Quadrant IV consists of the points with positive -coordinates and negative -coordinates. Therefore is the correct choice.

The line of symmetry of the parabola of the equation

is the vertical line

Substitute :

The line of symmetry is

That is, the line of the equation .

Remember that the basic form of the equation of a circle is:

This means that the center point  is defined by the two values subtracted in the squared terms.  We could rewrite our equation as:

Therefore, the center is 

For this question, we will need to do three things:

1. Determine the point in question.
2. Use trigonometry to find the area of the angle in question.
3. Use the equation for finding a sector area to finalize our answer.

Let us first solve for the coordinate by substituting into our equation:

Our point is, therefore: 

Now, we need to calculate the angle formed between the origin and the point that we were given. We can do this using the inverse tangent function. The ratio of  to  is here: 

Therefore, the angle is:

To solve for the sector area, we merely need to use our standard geometry equation. Note that the radius of the circle, based on the equation, is .

This rounds to .

For this question, we will need to do three things:

1. Determine the point in question.
2. Use trigonometry to find the area of the angle in question.
3. Use the equation for finding a sector area to finalize our answer.

Let us first solve for the coordinate by substituting into our equation:

Our point is, therefore: 

Now, we need to calculate the angle formed between the origin and the point that we were given. We can do this using the inverse tangent function. The ratio of  to  is here: 

Therefore, the angle is:

To solve for the sector area, we merely need to use our standard geometry equation. Note that , based on the equation, is .

This rounds to .

Since you are looking for a point on the , your  value will be zero.  

The center of the circle is at the origin and radius is the distance from the center, so that means the point you are looking for must be  points away from .  

This can be two points on the  but since you are looking for a positive one, your answer must be .

The displacement between negative three and positive one is four.

This mean that after flipping the point, it must be symmetrical to its original location.   The new point must also be 4 units to the right of the line.

The new point would be located at:  

The answer is:  

The relation 

is a circle with center  and radius  .

In other words, it is a circle with center at the origin, translated right  units and up  units (the radius is irrelevant to the question).

or

is this circle translated right zero units and up 2 units. The upshot is that the circle moves along the -axis only, and therefore is symmetric with respect to the -axis, but not the -axis. Also, as a consequence, it is not symmetric with respect to the origin.

Reflecting the point across the vertical line  will only change the x-value, but not the y-value.

The point after this reflection is: 

Rotating this point across the origin will swap the x and y-values.

The new point is: 

When a function  is flipped across the x-axis, the new function  is equal to . Therefore, our function  is equal to:

Our final answer is therefore