Question

figure ghij was transformed to create figure ghij. the table shows the coordinated of the transformed figure ghij.
image of grid with figure ghij

g	(5, -1)
h	(5, -1)
i	(5, -1)
j	(0, -2)

options:

• $(x, y) \to (x, -y)$
• $(x, y) \to (x, y - 6)$
• $(x, y) \to (x - 2, y - 2)$
• $(x, y) \to (y, -x)$

Explanation:

First, we need to find the original coordinates of the figure GHIJ from the graph. Let's assume the original coordinates:
- From the graph, G seems to be at (1, 5) (since on the grid, x=1, y=5), H at (3, 3), I at (5, 4), J at (2, 0). Wait, maybe better to check the transformed coordinates and test each transformation.

Wait, the transformed coordinates are G'(5, -1), H'(5, -1)? Wait, no, maybe the table has a typo? Wait, maybe the original figure: looking at the graph, G is at (1, 5), H at (3, 3), I at (5, 4), J at (2, 0). Let's check each transformation:

1. Test \((x,y)\to(x,-y)\): For G(1,5), it would be (1, -5), but G' is (5, -1), so no.

2. Test \((x,y)\to(x, y - 6)\): For G(1,5): \(y - 6 = 5 - 6 = -1\), x remains 1? But G' is (5, -1). Wait, maybe I misread the original coordinates. Wait, maybe the original G is at (1, 5), but G' is (5, -1). Wait, maybe the original J is at (2, 0), J' is (0, -2). Wait, let's check J:

Original J: from the graph, J is at (2, 0) (since x=2, y=0). J' is (0, -2). Let's test each transformation on J(2,0):

- \((x,y)\to(x,-y)\): (2, 0) becomes (2, 0), not (0, -2). No.

- \((x,y)\to(x, y - 6)\): (2, 0) becomes (2, -6), not (0, -2). No.

- \((x,y)\to(x - 2, y - 2)\): (2, 0) becomes (0, -2). Hey, that works! Let's check G. Wait, maybe original G is at (1, 5)? No, wait, maybe original G is at (1, 5), but G' is (5, -1). Wait, no, maybe I made a mistake. Wait, let's re-examine the graph. The original figure: G is at (1, 5)? Wait, the x-axis: from -6 to 6, y-axis from -4 to 4? Wait, no, the grid: the y-axis has 4 at the top, so G is at (1, 5)? No, maybe the y-axis is from -4 to 4, but the graph shows G above y=4. Wait, maybe the original coordinates:

Looking at the graph, G is at (1, 5) (x=1, y=5), H at (3, 3), I at (5, 4), J at (2, 0). Now, let's check the transformation \((x,y)\to(x - 2, y - 2)\) on J(2,0): 2 - 2 = 0, 0 - 2 = -2, which matches J'(0, -2). Now check G(1,5): 1 - 2 = -1, 5 - 2 = 3, but G' is (5, -1). Wait, that's not matching. Wait, maybe I got the original coordinates wrong. Wait, maybe G is at (1, 5), but G' is (5, -1). Let's check the other transformation: \((x,y)\to(y, -x)\). For G(1,5): (5, -1), which matches G'(5, -1)! Let's check H: original H, let's say H is at (3, 3). Then (y, -x) would be (3, -3), but H' is (5, -1). No, that's not. Wait, maybe original H is at (3, 3), but H' is (5, -1). No. Wait, maybe the table has a typo, and G', H', I' have different coordinates. Wait, the table shows G'(5, -1), H'(5, -1), I'(5, -1), J'(0, -2). That seems odd. Wait, maybe the original figure: J is at (2, 0), J' is (0, -2). Let's check the transformation \((x,y)\to(x - 2, y - 2)\) on J: (2 - 2, 0 - 2) = (0, -2), which matches. Now check G: suppose original G is at (7, 1)? No, that doesn't make sense. Wait, maybe the original figure is G(1,5), H(3,3), I(5,4), J(2,0). Let's check the transformation \((x,y)\to(x, y - 6)\) on G(1,5): (1, 5 - 6) = (1, -1), not (5, -1). No. Wait, maybe the original G is at (5, 5), then (5, 5 - 6) = (5, -1), which matches G'. H: (5, 3 - 6) = (5, -3), but H' is (5, -1). No. Wait, maybe the original H is at (5, 3), then (5, 3 - 6) = (5, -3), no. Wait, J: original J is (2, 0), (2, 0 - 6) = (2, -6), no. Wait, the transformation \((x,y)\to(x - 2, y - 2)\) on J(2,0) gives (0, -2), which matches. Let's check G: if original G is (7, 1), then (7 - 2, 1 - 2) = (5, -1), which matches G'. H: (7, 3) → (5, 1), no. Wait, this is confusing. Wait, maybe the correct transformation is \((x,y)\to(x - 2, y - 2)\) because J(2,0) becomes (0, -2). Let's check the options again. The options are:

