Question

which notation describes this transformation?
a. $(x,y)=(-x,-y)$
b. $(x,y)=(x - 8,y - 8)$
c. $(x,y)=(x - 4,y - 8)$
d. $(x,y)=(x,-y)$

Explanation:

Step1: Analyze the transformation type

Looking at the original figure (shape 1) and the transformed figure (shape 2), we observe that the x - coordinates seem to remain the same (or have a reflection that flips the sign? Wait, no, let's check the symmetry. The transformation appears to be a reflection over the x - axis. For a reflection over the x - axis, the rule is that the x - coordinate of a point \((x,y)\) in the original figure remains the same, and the y - coordinate is negated. So if the original point is \((x,y)\), the transformed point \((x',y')\) is given by \((x',y')=(x, - y)\).

Let's verify with a point from shape 1. Let's take a vertex of shape 1. For example, one of the vertices of shape 1: let's say a point \((2,6)\) (approximate, looking at the grid). The corresponding point in shape 2 should be \((2, - 6)\)? Wait, no, looking at the grid, shape 1 is in the first quadrant (positive y) and shape 2 is in the fourth? No, shape 2 is in the third? Wait, no, shape 1 is at positive x and positive y, shape 2 is at negative x and negative y? Wait, no, maybe I made a mistake. Wait, let's check the coordinates. Let's take a point from shape 1: let's say the bottom - left vertex of shape 1 is at \((2,3)\) (from the grid: x = 2, y = 3). The corresponding point in shape 2: looking at shape 2, the bottom - left vertex seems to be at \((-2,-3)\)? Wait, no, maybe it's a reflection over the origin? Wait, no, let's check the options.

Option A: \((x',y')=(-x,-y)\) is a reflection over the origin. Option D: \((x',y')=(x, - y)\) is a reflection over the x - axis.

Wait, let's take a point from shape 1. Let's take the top - left vertex of shape 1: let's say (2,6). The corresponding point in shape 2: looking at shape 2, the top - left (but in the lower half) vertex: if we use \((x',y')=(x, - y)\), then \((2, - 6)\)? But shape 2 is at x negative? Wait, no, maybe the x - coordinate is also negated? Wait, no, let's look at the x - coordinates. Shape 1 is at x from 2 to 6 (approx), shape 2 is at x from - 6 to - 2 (approx). So the x - coordinate of a point in shape 1 (x) is transformed to - x, and the y - coordinate (y) is transformed to - y. Wait, that would be \((x',y')=(-x,-y)\), which is option A. Wait, I think I made a mistake earlier.

Wait, let's take a point (x,y) in shape 1. Let's say (2,6) (x = 2, y = 6). The corresponding point in shape 2: looking at shape 2, the point should be (-2,-6). So \(x'=-x\) (since 2 becomes - 2) and \(y'=-y\) (since 6 becomes - 6). So the transformation rule is \((x',y')=(-x,-y)\), which is option A? Wait, no, wait the options:

Wait, the problem is about the transformation from shape 1 to shape 2. Let's check the x and y signs. Shape 1 is in the first quadrant (x>0,y>0), shape 2 is in the third quadrant (x<0,y<0). So for a point \((x,y)\) in shape 1, the transformed point \((x',y')\) has \(x'=-x\) and \(y'=-y\), which is the rule for a reflection over the origin, given by \((x',y')=(-x,-y)\) (option A) or is it a translation?

Wait, option B: \((x',y')=(x - 8,y - 8)\) is a translation. Let's check the horizontal and vertical shifts. If we take a point (2,6) from shape 1, applying \(x-8=2 - 8=-6\), \(y - 8=6 - 8=-2\). But the point in shape 2 is not (-6,-2). So B is wrong.

Option C: \((x',y')=(x - 4,y - 8)\). For (2,6), \(x-4=-2\), \(y - 8=-2\). Not matching.

Option D: \((x',y')=(x, - y)\). For (2,6), \(x' = 2\), \(y'=-6\). But shape 2 is at x negative, so D is wrong.

Option A: \((x',y')=(-x,-y)\). For (2,6), \(x'=-2\), \(y'=-6\). Which matches the position of shape 2 (since shape 2 is at negative x and negative y). So the correct transformation is \((x',y')=(-x,-y)\), which is option A? Wait, no, wait the x - coordinates of shape 1 are positive, shape 2 are negative, and y - coordinates of shape 1 are positive, shape 2 are negative. So the transformation is a reflection over the origin, which is \((x',y')=(-x,-y)\).

Wait, but let's check again. Let's take the top - right vertex of shape 1: (6,6). Applying option A: \((-6,-6)\), which should be the top - right (but in the third quadrant) of shape 2. Which seems to match.

So the correct notation is \((x',y')=(-x,-y)\), which is option A.

Answer:

A. \((x',y') = (-x,-y)\)