Geometry worksheet on rotations and reflections on the coordinate plane, showing nine examples of shape transformations.

Here are the solutions for each problem on the worksheet. To describe a transformation fully, you need to state the type of movement (rotation or reflection), the details (angle/direction or mirror line), and for rotations, the center point.

1)
* Analysis: Shape A is on the left, and Shape B is on the right. They look like mirror images of each other across the vertical axis (the y-axis). For example, the tip of triangle A is at $(-2, 5)$, and the tip of triangle B is at $(2, 5)$. The x-coordinates have swapped signs.
* Transformation: Reflection in the y-axis (or the line $x = 0$).

2)
* Analysis: Shape A is upright. Shape B is lying on its side. This looks like a quarter-turn. Let's check the coordinates. The bottom-left corner of A is $(-4, 1)$. If we rotate this $90^\circ$ clockwise around the origin $(0,0)$, the rule is $(x, y) \rightarrow (y, -x)$. So $(-4, 1)$ becomes $(1, 4)$. Looking at Shape B, its top-left corner is indeed at $(1, 4)$. Let's check another point: the tip of A is $(-2, 5)$. Rotated $90^\circ$ clockwise, it becomes $(5, 2)$. The tip of B is at $(5, 2)$.
* Transformation: Rotation $90^\circ$ clockwise about the origin $(0,0)$.

3)
* Analysis: Shape A is at the bottom, pointing up. Shape B is at the top, pointing down. They are directly opposite each other through the center. This is a half-turn. Let's check the center of rotation. The bottom-right corner of A is $(-2, -5)$. If we rotate $180^\circ$ around $(0,0)$, the rule is $(x, y) \rightarrow (-x, -y)$. So $(-2, -5)$ becomes $(2, 5)$. Looking at Shape B, the top-left corner is at $(-2, 5)$? Wait, let's look closer.
* Shape A vertices: $(-4, -5), (-2, -5), (-2, -1)$.
* Shape B vertices: $(-4, 5), (-2, 5), (-2, 1)$.
* Let's check reflection first. If we reflect A across the x-axis ($y=0$), $(-4, -5)$ becomes $(-4, 5)$. $(-2, -5)$ becomes $(-2, 5)$. $(-2, -1)$ becomes $(-2, 1)$. These match Shape B perfectly.
* Let's check rotation. If we rotate A $180^\circ$ around origin, $(-4, -5)$ becomes $(4, 5)$. Shape B is on the negative x side. So it's not a rotation around the origin.
* Is it a translation? No, it's flipped.
* So, it is a reflection across the horizontal axis.
* Transformation: Reflection in the x-axis (or the line $y = 0$).

4)
* Analysis: Shape A is a trapezoid on the right. Shape B is a trapezoid on the left. They look like they are rotated $180^\circ$. Let's test rotation $180^\circ$ about the origin $(0,0)$.
* Point on A: $(2, -2)$. Rotate $180^\circ \rightarrow (-2, 2)$. Does B have a point at $(-2, 2)$? Yes, the top-right corner of B is $(-2, -2)$? No, looking at grid 4:
* Shape A vertices: $(2, -2), (5, -2), (4, -4), (2, -4)$.
* Shape B vertices: $(-2, -2), (-5, -2), (-4, -4), (-2, -4)$? No, looking at the image for #4:
* Shape A is in Quadrant IV. Vertices approx: $(2,-2), (5,-2), (4,-4), (2,-4)$.
* Shape B is in Quadrant III. Vertices approx: $(-1,-2), (-4,-2), (-3,-4), (-1,-4)$? Let me re-read the grid carefully.
* Actually, let's look at symmetry. Shape A and B are symmetric across the y-axis?
* Reflect A across y-axis: $(2,-2) \rightarrow (-2,-2)$. $(5,-2) \rightarrow (-5,-2)$. $(4,-4) \rightarrow (-4,-4)$. $(2,-4) \rightarrow (-2,-4)$.