1. \((x,y)\to(x,-y)\): reflection over x-axis.

2. \((x,y)\to(x, y - 6)\): vertical translation down 6 units.

3. \((x,y)\to(x - 2, y - 2)\): translation left 2, down 2.

4. \((x,y)\to(y, -x)\): rotation 90 degrees counterclockwise.

Wait, let's take J: original J is (2, 0) (from the graph, J is at (2, 0)). J' is (0, -2). So (2 - 2, 0 - 2) = (0, -2), which is the third transformation. Now check G: suppose original G is (1, 5)? No, (1 - 2, 5 - 2) = (-1, 3), not (5, -1). Wait, maybe original G is (7, 1), (7 - 2, 1 - 2) = (5, -1), which matches. H: (7, 3) → (5, 1), but H' is (5, -1). No, that's not. Wait, maybe the table has a mistake, and H' and I' are different. Alternatively, maybe the correct transformation is \((x,y)\to(x, y - 6)\). Let's check J(2,0): (2, 0 - 6) = (2, -6), no. J' is (0, -2). Wait, maybe the original J is (2, 0), and the transformation is \((x,y)\to(x - 2, y - 2)\), which gives (0, -2). For G, if original G is (1, 5), then (1 - 2, 5 - 2) = (-1, 3), but G' is (5, -1). This is conflicting. Wait, maybe the original figure's coordinates are different. Let's look at the graph again. The original figure: G is at (1, 5) (x=1, y=5), H at (3, 3), I at (5, 4), J at (2, 0). The transformed figure G'H'I'J' has G'(5, -1), H'(5, -1), I'(5, -1), J'(0, -2). Wait, that can't be, unless H' and I' are miswritten. Maybe it's a typo, and G' is (1, -1), H' is (3, -3), I' is (5, -4), J' is (2, -6), but that's not the case. Alternatively, maybe the correct transformation is \((x,y)\to(x, y - 6)\). Let's check G(1,5): (1, 5 - 6) = (1, -1), no. G' is (5, -1). Wait, maybe the original G is (5, 5), then (5, 5 - 6) = (5, -1), which matches. H(5, 3): (5, 3 - 6) = (5, -3), but H' is (5, -1). No. J(2, 0): (2, 0 - 6) = (2, -6), no. J' is (0, -2). This is really confusing. Wait, let's check the transformation \((x,y)\to(x - 2, y - 2)\) on J(2,0): (0, -2), which matches. For G, if original G is (7, 1), then (5, -1), which matches. H(7, 3) → (5, 1), but H' is (5, -1). No. Maybe the answer is \((x,y)\to(x - 2, y - 2)\) because J matches. Alternatively, maybe the correct transformation is \((x,y)\to(y, -x)\) for G: (5, -1) from (1,5) (since y=5, -x=-1), which is (5, -1). Then H(3,3) would be (3, -3), but H' is (5, -1). No. Wait, maybe the original G is (1,5), so (y, -x) is (5, -1), which matches G'. H(3,3) would be (3, -3), but H' is (5, -1). No. J(2,0) would be (0, -2), which matches J'! Oh! Wait, J is (2,0), so (y, -x) is (0, -2), which is J'. G is (1,5), (y, -x) is (5, -1), which is G'. H: let's say H is (3,3), (y, -x) is (3, -3), but H' is (5, -1). Wait, maybe H is (3,5)? No, the graph shows H at (3,3). Wait, maybe the table has a typo, and H' is (3, -3), I' is (4, -5). But the table says H' and I' are (5, -1). This is a problem. But since J(2,0) transforms to (0, -2) via \((x,y)\to(y, -x)\) (since 2,0 → 0, -2) and G(1,5) transforms to (5, -1) via \((x,y)\to(y, -x)\) (1,5 → 5, -1), maybe H and I have different original coordinates. Maybe H is (3,5), then (5, -3), but no. Alternatively, maybe the correct answer is \((x,y)\to(y, -x)\) because G and J match. Let's check the options again. The options are:

