Question

square a was transformed to create square b. which of the following rules could describe this transformation? move the checkmarks to select three correct answers.
$(x, y) \
ightarrow (x - 7, y)$
$(x, y) \
ightarrow (x + 7, y)$
$(x, y) \
ightarrow (-x, y)$
$(x, y) \
ightarrow (-y, x)$
$(x, y) \
ightarrow (y, -x)$
image: coordinate grid with yellow square a (left) and blue square b (right), x-axis from -5 to 5, y-axis from -5 to 5

Explanation:

Step1: Analyze horizontal translation

Square A is on the left side of the y - axis and Square B is on the right side. For a horizontal translation, if we consider a point \((x,y)\) in Square A, to get to Square B, we can either translate it 7 units to the right (using \((x,y)\to(x + 7,y)\)) or reflect it over the y - axis (using \((x,y)\to(-x,y)\)) or rotate it. Let's check the rotation options. A 90 - degree rotation or 270 - degree rotation? Let's take a vertex of Square A, say \((-4,3)\). If we apply \((x,y)\to(y,-x)\), we get \((3,4)\), which is not in Square B. If we apply \((x,y)\to(-y,x)\), we get \((-3,-4)\), not in Square B. Wait, maybe I made a mistake. Wait, let's take a vertex of Square A: let's say the top - left vertex of A is \((-5,5)\), top - right is \((-2,5)\), bottom - left is \((-5,2)\), bottom - right is \((-2,2)\). For Square B, top - left is \((2,5)\), top - right is \((5,5)\), bottom - left is \((2,2)\), bottom - right is \((5,2)\).

- For \((x,y)\to(x + 7,y)\): Take \((-5,5)\), \(x+7=-5 + 7 = 2\), \(y = 5\), which is the top - left of B. \((-2,5)\to(-2 + 7,5)=(5,5)\), top - right of B. So this works.
- For \((x,y)\to(-x,y)\): Take \((-5,5)\to(5,5)\)? No, wait \((-5,5)\to(5,5)\) is not correct. Wait, \((-5,5)\to(5,5)\) is not a vertex of B. Wait, maybe I took the wrong vertex. Wait, the center of Square A: let's find the center of A. The x - coordinates of A range from - 5 to - 2, so the mid - x is \(\frac{-5-2}{2}=\frac{-7}{2}=-3.5\). The y - coordinates range from 2 to 5, mid - y is \(\frac{2 + 5}{2}=3.5\). Center of B: x - coordinates from 2 to 5, mid - x is \(\frac{2 + 5}{2}=3.5\), y - coordinates from 2 to 5, mid - y is 3.5. So the center of A is \((-3.5,3.5)\), center of B is \((3.5,3.5)\). So the horizontal distance between centers is \(3.5-(-3.5)=7\), so translation \((x,y)\to(x + 7,y)\) works. Also, reflection over the y - axis: the rule \((x,y)\to(-x,y)\). Let's take a point in A: \((-4,3)\), \(-x = 4\), \(y = 3\), which is in B? Wait, Square B's x - coordinates are from 2 to 5, y from 2 to 5. \((4,3)\) is in B. Another point: \((-3,4)\to(3,4)\), which is in B. So reflection over y - axis \((x,y)\to(-x,y)\) works. Now, what about rotation? Wait, maybe a 90 - degree rotation? Wait, no, let's check the last option \((x,y)\to(y,-x)\). Take \((-5,5)\to(5,5)\), which is not in B. Wait, \((-5,2)\to(2,5)\), which is a vertex of B! Wait, \((-5,2)\) is a bottom - left vertex of A. \((y,-x)=(2,5)\), which is top - left vertex of B? No, \((2,5)\) is top - left of B? Wait, Square B: top - left is (2,5), top - right (5,5), bottom - left (2,2), bottom - right (5,2). So \((-5,2)\to(2,5)\) (applying \((x,y)\to(y,-x)\)): \(y = 2\), \(-x=5\) (since \(x=-5\)), so \((2,5)\), which is top - left of B. \((-2,5)\to(5,2)\) (bottom - right of B). \((-2,2)\to(2,2)\) (bottom - left of B). \((-5,5)\to(5,5)\) (top - right of B). Wait, that works! Wait, I made a mistake earlier. Let's re - check:

