I'll assume that the problem relates to points in a Cartesian plane with an x-axis running "from left to right" and a y-axis running "from bottom to top", so that "right" refers to the direction of increasing x. Any other letter I use will be a real constant.

The problem doesn't state that a parent function exists. Any set of points in the plane can be reflected, etc., whether or not they represent a function. For any original point (x, y), there is some new point that results from the reflection, dilation or translation.

Let's start by dealing with translation, reflection, and dilation separately. If more than one of these is applied, **the order in which they are applied may affect the overall result.**

A translation maps each point (x, y) to the point (x + a, y + b).

I'll assume that the reflection is in a line with equation

x = c, y = c, y = x + c, or y = -x + c.

Reflection in a general line is more complicated.

Reflection in the line x = c maps the point (x, y) to the point (2c - x, y).

Reflection in the line y = c maps the point (x, y) to the point (x, 2c - y).

Reflection in the line y = x + c maps the point (x, y) to the point (y - c, x + c).

Reflection in the line y = -x + c maps the point (x, y) to the point (-y + c, - x + c).

I'll assume that "dilation" doesn't include a stretch in relation to just a particular direction. Thus, the dilation is by a factor of r in relation to a fixed point (a, b).

Such a dilation maps the point (x, y) to the point (rx - ra + a, ry - rb + b),

so if a = b = 0 and r = 2, the point (x, y) is mapped to the point (2x, 2y).

Any questions about the above?

Yes, the bolded quote is correct: the order in which to apply the reflection/dilation/translation can affect the final outcome.

I'll I start with a required radical function of choice: f(x)= square root (-x), which has points (0,0), (-1,1), (-4,2) and (-9, 3).

Next, reflect it

**to the right** across the y-axis with f(x)= -x, which is f(x)= sqrt[-(-x)] and has the points (0,0), (1,1), (4,2) and (9,3).

Then, translate it

**to the right** two units with f(x)=sqrt[-(-(x-2))], which simplifies to f(x)=sqrt(x-2) and has the points (2,0), (3,1), (6,2) and (11,3).

Finally, set the center of dilation at the origin (so it dilates

**to the right**) with a scale factor of 2.

The point (2,0) is 2 away from the origin, multiplied by scale factor of 2 = 4 for the x value. The y value is 0 away from the origin, so the point is (4,0).

The point (3,1) is 3 away from the origin, multiplied by 2 equals 6 for the x value. The y value is 1 away, multiplied by 2 is 2, so the point is (6,2).

The point (6,2) is 6 away from the origin, multiplied by 2 equals 12 for the x value. The y value is 2 away, multiplied by 2 is 4, so the point is (12,4)

The point (11,3) is 11 away from the origin, multiplied by 2 is 22 for x. The y value is 3 away, times 2 is 6, so the point is (22, 6).

HOWEVER, if I reflect and next dilate using the origin & scale factor of 2, I get (0,0), (2,2), (8,4) and (18,6); and then I translate right two units last, arriving at the final points of (2,0), (4,2), (10,4) and (20,6).

Interesting exercise.