Practice worksheet for identifying and applying geometric transformations on graphs.

Let’s solve each transformation problem step by step. We’ll go one at a time, carefully tracking how each point moves based on the rule given.

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Problem 1: Rotate 270° about origin

Rule for 270° rotation (clockwise or counterclockwise? In math, unless specified, we assume counterclockwise).
But note: rotating 270° CCW is same as 90° CW.

Standard rules:
- 90° CCW: (x, y) → (-y, x)
- 180°: (x, y) → (-x, -y)
- 270° CCW: (x, y) → (y, -x)

Wait — let’s double-check with standard convention:

Actually, in most school curricula, when they say “rotate 270°” without direction, it’s often counterclockwise, and the rule is:

> 270° CCW: (x, y) → (y, -x)

Original points:
A = (-5, 4)
B = (-3, 4)
C = (-3, 6)
D = (-5, 6)

Apply (x, y) → (y, -x):

A’ = (4, -(-5)) = (4, 5)
B’ = (4, -(-3)) = (4, 3)
C’ = (6, -(-3)) = (6, 3)
D’ = (6, -(-5)) = (6, 5)

✔ Check: This makes sense — original square in Q2, after 270° CCW should be in Q1, which matches.

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Problem 2: Reflect over x-axis

Rule: (x, y) → (x, -y)

Original:
K = (-4, 4)
L = (2, 6)
M = (4, 2)

Reflect:

K’ = (-4, -4)
L’ = (2, -6)
M’ = (4, -2)

✔ Simple flip over horizontal axis — y-values change sign.

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Problem 3: Reflect across line y = x

Rule: (x, y) → (y, x)

Original:
J = (-6, 2)
K = (-6, 5)
L = (-2, 5)

Swap coordinates:

J’ = (2, -6)
K’ = (5, -6)
L’ = (5, -2)

✔ Swapping x and y reflects over diagonal line y=x.

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Problem 4: Rotate 90° counterclockwise about origin

Rule: (x, y) → (-y, x)

Original:
P = (-2, 2)
Q = (-1, -1)
R = (1, -1)

Apply:

P’ = (-2, -2) → Wait! Let’s do it right:

(x, y) → (-y, x)

P = (-2, 2) → (-2, -2)? No:

-y = -(2) = -2
x = -2
So P’ = (-2, -2)? That can’t be right.

Wait — let’s recalculate:

P = (-2, 2)
→ new x = -y = -(2) = -2
→ new y = x = -2
So P’ = (-2, -2)

Q = (-1, -1)
→ new x = -(-1) = 1
→ new y = -1
So Q’ = (1, -1)

R = (1, -1)
→ new x = -(-1) = 1
→ new y = 1
So R’ = (1, 1)

Wait — that seems off. Let me verify with a known example.

Take point (1,0), rotate 90° CCW → should become (0,1)

Using rule: (x,y) → (-y, x) → (0,1) ✔

Another: (0,1) → (-1, 0) ✔

So our rule is correct.

Back to P = (-2, 2) → (-2, -2)? Let’s plot mentally:

Original P is left 2, up 2. After 90° CCW, it should be down 2, left 2 → (-2, -2) ✔

Q = (-1,-1) → (1, -1) → right 1, down 1 → yes, that’s where it goes.

R = (1,-1) → (1,1) → right 1, up 1 → yes.

Actually, wait — R = (1, -1) → (-(-1), 1) = (1, 1) ✔

So:

P’ = (-2, -2)
Q’ = (1, -1)
R’ = (1, 1)

But looking back at the image, maybe I misread original points? The user didn’t provide coordinates explicitly beyond what’s written. But since this is text-based, we go by what’s written.

In the problem statement for #4, it says:

Original Coordinates:
P ___ ___
Q ___ ___
R ___ ___

And from context, likely:

Assuming from typical problems, but since not fully visible, perhaps we should rely on standard interpretation.

Wait — actually, in the initial prompt, the user included an image description, but we are to solve based on content. Since the OCR might have missed some, but in the text provided earlier, for problem 4, it shows:

“Rotate 90° counterclockwise about the origin”

With triangle having points probably at:

Looking at common versions of this worksheet, often:

P = (-2, 2), Q = (-1, -1), R = (1, -1) — as above.

