A rectangle with vertices (3,-2),(3,-4), (7, -2), (7, -4) is reflected across the x-axis and then rotated 90° counterclockwise. What other transformations produce the sameimage?

A reflection about the y-axis and a clockwise rotation of 90° around the origin.

another example is:

A counterclockwise rotation of 90° around the origin, and then a reflection about the y-axis.

Step-by-step explanation:

Let's do this for a single point because it is essentially the same.

Let's use the point (3, -2)

First, we have a reflection across the x-axis, this only changes the sign of the y-component.

Then the new point will be: (3, -(-2)) = (3, 2)

Notice that this point is on the first quadrant.

Now we do a 90° counterclockwise rotation.

Then we move to the second quadrant, and we change the order of the components in the point (and because we are in the second quadrant, the x-component is negative and the y-component is positive)

Then the new point is (-2, 3)

Now, another transformation that is equivalent to this one is to do first a reflection about the y-axis, so we only change the sign of the y-component

Then the new point is (-3, -2)

And now we are in the third quadrant.

and then we do a rotation of 90° clockwise, which moves our point to the second quadrant, changes the order of the components and leaves the y-component positive and the x-component negative, then the new point is:

(-2, 3)

Same as before.

(notice that we used the same transformations, but applied to different line and direction of rotation)

Another example is if we first start with a counterclockwise rotation of 90° around the origin, the original point (3, -2) is on the fourth quadrant, so this rotation leaves our point in the first quadrant, so we change the order of the components and both of them will have a positive sign.

The new point is: (2, 3)

Now let's do a reflection about the y-axis, which changes the sign of the x-component.

Then the final point is (-2, 3), same as before.

And there are a lot of other transformations that will be equivalent to this ones, these are just two examples.

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