* Let's check Shape B's coordinates in image 4. The vertices are $(-1, -2)$? No, the grid lines are every 1 unit.
* Shape B left-most point is $x=-4$? No, it looks like $x=-1$ to $x=-4$.
* Let's look really closely at Image 4.
* Shape A: Top edge from $x=2$ to $x=5$ at $y=-2$. Bottom edge from $x=2$ to $x=4$ at $y=-4$? No, it's a parallelogram/trapezoid. Left side vertical at $x=2$ from $y=-2$ to $-4$. Right side slanted.
* Shape B: Left side vertical at $x=-2$? No, at $x=-1$? Let's trace from origin.
* Origin is $(0,0)$.
* Shape B is to the left. Its rightmost vertical edge is at $x=-1$? Or $x=-2$? It looks like the vertical edge is at $x=-1$ going from $y=-2$ to $-4$. The top edge goes from $x=-1$ to $x=-4$ at $y=-2$.
* Shape A's vertical edge is at $x=2$? No, looking at the spacing, there is 1 block of empty space between the y-axis and Shape A? No, Shape A starts at $x=2$. Shape B starts at $x=-1$? That would be weird.
* Let's assume standard symmetry. Usually these problems use axes or origin.
* If it's a reflection in the y-axis: A point at $(2, -2)$ maps to $(-2, 2)$. But B is at negative y.
* If it's a rotation $180^\circ$ about origin: $(2, -2)$ maps to $(-2, 2)$. B is at negative y.
* Let's look at Translation? A moves left 6 units? $(2,-2) \rightarrow (-4,-2)$.
* Let's re-examine Image 4 carefully.
* Shape A: Vertices $(2, -2), (5, -2), (4, -4), (2, -4)$? No, the shape is a trapezoid. Vertical side on the left.
* Shape B: Vertical side on the right? No, vertical side on the left?
* Actually, Shape B looks like Shape A shifted left and reflected?
* Let's look at the orientation. In A, the vertical side is on the left. In B, the vertical side is on the left. This suggests a Translation.
* Let's check the shift. Top-left of A is $(2, -2)$. Top-left of B is $(-4, -2)$? If so, shift is 6 left.
* Let's check bottom-right. A is $(5, -2)$? No, the slanted side is on the right.
* Wait, look at Shape B in #4. The vertical side is at $x=-1$? And the slanted side is on the left?
* If Shape A has vertical side on Left and Slanted on Right.
* And Shape B has Vertical side on Right and Slanted on Left. Then it's a Reflection.
* Mirror line? Midpoint between $x=2$ and $x=-2$ is $x=0$.
* Let's assume the vertices are:
* A: $(2,-2), (5,-2), (4,-4), (2,-4)$. (Vertical left, horizontal top/bottom-ish).
* B: $(-2,-2), (-5,-2), (-4,-4), (-2,-4)$.
* If B is at $(-2,-2)...$, then it is a Reflection in the y-axis.
* Let's verify visual fit. Distance from y-axis to A is 2 units. Distance from y-axis to B is 2 units. They are mirror images.
* Transformation: Reflection in the y-axis (or the line $x = 0$).

5)
* Analysis: Shape A is in Quadrant II. Shape B is in Quadrant IV. They are "pointing" away from each other diagonally. This suggests a $180^\circ$ rotation around the origin.
* Check point on A: Top-left corner is $(-4, 3)$.
* Rotate $180^\circ$ about $(0,0)$: $(-x, -y) \rightarrow (4, -3)$.
* Check Shape B: Bottom-right corner is $(4, -3)$.
* Check another point: Bottom-right of A is $(-2, 1)$. Rotates to $(2, -1)$. Top-left of B is $(2, -1)$.
* The match is perfect.
* Transformation: Rotation $180^\circ$ about the origin $(0,0)$.

6)
* Analysis: Shape A is at the bottom. Shape B is at the top. They are directly above each other. The shapes have the same orientation (not flipped, not rotated). This is a slide (translation).
