Write any radical equation that reflects, dilates, and translates right?

Nov 2016
42
1
USA
The open ended question states: "Write your own radical equation that reflects, dilates, and translates right."

If possible, let's use the parent function f(x) = radical x.

First: I'm not sure whther the wording implies that the equation has to reflect to the right, dilate to the right, and translate to the right,
or whether one is permitted to reflect it across the x-axis, dilate in any direction, etc. (as long as it is translated to the right)?

Second: for the dilation, should the answer I develop state a coordinate for the center of dilation and a scale factor?
If so, then let's use the origin as the center and a scale factor of 2.

Thank you for assistance.
 
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skipjack

Forum Staff
Dec 2006
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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?
 
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Nov 2016
42
1
USA
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.
 
Nov 2016
42
1
USA
A quick answer is f(x)= -0.5sqrt(x-4)
Required radical; parent function f(x)=sqrt(x)
The 0.5 dilates.
The negative sign reflects.
The -4 translates to the right.
 

skipjack

Forum Staff
Dec 2006
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2,470
If the original set of points (x, y) is specified by giving y as a function of x, it's a good idea to specify the domain of y explicitly (especially if it's an interval), as it may change as a result of a translation, reflection or dilation.

If, say, y = √(x - 2), and the initial domain is [2, 11], dilating from the origin by a factor of 2 changes the domain to [4, 22] and the equation for y becomes y = √(2x - 8).

For example the original point (11, 3) is mapped by the dilation to the point (22, 6) because √(2*22 - 8) = √36 = 6.
 
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