So we’ll stick with:

P’ = (-2, -2)
Q’ = (1, -1)
R’ = (1, 1)

But let me check if that forms a proper rotated triangle.

Original: P(-2,2), Q(-1,-1), R(1,-1) — base QR horizontal at y=-1, P above.

After 90° CCW: P’(-2,-2), Q’(1,-1), R’(1,1) — now vertical side from Q’ to R’, P’ to the left-down. Seems plausible.

Alternatively, maybe the original points are different? But without clearer data, we proceed.

Wait — perhaps I made a mistake in Q.

Q = (-1, -1) → (-y, x) = (-(-1), -1) = (1, -1) ✔

Yes.

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Problem 5: Reflect across y-axis

Rule: (x, y) → (-x, y)

Original:
X = (-5, 1)
Y = (-2, 2)
Z = (-2, 1)

Reflect:

X’ = (5, 1)
Y’ = (2, 2)
Z’ = (2, 1)

✔ Flip over vertical axis — x changes sign.

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Problem 6: Reflect across x-axis

Same as Problem 2.

Original:
G = (1, 2)
H = (4, 2)
I = (3, 4)

Reflect:

G’ = (1, -2)
H’ = (4, -2)
I’ = (3, -4)

✔ Y-values negate.

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Problem 7: Translate 4 units left, 5 units up

Translation rule: (x, y) → (x - 4, y + 5)

Original quadrilateral vertices (from grid, assuming):

Typically in such grids, if not labeled, we estimate.

From common version: suppose points are:

A = (1, -1)
B = (3, -1)
C = (4, -3)
D = (2, -3)

Then translate:

A’ = (1-4, -1+5) = (-3, 4)
B’ = (3-4, -1+5) = (-1, 4)
C’ = (4-4, -3+5) = (0, 2)
D’ = (2-4, -3+5) = (-2, 2)

✔ Move every point left 4, up 5.

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Problem 8: Rotate 180° about origin

Rule: (x, y) → (-x, -y)

Original triangle: assume points from grid.

Commonly: A = (1, 1), B = (3, 2), C = (2, 4) — but let's see.

Actually, in many worksheets, for problem 8, it’s a triangle with:

Say: P = (1, 2), Q = (3, 1), R = (2, 4)

Then:

P’ = (-1, -2)
Q’ = (-3, -1)
R’ = (-2, -4)

✔ 180° rotation flips both signs.

To confirm: any point (a,b) becomes (-a,-b) — straight through origin.

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Now, compiling all answers neatly.

Note: For problems where exact original coordinates weren't fully clear from text (like 4,7,8), I used standard assumptions based on typical worksheet layouts. If your actual worksheet has different coordinates, adjust accordingly — but the method remains the same.

Final Answer:

1. A'(4,5), B'(4,3), C'(6,3), D'(6,5)
2. K'(-4,-4), L'(2,-6), M'(4,-2)
3. J'(2,-6), K'(5,-6), L'(5,-2)
4. P'(-2,-2), Q'(1,-1), R'(1,1)
5. X'(5,1), Y'(2,2), Z'(2,1)
6. G'(1,-2), H'(4,-2), I'(3,-4)
7. Assuming original A(1,-1), B(3,-1), C(4,-3), D(2,-3): A'(-3,4), B'(-1,4), C'(0,2), D'(-2,2)
8. Assuming original P(1,2), Q(3,1), R(2,4): P'(-1,-2), Q'(-3,-1), R'(-2,-4)

If your worksheet has specific coordinates written in the blanks, use those with the same rules.

For accuracy, always apply:

- Rotation 90° CCW: (x,y) → (-y,x)
- Rotation 180°: (x,y) → (-x,-y)
- Rotation 270° CCW: (x,y) → (y,-x)
- Reflection over x-axis: (x,y) → (x,-y)
- Reflection over y-axis: (x,y) → (-x,y)
- Reflection over y=x: (x,y) → (y,x)
- Translation left a, up b: (x,y) → (x-a, y+b)

You’ve got this! Just plug in your numbers.