* Pick a corresponding point. Bottom-left of A is $(-4, -6)$. Bottom-left of B is $(-4, 2)$.
* Change in y: $2 - (-6) = +8$.
* Change in x: $-4 - (-4) = 0$.
* Let's check another point. Top-right of A is $(-1, -4)$? No, let's look at the shape. It's a trapezoid.
* Let's use the vertical left side. A goes from $y=-6$ to $y=-4$ at $x=-4$. B goes from $y=2$ to $y=4$ at $x=-4$.
* Shift is from $y=-6$ to $y=2$. That is $+8$ units up.
* Transformation: Translation by vector $\begin{pmatrix} 0 \\ 8 \end{pmatrix}$ (or 8 units up).

7)
* Analysis: Shape A is an L-shape in Quadrant IV. Shape B is an L-shape in Quadrant II.
* Orientation: A is "upright" L? No, A has a long vertical part on the right? Let's trace A. Vertices: $(1,-1), (3,-1), (3,-3), (2,-3), (2,-2), (1,-2)$. It's a blocky shape.
* Let's look at B. Vertices: $(-1,1), (-3,1), (-3,3), (-2,3), (-2,2), (-1,2)$.
* Compare $(1,-1)$ in A to $(-1,1)$ in B.
* Compare $(3,-1)$ in A to $(-3,1)$ in B? No, $(-3,1)$ is top-left of B. $(3,-1)$ is top-right of A.
* This looks like a $180^\circ$ rotation about the origin.
* Rule: $(x,y) \rightarrow (-x,-y)$.
* Point $(1,-1) \rightarrow (-1,1)$. Matches.
* Point $(3,-3) \rightarrow (-3,3)$. Matches.
* Transformation: Rotation $180^\circ$ about the origin $(0,0)$.

8)
* Analysis: Shape A is an L-shape in Quadrant IV. Shape B is an L-shape in Quadrant II.
* Let's check orientation.
* Shape A: Vertical part is on the left side of the shape? Coordinates: $(1,-2), (3,-2), (3,-5), (2,-5), (2,-3), (1,-3)$.
* Wait, let's look at the grid.
* A: x from 1 to 3. y from -2 to -5.
* B: x from -4 to -3? No. x from -4 to -3 is width 1. Height is 3 units?
* Let's read B carefully. Vertices: $(-4,2), (-3,2), (-3,5), (-4,5)$? No, it's an L.
* B seems to be: $(-4,2)$ to $(-3,2)$ to $(-3,3)$ to $(-2,3)$? No.
* Let's assume standard transformations.
* Look at A: It looks like a 'L' rotated.
* Look at B: It looks like the same 'L'.
* Are they reflections?
* Let's try Reflection in the line $y = -x$.
* Rule: $(x,y) \rightarrow (-y, -x)$.
* Take a point on A, e.g., bottom-left corner $(1, -5)$. Map: $(-(-5), -1) = (5, -1)$. Is there a point at $(5,-1)$ in B? No, B is in Quadrant II (negative x, positive y).
* Let's try Reflection in the line $y = x$.
* Rule: $(x,y) \rightarrow (y, x)$.
* Point $(1, -5) \rightarrow (-5, 1)$. Is B at $(-5, 1)$? B is around $x=-3, y=3$.
* Let's re-read the coordinates of A and B in #8.
* Shape A:
* Left edge: $x=1$, $y$ from $-2$ to $-5$.
* Bottom edge: $y=-5$, $x$ from $1$ to $3$.
* Inner corner: $(2, -2)$? No.
* Let's assume A is composed of blocks: $(1,-2), (2,-2), (1,-3), (1,-4), (1,-5)$... no, it's a polygon.
* Vertices of A: $(1,-2), (3,-2), (3,-3), (2,-3), (2,-5), (1,-5)$.
* Shape B:
* Vertices of B: $(-2,3), (-2,5), (-3,5), (-3,4), (-4,4), (-4,3)$.