1. \((x,y)\to(x,-y)\): G(1,5)→(1,-5)≠(5,-1)

2. \((x,y)\to(x, y - 6)\): G(1,5)→(1,-1)≠(5,-1)

3. \((x,y)\to(x - 2, y - 2)\): G(1,5)→(-1,3)≠(5,-1)

4. \((x,y)\to(y, -x)\): G(1,5)→(5,-1), J(2,0)→(0,-2). This matches G' and J'. So even though H' and I' seem off, maybe it's a typo, and the correct transformation is \((x,y)\to(y, -x)\). Wait, but H(3,3) would be (3,-3), but H' is (5,-1). Maybe H is (3,5), then (5,-3), no. Alternatively, maybe the original H is (3,5), but the graph shows H at (3,3). This is confusing, but based on G and J, the transformation \((x,y)\to(y, -x)\) works for G and J. Wait, no, J(2,0)→(0,-2) is (y, -x) because y=0, -x=-2, so (0, -2). Yes! So J(2,0)→(0, -2) is (y, -x). G(1,5)→(5, -1) is (y, -x) (y=5, -x=-1). So that works. So the correct transformation is \((x,y)\to(y, -x)\)? Wait, no, wait: (x,y)→(y, -x) is a 90-degree counterclockwise rotation. Let's confirm:

For a point (a,b), rotating 90 degrees counterclockwise around the origin gives (-b, a). Wait, no, I think I mixed up. The correct rotation 90 degrees counterclockwise is (x,y)→(-y, x). Wait, no, let's recall:

Rotation 90 degrees counterclockwise: (x, y) → (-y, x)

Rotation 90 degrees clockwise: (x, y) → (y, -x)

Ah! Yes, 90 degrees clockwise is (x,y)→(y, -x). So G(1,5) rotated 90 degrees clockwise: (5, -1), which matches G'. J(2,0) rotated 90 degrees clockwise: (0, -2), which matches J'. So that's correct. So the transformation is \((x,y)\to(y, -x)\), which is the fourth option. Wait, but let's check H. If H is (3,3), rotated 90 degrees clockwise: (3, -3). But the table says H' is (5, -1). Maybe H is (3,5), then (5, -3), no. Alternatively, maybe the table has a typo, and H' is (3, -3), I' is (4, -5). But based on G and J, the fourth option works. So the correct answer is \((x,y)\to(y, -x)\), which is the last option. Wait, but let's recheck the options:

The options are:

- \((x,y)\to(x,-y)\)

- \((x,y)\to(x, y - 6)\)

- \((x,y)\to(x - 2, y - 2)\)

- \((x,y)\to(y, -x)\)

Yes, the fourth option is \((x,y)\to(y, -x)\), which works for G and J. So that's the correct transformation.

Answer:

\(\boldsymbol{(x,y)\to(y,-x)}\) (the fourth option)