- For \((x,y)\to(x + 7,y)\): Translating 7 units right. A point \((x,y)\) in A, \(x\) ranges from - 5 to - 2, \(x + 7\) ranges from 2 to 5, which is the x - range of B, y remains same (2 - 5), so this works.
- For \((x,y)\to(-x,y)\): Reflecting over y - axis. \(x\) in A is negative, \(-x\) is positive, y same. So \(-x\) ranges from 2 to 5 (since \(x\) is - 5 to - 2), y 2 - 5, which is B's range, so this works.
- For \((x,y)\to(y,-x)\): Let's take all four vertices of A:
- Vertex 1: \((-5,5)\): \(y = 5\), \(-x = 5\), so \((5,5)\) (top - right of B)
- Vertex 2: \((-2,5)\): \(y = 5\), \(-x = 2\), so \((5,2)\) (bottom - right of B)
- Vertex 3: \((-2,2)\): \(y = 2\), \(-x = 2\), so \((2,2)\) (bottom - left of B)
- Vertex 4: \((-5,2)\): \(y = 2\), \(-x = 5\), so \((2,5)\) (top - left of B)
So this rotation (90 - degree clockwise? Wait, the rule \((x,y)\to(y,-x)\) is a 90 - degree clockwise rotation) works.

Wait, but the first option \((x,y)\to(x - 7,y)\) would translate left, which is opposite, so it's wrong. The second option \((x,y)\to(x + 7,y)\) (translation right) works. The third option \((x,y)\to(-x,y)\) (reflection over y - axis) works. The fifth option \((x,y)\to(y,-x)\) (90 - degree clockwise rotation) works. Wait, but the problem says to select three correct answers. Wait, maybe I misread the graph. Let me check again.

Wait, Square A is at x from - 5 to - 2, y from 2 to 5. Square B is at x from 2 to 5, y from 2 to 5. So the horizontal distance between corresponding points: for example, the right - most point of A is \((-2,y)\), the left - most point of B is \((2,y)\), so the distance is \(2-(-2)=4\)? Wait, no, \(-2\) to 2 is 4 units? Wait, maybe my initial vertex selection was wrong. Let's count the grid squares. From x=-4 to x = 3: wait, the graph has x - axis from - 5 to 5, y from - 5 to 5. Let's take a point in A: let's say the top - left corner of A is at (-4,5), top - right at (-1,5), bottom - left at (-4,2), bottom - right at (-1,2). Then Square B: top - left at (3,5), top - right at (6,5)? No, the x - axis only goes to 5. Wait, the graph shows x from - 5 to 5, so Square B's x should be from 2 to 5 (since 5 is the maximum x). So maybe the left - most point of A is at x=-5, right - most at x=-2, so the distance between x=-2 and x = 2 is 4? No, I think I made a mistake in the horizontal translation. Wait, the correct horizontal translation: if we take a point \((x,y)\) in A, to get to B, the x - coordinate changes by \(x_{B}-x_{A}\). Let's take the center of A: mid - x of A is \(\frac{-5-2}{2}=-3.5\), mid - x of B is \(\frac{2 + 5}{2}=3.5\), so the change is \(3.5-(-3.5)=7\), so \(x+7\) is correct.

Now, reflection over y - axis: \((x,y)\to(-x,y)\). For a point \((-3.5,3.5)\) (center of A), \(-x = 3.5\), \(y = 3.5\), which is the center of B, so that works.

Rotation: Let's take a point \((-4,3)\) in A. Applying \((x,y)\to(y,-x)\): \(y = 3\), \(-x = 4\), so \((3,4)\), which is in B (since B is from x=2 - 5, y=2 - 5). Another point \((-2,4)\to(4,2)\), which is in B.

Wait, but the options are:

1. \((x,y)\to(x - 7,y)\): Translates left, wrong.

2. \((x,y)\to(x + 7,y)\): Translates right, correct.

3. \((x,y)\to(-x,y)\): Reflects over y - axis, correct.

4. \((x,y)\to(-y,x)\): Let's test \((-4,3)\to(-3,-4)\), which is below the x - axis, not in B, wrong.

5. \((x,y)\to(y,-x)\): Test \((-4,3)\to(3,4)\), which is in B, correct.

So the three correct answers are \((x,y)\to(x + 7,y)\), \((x,y)\to(-x,y)\), and \((x,y)\to(y,-x)\).

Answer:

B. \((x,y)\to(x + 7,y)\)
C. \((x,y)\to(-x,y)\)
E. \((x,y)\to(y,-x)\)