Practice worksheet for identifying and describing geometric transformations on coordinate grids.

Let’s go step by step to describe each transformation.

We are looking at how shape A moves or changes to become shape B in each grid. The possible transformations are:
- Translation (slide)
- Reflection (flip)
- Rotation (turn)
- Dilation (resize — but none of these look like resizing, so we’ll ignore that for now)

We’ll check each one carefully.

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i)
Shape A is a small rectangle with a dot on top-left corner. Shape B is the same size and orientation, just moved right and down.
Count squares: From A’s bottom-right corner to B’s bottom-right corner → 4 units right, 2 units down.
So this is a translation: 4 units right, 2 units down.

But let’s be precise — better to pick a point. Let’s take the bottom-left corner of A: it’s at (-3, -1). Bottom-left of B is at (1, -3).
Change in x: 1 - (-3) = +4
Change in y: -3 - (-1) = -2
→ So translation: (x + 4, y - 2)

✔ Correct.

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ii)
Shape A is an L-shape pointing up-right. Shape B is the same L-shape but flipped horizontally and shifted. Wait — actually, if you flip A over the vertical line x=0 (y-axis), does it match B? Let’s see.

Original A: has points like (1,1), (1,2), (2,1) — wait, better to visualize.

Actually, looking closely: A is in quadrant I, B is in quadrant IV. But they’re not rotated — they’re mirrored.

Try reflecting A over the x-axis? No, because then it would be upside-down.

Wait — try reflecting over the line y = x? Not quite.

Actually, notice: if you reflect A over the vertical line x = 3? Too messy.

Better idea: Compare coordinates.

Assume key point of A: say the “corner” of the L — maybe (1,1). In B, the corresponding corner seems to be at (5,-1)? That doesn’t help.

Wait — perhaps it’s a reflection over the horizontal line y = 0.5? Not clean.

Alternative approach: Look at orientation.

In A, the long part goes up from base. In B, the long part goes down from base — so it’s flipped vertically.

Also, it’s shifted right.

Actually, let’s do this: Take the entire shape A and flip it over the x-axis (reflect over y=0). Then slide it right.

After reflection over x-axis: point (1,1) becomes (1,-1); (1,2)→(1,-2); (2,1)→(2,-1). Now compare to B: B appears to have points around (4,-1), (4,-2), (5,-1). So after reflection, we need to shift right by 3 units.

Check: (1,-1) + (3,0) = (4,-1) — matches B’s position.

So transformation: Reflection over x-axis, then translation 3 units right.

But can we write as single rule? Usually we combine.

Rule: First reflect: (x,y) → (x, -y), then translate: (x+3, -y)

So overall: (x + 3, -y)

Let me verify with another point. Say A has point (2,2) — assuming the top of the L. After transform: (2+3, -2) = (5,-2). Does B have a point there? Looking at diagram — yes, likely.

✔ So ii) is: (x + 3, -y) — which is reflection over x-axis followed by translation right 3.

But sometimes teachers accept describing both steps. However, since the question says “write a rule”, we should give coordinate rule.

Alternatively, is it a glide reflection? Yes, but rule is still (x+3, -y)

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iii)
Shape A is a rectangle in left side. Shape B is same rectangle, same orientation, just moved right.

From A’s bottom-left: say (-4, -2) to B’s bottom-left: (2, -2) → change in x: +6, y: 0.

So translation: 6 units right → rule: (x + 6, y)

✔ Simple.

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iv)
Triangle A and triangle B. They look congruent, but oriented differently.

A is pointing up, B is pointing right? Let’s see.

Actually, A has vertices roughly at (0,0), (1,2), (2,0) — wait, better to assume.

Notice: If you rotate A 90 degrees clockwise around origin, where does it go?

Take point (1,2) in A — rotate 90° CW: (x,y) → (y, -x) → (2, -1)

Does B have a point at (2,-1)? Looking at diagram — B is in fourth quadrant, and its “top” vertex might be at (2,-1)? Possibly.

Another point: A’s (0,0) → (0,0) under rotation? But B doesn’t include origin.

Wait — perhaps center of rotation is not origin.

Look: A is centered near (1,1), B is centered near (3,-1). Maybe rotation about some point.

Alternative: Try reflection.

If you reflect A over the line y = x? (x,y)→(y,x). Point (1,2)→(2,1). But B is below x-axis.

Not matching.

Try rotation 90° clockwise about point (2,0)? Complicated.

Wait — here’s a better way: Notice that triangle A and B are congruent and appear to be rotated 90 degrees relative to each other.

Specifically, if you take A and rotate it 90 degrees clockwise around the point (2,0), does it map to B?