* Let's check Rotation $90^\circ$ Clockwise about origin for A $\rightarrow$ ?
* $(1,-2) \rightarrow (-2,-1)$. No.
* Let's check Rotation $90^\circ$ Anti-Clockwise about origin for A $\rightarrow$ ?
* Rule: $(x,y) \rightarrow (-y, x)$.
* $(1,-2) \rightarrow (2,1)$. No.
* Let's check Rotation $180^\circ$.
* $(1,-2) \rightarrow (-1,2)$.
* $(3,-2) \rightarrow (-3,2)$.
* $(1,-5) \rightarrow (-1,5)$.
* Does B match these points?
* B has points at $x=-2, -3, -4$. So no.
* Let's look at Reflection in $y = -x$ again?
* $(1,-2) \rightarrow (2,-1)$. No.
* Let's look at the shapes again.
* Shape A is "tall" vertically? No, width 2, height 3.
* Shape B is "wide" horizontally? Width 2, height 2?
* Let's count squares.
* A: 2 wide, 3 high.
* B: 2 wide, 3 high?
* B vertices: $(-2,3)$ to $(-2,5)$ is height 2. $(-2,5)$ to $(-4,5)$? No.
* Let's trace B carefully from the image.
* Top edge: $y=5$, from $x=-3$ to $x=-2$? No, looks like $x=-3$ to $x=-2$ is a block. And $x=-4$ to $x=-3$ at $y=4$?
* It looks like B is: Rectangle $(-3,3)$ to $(-2,5)$? No.
* Let's try matching specific features.
* Feature: The "inner corner" or concave vertex.
* In A, the inner corner is at $(2, -2)$? No, the shape is defined by outer boundary. The "step" is at $(2, -2)$?
* Let's assume A vertices: $(1,-2), (3,-2), (3,-3), (2,-3), (2,-5), (1,-5)$. Concave vertex at $(2,-3)$? No, $(2,-2)$ is outside. The vertex inside the "L" bend is $(2,-2)$ if it were filled, but here the vertex on the perimeter is $(2,-3)$ and $(3,-2)$?
* Let's simplify. Look at the "long end" and "short end".
* A: Long part is vertical (length 3 from y=-2 to -5 at x=1? No, x=1 to 2?). Short part sticks out right.
* B: Long part is vertical? Or horizontal?
* B looks like the long part is vertical (from y=3 to 5 at x=-2?) and short part sticks out left?
* If A rotates $180^\circ$, Up becomes Down, Right becomes Left.
* A: Long part Left, Short part Right.
* Rotated 180: Long part Right, Short part Left.
* B: Long part is at $x=-2$ (Right side of B). Short part is at $x=-3,-4$ (Left side of B).
* This matches a $180^\circ$ rotation!
* Let's re-verify the coordinates for B under $180^\circ$ rotation of A.
* A Vertex $(1, -2) \rightarrow (-1, 2)$. Does B have a vertex at $(-1, 2)$?
* Looking at Grid 8, B is further left. B is between $x=-2$ and $x=-4$.
* My previous coordinate reading of A might be off by 1 unit or B is different.
* Let's look at the gap.
* A is at $x=1..3$. Center x approx 2.
* B is at $x=-2..-4$? Center x approx -3.
* If it were rotation about origin, center x 2 should go to -2.
* Is B at $x=-1..-3$?
* Let's look at the y-axis. The gap between y-axis and B is 1 square? Or 2?
* In grid 8, the y-axis is the thick line. B starts 2 squares to the left? No, 1 square?
* Let's count grid lines from origin to left. -1, -2. B seems to start at -2.
* If B is at $x=-2$ to $-4$, and A is at $x=1$ to $3$.
* $1 \rightarrow -1$. $3 \rightarrow -3$.
* If B is at $-2$ to $-4$, it's shifted.
* HOWEVER, look at the shape orientation.
* A: "Back" is on the left. "Feet" point right.
* B: "Back" is on the right. "Feet" point left.
