Question

which notation describes this transformation?
a. ((x, y) = (x + 9, y - 2))
b. ((x, y) = (-y, x))
c. ((x, y) = (-x, y))
d. ((x, y) = (x - 9, y + 2))

Explanation:

Step1: Identify a vertex of shape 2

Take a vertex of the pink shape (shape 2), e.g., the rightmost vertex at \((-1, -2)\) (wait, no, looking at the graph, let's pick a clear vertex. Let's take the top - left vertex of shape 2: \((-6, 0)\), and the corresponding vertex of shape 1. Wait, maybe better to take the right - most vertex of shape 2: let's see, shape 2 has a vertex at \((-1, -2)\)? No, looking at the grid, shape 2 (pink) has a vertex at \((-6, 0)\), and shape 1 (red) has a vertex at \((3, - 2)\)? Wait, no, let's check the transformation. Let's take a vertex of shape 2: say the vertex at \((-6, 0)\) (on the x - axis, y = 0). The corresponding vertex of shape 1: let's see, if we apply \((x',y')=(x + 9,y - 2)\) to \((-6,0)\): \(x'=-6 + 9=3\), \(y'=0-2=-2\). Now check the red shape (shape 1), does it have a vertex at \((3, - 2)\)? Looking at the graph, yes, that seems to match. Let's check another vertex. Take the bottom - most vertex of shape 2: say \((-4, - 4)\). Apply \((x',y')=(x + 9,y - 2)\): \(x'=-4 + 9 = 5\), \(y'=-4-2=-6\). Check shape 1, it has a vertex around there. Now check other options. Option B: \((x',y')=(-y,x)\). For \((-6,0)\), \(x'=-0 = 0\), \(y'=-6\), which doesn't match. Option C: \((x',y')=(-x,y)\). For \((-6,0)\), \(x' = 6\), \(y'=0\), not matching. Option D: \((x',y')=(x - 9,y + 2)\). For \((-6,0)\), \(x'=-6-9=-15\), \(y'=0 + 2 = 2\), not matching. So option A works.

Step2: Verify with another vertex

Take the right - most vertex of shape 2: let's say the vertex at \((-1, - 2)\) (wait, no, let's take the vertex at \((-1, - 2)\) for shape 2. Apply option A: \(x'=-1+9 = 8\), \(y'=-2-2=-4\)? Wait, maybe my first vertex was wrong. Let's re - examine. Let's take the vertex of shape 2 at \((-6,0)\) (x=-6, y = 0). After applying \((x + 9,y - 2)\), we get \(x=-6 + 9=3\), \(y=0-2=-2\). Now look at shape 1 (red), it has a vertex at \((3, - 2)\), which matches. Another vertex: shape 2 has a vertex at \((-1, - 2)\) (wait, no, shape 2's right - most vertex: let's count the grid. Each square is 1 unit. Shape 2 (pink) has a vertex at \((-1, - 2)\)? No, shape 2 is on the left, shape 1 on the right. Let's take the vertex of shape 2 at \((-4, - 4)\) (x=-4, y=-4). Apply \((x + 9,y - 2)\): \(x=-4 + 9 = 5\), \(y=-4-2=-6\). Shape 1 has a vertex at \((5, - 6)\), which matches. So option A is correct.

Answer:

A. \((x',y')=(x + 9,y - 2)\)