Test point: Suppose A has vertex at (1,1). Rotate 90° CW around (2,0):

Vector from center: (1-2, 1-0) = (-1,1)

Rotate 90° CW: (x,y) → (y, -x) → (1, 1) [wait no: vector (-1,1) rotated 90° CW is (1,1)? Let's recall:

Rotation 90° CW: (a,b) → (b, -a)

So vector (-1,1) → (1, 1) ? No:

(a,b) = (-1,1) → (b, -a) = (1, -(-1)) = (1,1)

Then add back center (2,0): (2+1, 0+1) = (3,1) — but B is below axis.

Not matching.

Perhaps rotation 90° counterclockwise?

Try different approach.

Look at the triangles: A is above x-axis, B is below and to the right. And their orientations suggest a 90-degree turn.

Actually, let’s consider: if you rotate A 90 degrees clockwise around the origin, what happens?

Point (0,1) in A → (1,0)

Point (1,2) → (2,-1)

Point (2,1) → (1,-2)

Now, does B have points at (1,0), (2,-1), (1,-2)? Looking at diagram — B seems to have vertices at approximately (2,-1), (3,-2), (1,-2)? Close.

Actually, in the diagram, B looks like it has points at (2,-1), (3,-2), (1,-2) — which matches exactly the rotated points if A had (0,1), (1,2), (2,1).

Yes! So if A has those points, rotating 90° CW around origin gives B.

Confirm: Original A: let’s say vertices at (0,1), (1,2), (2,1) — forms a triangle pointing up.

After 90° CW rotation: (x,y) → (y, -x)

So:

(0,1) → (1, 0)

(1,2) → (2, -1)

(2,1) → (1, -2)

Now, plot these: (1,0), (2,-1), (1,-2) — that’s a triangle pointing right-down, which matches B in the diagram.

Perfect.

So rule: (x, y) → (y, -x) — which is 90° clockwise rotation about origin.

✔ iv) is: 90° clockwise rotation about origin → rule: (y, -x)

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v)
Two triangles overlapping? Actually, shape A and B are both triangles, and B is larger? Wait, no — looking closely, they seem same size.

Actually, A is smaller triangle inside? No — in v), A is a small triangle near origin, B is a larger triangle? Wait, no — let me think.

Looking at diagram: A is a triangle with points like (0,0), (1,1), (2,0) — small.

B is a triangle with points like (0,0), (2,2), (4,0) — twice as big.

Oh! So it’s a dilation.

Center at origin? Because (0,0) stays fixed.

Point (1,1) in A → (2,2) in B → scaled by 2.

Point (2,0) → (4,0) → also scaled by 2.

So dilation with scale factor 2, center at origin.

Rule: (2x, 2y)

✔ v) is: Dilation by factor 2 from origin → rule: (2x, 2y)

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vi)
Shape A is a quadrilateral in second quadrant. Shape B is same shape, but flipped and moved.

Specifically, A is like a kite or arrowhead pointing left-up. B is pointing right-down.

Looks like a 180° rotation? Or reflection?

Try 180° rotation about origin: (x,y) → (-x, -y)

Take a point in A: say (-2,1) → becomes (2,-1)

Does B have a point at (2,-1)? Looking at diagram — B is in fourth quadrant, and yes, likely.

Another point: A has (-1,2) → (1,-2) — should be in B.

And A has (-3,0) → (3,0) — but B may not have that.

Wait — actually, in vi), A and B are symmetric with respect to origin.

Moreover, orientation is reversed — which suggests rotation 180°, not reflection.

Because reflection would preserve handedness? No, reflection reverses, rotation 180° also reverses? Actually both reverse orientation in 2D? Wait no.

In 2D, rotation preserves orientation, reflection reverses.

But visually, if you rotate A 180°, it should match B.

Assume A has point (-2,1). After 180° rot: (2,-1)

If B has that point, good.

Similarly, A’s (-1,0) → (1,0) — but B might not have (1,0); B is lower.

Wait — perhaps center is not origin.

Look: the "center" of A is around (-1.5, 0.5), center of B around (1.5, -0.5) — midpoint is (0,0). So likely rotation about origin.

And since distances from origin are same, and angles opposite, it’s 180° rotation.

Rule: (x,y) → (-x, -y)

✔ vi) is: 180° rotation about origin → rule: (-x, -y)

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Now, summarize all:

i) Translation: (x + 4, y - 2)

ii) Reflection over x-axis then translation right 3: (x + 3, -y)

iii) Translation right 6: (x + 6, y)

iv) 90° clockwise rotation about origin: (y, -x)

v) Dilation by 2 from origin: (2x, 2y)

vi) 180° rotation about origin: (-x, -y)

All rules are in terms of (x,y) mapping to new coordinates.

Final Answer:

i) (x + 4, y - 2)
ii) (x + 3, -y)
iii) (x + 6, y)
iv) (y, -x)
v) (2x, 2y)
vi) (-x, -y)