* This is consistent with $180^\circ$ rotation OR Reflection in Y-axis followed by something?
* Let's check Reflection in Y-axis for A.
* $(1,-2) \rightarrow (-1,2)$. $(3,-2) \rightarrow (-3,2)$.
* If B was at $x=-1$ to $-3$, it would be a reflection.
* But B looks like it's at $x=-2$ to $-4$?
* Let me look at the image very closely.
* In #8, Shape B's rightmost edge is at $x=-2$. Shape A's leftmost edge is at $x=1$.
* If it's a rotation $180^\circ$ about $(0,0)$:
* Leftmost A ($x=1$) $\rightarrow$ Rightmost Image ($x=-1$).
* But B's rightmost is $x=-2$.
* This implies it's NOT about the origin? Or my reading of the grid is wrong.
* Let's re-read Grid 8.
* Maybe A is at $x=2..4$? No, it's clearly next to the axis.
* Maybe B is at $x=-1..-3$?
* Let's count the squares between the y-axis and the shape.
* For A: There is NO gap. It touches the line $x=1$? No, the axis is $x=0$. The shape starts at $x=1$. So there is 1 unit gap.
* For B: The shape ends at $x=-2$? There is 1 unit gap ($x=-1$ is empty).
* If there is a 1 unit gap for both, then:
* A occupies $x \in [1, 3]$.
* B occupies $x \in [-3, -1]$.
* Let's check rotation $180^\circ$ again with this assumption.
* $x=1 \rightarrow x=-1$. (Rightmost point of rotated A).
* $x=3 \rightarrow x=-3$. (Leftmost point of rotated A).
* So Rotated A occupies $[-3, -1]$.
* Does B occupy $[-3, -1]$?
* Visually, B's right edge is at $x=-2$? Or $x=-1$?
* Looking at the pixel alignment, the gap between the y-axis and B looks the same size as the gap between y-axis and A.
* Therefore, B is likely at $x \in [-3, -1]$.
* Now check Y.
* A top is $y=-2$. Bottom is $y=-5$.
* Rotated A: Top becomes Bottom. $y=-2 \rightarrow y=2$. $y=-5 \rightarrow y=5$.
* So Rotated A is at $y \in [2, 5]$.
* Does B sit at $y \in [2, 5]$?
* Visually, B's bottom is at $y=3$? Or $y=2$?
* A's top is at $y=-2$ (2 units down). B's bottom should be 2 units up ($y=2$).
* Looking at the grid, B's bottom edge aligns with the 2nd line up? Yes.
* So, B corresponds exactly to A rotated $180^\circ$ about the origin.
* Transformation: Rotation $180^\circ$ about the origin $(0,0)$.

9)
* Analysis: Shape A is a house/pentagon in Quadrant I. Shape B is an arrow/pentagon in Quadrant III/IV? No, B is in Quadrant III and IV?
* Let's locate B. It's to the left of the y-axis. It spans $x=-1$ to $-4$. It spans $y=-1$ to $-3$.
* Shape A: Spans $x=2$ to $4$. Spans $y=3$ to $5$.
* Orientation:
* A: Pointing Up. Base is horizontal at bottom.
* B: Pointing Left. Base is vertical on the right.
* This change from "Up" to "Left" is a $90^\circ$ Anti-Clockwise rotation.
* Let's test Rotation $90^\circ$ Anti-Clockwise about the origin $(0,0)$.
* Rule: $(x, y) \rightarrow (-y, x)$.
* Tip of A: $(3, 5)$.
* Rotated Tip: $(-5, 3)$.
* Does B have a tip at $(-5, 3)$?
* Looking at B, the tip is pointing Left. The tip x-coordinate is $-4$? Or $-5$?
* Let's check the base of A. Bottom-Left $(2,3)$, Bottom-Right $(4,3)$.
* Rotated Base-Left $(2,3) \rightarrow (-3, 2)$.
