A circle is translated 4 units to the right and then reflected over the x-axis. Complete the statement so that it will always be true of the circle at the new location.

The statement is now presented as:

In other words, this mathematical statement can be translated as:

There is a point (h, k) in the set of real ordered pairs so that a circumference centered at (h,k) and with a radius r implies a equivalent circumference centered at (h+4,-k) and with a radius r.

Step-by-step explanation:

Let the coordinates of the center of the circle, which must be transformed into by operations of translation and reflection. From Analytical Geometry we understand that circles are represented by the following equation:

Where is the radius of the circle, which remains unchanged in every operation.

Now we proceed to describe the series of operations:

1) Center of the circle is translated 4 units to the right (+x direction):

(Eq. 1)

Where is the translation vector, dimensionless.

If we know that and , then:

2) Reflection over the x-axis:

(Eq. 2)

Where is the reflection point, dimensionless.

If we know that and , the new point is:

And thus, and . The statement is now presented as:

In other words, this mathematical statement can be translated as:

There is a point (h, k) in the set of real ordered pairs so that a circumference centered at (h,k) and with a radius r implies a equivalent circumference centered at (h+4,-k) and with a radius r.