Question

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Answer 1

We have to identify the transformation that maps the triangle DEF to its image D'E'F'.

We can graph the points but if we look at the changes in the coordinate we may know what transformations happened.

For example, we can see that the y-coordinate stays the same for preimage and image points.

The x-coordinate changes but not a fixed length: D is translated 3-1 = 2 units, E is translated -4-(-8) = -12 units and F is translated 6-(-2) = 8 units.

This has to be a reflection across a vertical axis. Now we can graph the points to see if we can find the x-coordinate of that axis:

We can easily see in the graph that the axis of reflection is the vertical line x = 2.

We could check this by calculating the midpoint of each preimage and image point.

We can now have to deduce the rule for this reflection.

For the y-coordinate, we know it does not change.

For x we have to define an auxiliary constant d that represents the distance of the preimage point to the reflection axis. Then, the image x-coordinate will be the x-coordinate of the axis plus that distance, where the distance can be negative or positive depending on the position of the preimage point.

We can then write

`\begin{gathered} x^(\prime)=2+d \\ d=-(x-2)=-x+2 \\ \Rightarrow x^(\prime)=2+(-x+2)=4-x \end{gathered}`

We can check with the points as:

`\begin{gathered} x^(\prime)_D=4-x_D=4-3=1 \\ x^(\prime)_E=4-x_E=4-(-4)=4+4=8 \\ x^(\prime)_F=4-x_F=4-6=-2 \end{gathered}`

The rule is then:

`(x,y)\to(4-x,y)`

Answer:

The rule of the transformation is (x,y) --> (4-x,y)