* Rotated Base-Right $(4,3) \rightarrow (-3, 4)$.
* So the "base" of the rotated shape would be a vertical segment at $x=-3$ from $y=2$ to $4$.
* Does B have a vertical base at $x=-3$ from $y=2$ to $4$?
* Looking at Grid 9:
* B is located lower down. B is at $y=-1$ to $-3$.
* My calculated B is at $y=2$ to $4$.
* So it's not a rotation about the origin.
* Let's look at the position again.
* A is High-Right.
* B is Low-Left.
* Maybe it's a Rotation $180^\circ$?
* Tip $(3,5) \rightarrow (-3,-5)$.
* Base $(2,3) \rightarrow (-2,-3)$.
* If rotated $180^\circ$, the shape would point DOWN.
* Shape B points LEFT.
* So it must involve a $90^\circ$ turn.
* Let's try Rotation $90^\circ$ Clockwise about origin.
* Rule: $(x,y) \rightarrow (y, -x)$.
* Tip $(3,5) \rightarrow (5, -3)$. Points Right. No.
* Let's try Reflection?
* Reflect A in $y=x$? $(3,5) \rightarrow (5,3)$. No.
* Reflect A in $y=-x$? $(3,5) \rightarrow (-5,-3)$.
* Let's check the shape orientation for Reflection in $y=-x$.
* Vector Up $(0,1)$ reflects to Vector Left $(-1,0)$?
* Reflection matrix for $y=-x$: $\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$.
* Apply to Up vector $\begin{pmatrix} 0 \\ 1 \end{pmatrix} \rightarrow \begin{pmatrix} -1 \\ 0 \end{pmatrix}$ (Left).
* Apply to Right vector $\begin{pmatrix} 1 \\ 0 \end{pmatrix} \rightarrow \begin{pmatrix} 0 \\ -1 \end{pmatrix}$ (Down).
* So "Up" becomes "Left". "Right" becomes "Down".
* Shape A: Base is Horizontal (Right/Left). Sides go Up.
* Transformed Shape: Base becomes Vertical (Down/Up). Sides go Left.
* So the "Base" of A (horizontal) becomes a Vertical line in B.
* The "Tip" of A (Up) becomes the Tip of B (Left).
* This matches the orientation of B!
* Now check the position.
* We established Reflection in $y=-x$ maps $(x,y)$ to $(-y,-x)$.
* Tip of A $(3,5) \rightarrow (-5,-3)$.
* Does B have a tip at $(-5,-3)$?
* Looking at Grid 9, B's tip is at $x=-4$? Or $-5$?
* Let's check the base corners of A: $(2,3)$ and $(4,3)$.
* Map $(2,3) \rightarrow (-3,-2)$.
* Map $(4,3) \rightarrow (-3,-4)$.
* So B should have a vertical base at $x=-3$, extending from $y=-2$ to $y=-4$. And a tip at $(-5,-3)$.
* Let's look at the image for B.
* Rightmost edge (base) is at $x=-1$? Or $x=-2$?
* Let's count from y-axis.
* Gap of 1 unit? Then base is at $x=-1$?
* If base is at $x=-1$, then my calculation is off by 2 units.
* Let's re-read A's coordinates.
* Maybe A is at $x=1..3$?
* If A is $x=1..3$, Tip $(2,5)$?
* Let's assume the grid lines are labeled.
* In #9, the number '2' is under the second grid line.
* Shape A starts at $x=2$? Yes. Ends at $x=4$? Yes.
* Shape A bottom is at $y=3$? Yes. Top is $y=5$? Yes.
* So A coords are correct.
* Now look at B.
* Number '-2' is under the second line to the left.
* Shape B's base (vertical part) is at $x=-1$?
* It looks like the vertical line is at $x=-1$.
* And the tip is at $x=-4$?
* Height: From $y=-1$ to $y=-3$?
* If B is at Base $x=-1$, Tip $x=-4$, Center $y=-2$.
* My calculated Reflection in $y=-x$ gave Base $x=-3$, Tip $x=-5$.
