end{align*} : x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*} values switch places
• reflections across the line y=x\begin{align*}y = -x\end{align*} . x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*} values switch places and are multiplied by __.

What is the reflection across the line y=x called?

#### Practice Questions:

1) Line AB¯¯¯¯¯¯¯¯\begin{align*}\overline{AB}\end{align*} drawn from (-5, 3) to (7, 3) has been reflected across the x\begin{align*}x\end{align*} -axis. Draw the preimage and image and properly label each.

2) he diamond ABCD\begin{align*}ABCD\end{align*} is reflected across the line y=x\begin{align*}y = x\end{align*} to form the image ABCD\begin{align*}A^\prime B^\prime C^\prime D^\prime \end{align*}. Find the coordinates of the reflected image. On the diagram, draw and label the reflected image.



3) Draw a triangle with vertices at points (-2, 3), (-3,1) and (1,1).  Then reflect the triangle across the line y=x\begin{align*}y=-x\end{align*}.

4) The purple pentagon is reflected across the yaxis\begin{align*}y-axis\end{align*}to make the new image. Find the coordinates of the purple pentagon. On the diagram, draw and label the reflected pentagon.



### Notation/Rules

We can generalize reflections by using the following template:

ryaxisAB=ryaxis(x,y)(x,y)\begin{align*}r_{y-axis}A \rightarrow B=r_{y-axis}(x,y) \rightarrow (-x,y)\end{align*}

Reflection notation helps us describe the movement of a figure more generally, generalizing the coordinates from (2,3) for example, to (x,y).

1) Describe the following reflection using reflection notation:



Solution:

ryaxis(x,y)(x,y)\begin{align*}r_{y-axis}(x,y) \rightarrow (-x,y)\end{align*}

2) Reflect Image A in the diagram below:

a) Across the y\begin{align*}y\end{align*} -axis and label it B\begin{align*}B\end{align*} .

b) Across the x\begin{align*}x\end{align*} -axis and label it O\begin{align*}O\end{align*} .

c) Across the line y=x\begin{align*}y=-x\end{align*} and label it Z\begin{align*}Z\end{align*} .



Write notation for each to indicate the type of reflection.

Solution:

a) Reflection across the y\begin{align*}y\end{align*} -axis: ryaxisAB=ryaxis(x,y)(x,y)\begin{align*}r_{y-axis}A \rightarrow B=r_{y-axis}(x,y) \rightarrow (-x,y)\end{align*}

b) Reflection across the x\begin{align*}x\end{align*} -axis: rxaxisAO=rxaxis(x,y)(x,y)\begin{align*}r_{x-axis}A \rightarrow O=r_{x-axis}(x,y) \rightarrow (x,-y)\end{align*}

c) Reflection across the y=x\begin{align*}y=-x\end{align*} : ry=xAZ=ry=x(x,y)(y,x)\begin{align*}r_{y=-x}A \rightarrow Z=r_{y=-x}(x,y) \rightarrow (-y,-x)\end{align*}