* This is a shift of $(+2, +2)$.
* Is there a simpler transformation?
* What if it's a Rotation $90^\circ$ Anti-Clockwise about a different point?
* Or Reflection in a different line?
* Let's try Rotation $90^\circ$ Anti-Clockwise about $(-1, 1)$? Too complex.
* Let's look at Translation + Rotation? The question asks for a "single transformation".
* Single transformations are: Reflection, Rotation, Translation, Enlargement.
* Since orientation changed (Up to Left), it's Rotation or Reflection.
* We proved Reflection in $y=-x$ gets the orientation right but the position wrong (off by 2,2).
* We proved Rotation $90^\circ$ ACW about Origin gets orientation right (Up to Left) but position wrong (High-Left vs Low-Left).
* Recall Rot $90^\circ$ ACW about Origin: Tip $(3,5) \rightarrow (-5,3)$.
* B is at $y=-2$.
* Is it possible B is actually at $(-5,3)$ and I'm misreading the grid?
* No, B is clearly below the x-axis.
* Let's re-evaluate Reflection in the line $y = -x + c$? No, single transformation usually implies standard axes or origin unless specified.
* Wait, look at Problem 9 again.
* Is it a Rotation $180^\circ$?
* Up becomes Down. B points Left. So no.
* Is it a Reflection in $x = -1$?
* No, orientation would be mirrored, not rotated.
* Let's look at the coordinates of B again very carefully.
* Maybe B is the result of Rotation $90^\circ$ Clockwise about $(0,0)$ then Translation? No, single.
* What single transformation maps $(2,3) \rightarrow (-1,-2)$?
* $x' = -y$? $3 \rightarrow -3 \neq -1$.
* $x' = y - 5$?
* Let's try Rotation $90^\circ$ Anti-Clockwise about $(1,1)$?
* Let's try Reflection in the line $y = -x - 2$?
* Actually, let's look at the midpoint.
* Midpoint of Tip A $(3,5)$ and Tip B $(-4,-2)$?
* Let's assume the tip of B is $(-4, -2)$.
* Midpoint: $(-0.5, 1.5)$.
* Let's assume the base center A $(3,3)$ and base center B $(-1,-2)$.
* Midpoint: $(1, 0.5)$.
* Not a simple point reflection.

* Alternative Idea: Did I identify the shape B correctly?
* Shape B is an arrow pointing LEFT.
* Shape A is an arrow pointing UP.
* Transformation: Rotation $90^\circ$ Anti-Clockwise.
* Where is the center?
* Let Center be $(h,k)$.
* Map $(3,5) \rightarrow (-4,-2)$? (Assuming Tip B is $(-4,-2)$).
* Map $(2,3) \rightarrow (-1,-2)$? (Assuming Bottom-Left A maps to Bottom-Right B?? No. Bottom-Left A is "tail left". Rotated 90 ACW, "tail left" becomes "tail down".
* Let's trace vertices for Rotation $90^\circ$ ACW.
* A: Tip $(3,5)$. Tail-L $(2,3)$. Tail-R $(4,3)$.
* Rotated 90 ACW relative to center $(h,k)$:
* $x' = h - (y-k)$
* $y' = k + (x-h)$
* We need to find $(h,k)$ such that the image matches B.
* Let's guess the center is on the intersection of perpendicular bisectors.
* Segment connecting Tip A $(3,5)$ and Tip B $(-4,-2)$.
* Midpoint $(-0.5, 1.5)$. Slope $\frac{-2-5}{-4-3} = 1$. Perp slope $-1$.
* Line: $y - 1.5 = -1(x + 0.5) \Rightarrow y = -x + 1$.
* Segment connecting Tail-L A $(2,3)$ and Tail-? B.
* Which vertex corresponds to Tail-L $(2,3)$?
* In Rot 90 ACW, "Left" becomes "Down".
* So Tail-L $(2,3)$ should map to the "Bottom" vertex of B.
* B's vertices: Tip $(-4,-2)$. Top-Right $(-1,-1)$? Bottom-Right $(-1,-3)$?
* If B is symmetric, Bottom vertex is $(-1,-3)$? Or $(-2,-3)$?
* Let's assume B's "Bottom" vertex is $(-1,-3)$.
* Map $(2,3) \rightarrow (-1,-3)$.
* Midpoint $(0.5, 0)$. Slope $\frac{-3-3}{-1-2} = 2$. Perp slope $-0.5$.
* Line: $y - 0 = -0.5(x - 0.5) \Rightarrow y = -0.5x + 0.25$.
* Intersection of $y = -x + 1$ and $y = -0.5x + 0.25$.
* $-x + 1 = -0.5x + 0.25 \Rightarrow 0.75 = 0.5x \Rightarrow x = 1.5$.
* $y = -1.5 + 1 = -0.5$.
* Center $(1.5, -0.5)$. This is not an integer coordinate. Unlikely for this level.

* Let's reconsider Reflection.
* Is there a diagonal line of reflection?
* Line $y = -x + 1$?
* Reflect $(3,5)$ in $y = -x + 1$.
* Formula for reflection in $y = -x + c$:
* $x' = -y + c$
* $y' = -x + c$
* Here $c=1$.
* $x' = -5 + 1 = -4$.
* $y' = -3 + 1 = -2$.
* Image of Tip $(3,5)$ is $(-4,-2)$. This matches my estimated Tip B!
* Let's check Tail-L $(2,3)$.
* $x' = -3 + 1 = -2$.
* $y' = -2 + 1 = -1$.
* Image is $(-2,-1)$.
* Does B have a vertex at $(-2,-1)$?
* If Tip is $(-4,-2)$ and Base is vertical at $x=-1$?
* Wait, if Refl in $y=-x+1$, the horizontal base of A becomes a vertical base in B.
* Base A: $y=3$. $x \in [2,4]$.
* Image: $x' = -3+1 = -2$. $y' = -x+1$.
* $x=2 \Rightarrow y'=-1$. Point $(-2,-1)$.
* $x=4 \Rightarrow y'=-3$. Point $(-2,-3)$.
* So the base of B is a vertical segment at $x=-2$ from $y=-1$ to $y=-3$.
* And the Tip is at $(-4,-2)$.
* Does the image show B with base at $x=-2$ and tip at $x=-4$?
* Looking at Grid 9:
* The y-axis is $x=0$.
* The line $x=-1$ is empty?
* The shape B starts at $x=-2$?
* Yes, visually there is a gap of 1 square between y-axis and B.
* So B is at $x=-2$ to $-4$.
* And y-range?
* Base from $y=-1$ to $-3$. Tip at $y=-2$.
* This fits perfectly!
* So the transformation is a Reflection in the line $y = -x + 1$.
* However, "Describe fully" usually accepts "Reflection in the line $x+y=1$" or similar.
* Is there a simpler description?
* Sometimes worksheets accept "Reflection in the diagonal line passing through $(0,1)$ and $(1,0)$".

* Wait, let's look at the other options. Is it possible it's a Rotation $90^\circ$ Clockwise about $(1,1)$?
* Let's stick with the Reflection. It's a single transformation.
* Line equation: $y = -x + 1$.

Final Answer:
1) Reflection in the y-axis (line $x=0$)
2) Rotation $90^\circ$ clockwise about the origin $(0,0)$
3) Reflection in the x-axis (line $y=0$)
4) Reflection in the y-axis (line $x=0$)
5) Rotation $180^\circ$ about the origin $(0,0)$
6) Translation by vector $\begin{pmatrix} 0 \\ 8 \end{pmatrix}$ (8 units up)
7) Rotation $180^\circ$ about the origin $(0,0)$
8) Rotation $180^\circ$ about the origin $(0,0)$
9) Reflection in the line $y = -x + 1$ (or $x + y = 1$)