Graphing and labeling translations of triangles on coordinate grids with corresponding equations.

Let’s solve each translation step by step.

We are given pre-image points and a translation rule:
(x, y) → (x + a, y + b) — this means we add “a” to the x-coordinate and “b” to the y-coordinate of each point.

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Part A – Left Graph: Translation is (x, y) → (x + 3, y - 5)

That means:
- Add 3 to every x-value.
- Subtract 5 from every y-value.

Given points:

A (-2, 1)
→ x: -2 + 3 = 1
→ y: 1 - 5 = -4
So A’ = (1, -4) ✔ (already filled in correctly)

B (-3, 0)
→ x: -3 + 3 = 0
→ y: 0 - 5 = -5
So B’ = (0, -5) ✘ The image says (-6, -5) — that’s wrong. Let me double-check:
Wait — maybe I misread? No, the rule is clearly (x+3, y-5). So for B(-3,0):
-3 + 3 = 0
0 - 5 = -5 → (0, -5)

But the student wrote (-6, -5) — which would be if they did x - 3 instead of x + 3. That’s an error.

C (1, 2)
→ x: 1 + 3 = 4
→ y: 2 - 5 = -3
So C’ = (4, -3) ✔ (correctly filled)

Wait — let me check again. Maybe the original problem has different coordinates? Looking at the graph visually might help, but since we can’t describe images, we go by what’s written.

Actually — looking back at the user’s input: In part A left, it says:

> A (-2,1) → A'(1,-4) ✔️
> B (-3,0) → B'(-6,-5) ← This must be a mistake in the student’s work or typo? But according to math, it should be (0, -5)

Hold on — perhaps the translation is not (x+3, y-5)? Wait no, it’s written clearly.

Alternatively — maybe point B is not (-3,0)? Let me re-read the problem as presented:

In the text provided by user:

For A left:
> (x,y) → (x + 3, y - 5)
> A (-2,1) → A'(1,-4)
> B (-3,0) → B'(-6,-5) ← this is likely incorrect
> C (1,2) → C'(4,-3)

But (-3 + 3 = 0, not -6). So unless there's a sign error...

Wait — could it be that the translation is (x - 3, y - 5)? Then:

B(-3,0) → x: -3 -3 = -6, y: 0 -5 = -5 → matches B'(-6,-5)

And A(-2,1) → x: -2 -3 = -5, but A’ is given as (1,-4) — doesn't match.

Hmm. Contradiction.

Wait — look at A: (-2,1) → (1,-4):
-2 + 3 = 1 ✔️
1 - 5 = -4 ✔️ → so translation IS (x+3, y-5)

Then B(-3,0) → must be (0, -5), NOT (-6,-5)

Similarly, C(1,2) → (4,-3) ✔️

So the student made a mistake on B’. It should be (0, -5)

But wait — in the image description, it says “Name KEY” and shows answers already filled? Or is that the student’s attempt?

The instruction says: “Graph and label each pre-image and image, then shade in the image.” And below each graph, some answers are filled in — possibly the correct ones? Or the student’s?

Looking at the structure: For example, in A right:

> (x,y) → (x - 6, y)
> C (2,0) → C'(-4,0) ✔️ because 2-6=-4, 0 unchanged
> F (5,4) → F'(-1,4) ✔️ 5-6=-1
> G (1,4) → G'(-5,4) ✔️ 1-6=-5

All correct.

Back to A left: If A and C are correct with (x+3,y-5), then B must be wrong as written.

Perhaps it’s a typo in the problem statement? Or maybe point B is actually (-9,0)? Unlikely.

Another possibility: Maybe the coordinate for B is (-3, 0), but the student applied the wrong operation.

Since A and C confirm the rule is (x+3, y-5), we’ll go with that.

So correction:

B (-3, 0) → B’ (0, -5)

But wait — let’s check the second graph in A (right side):

Translation: (x,y) → (x - 6, y)

Points:

C (2,0) → (2-6, 0) = (-4,0) ✔️
F (5,4) → (5-6,4) = (-1,4) ✔️
G (1,4) → (1-6,4) = (-5,4) ✔️

All good.

Now Part B – Left Graph: Translation is (x, y) → (x + 2, y - 3)

Points:

A (-3, 1) → x: -3 + 2 = -1, y: 1 - 3 = -2 → A’(-1, -2) ✔️ (given)

B (-2, 0) → x: -2 + 2 = 0, y: 0 - 3 = -3 → B’(0, -3) ✘ Student wrote (-4, -3) — again, seems like they subtracted 2 instead of adding? Because -2 -2 = -4.

C (2, 2) → x: 2 + 2 = 4, y: 2 - 3 = -1 → C’(4, -1) ✔️ (given)

So again, B is miscalculated. Should be (0, -3), not (-4, -3)

Part B – Right Graph: Translation is (x, y) → (x + 1, y + 0) → same as (x+1, y)

Points:

C (2, 0) → (2+1, 0) = (3, 0) ✔️
F (3, 5) → (3+1, 5) = (4, 5) ✔️
G (-1, 4) → (-1+1, 4) = (0, 4) ✔️

All correct.

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So summarizing corrections needed:

In A left:
B (-3, 0) → should be (0, -5), not (-6, -5)

In B left:
B (-2, 0) → should be (0, -3), not (-4, -3)

All others are correct.

But wait — the user may have intended these as the correct answers? Or perhaps I’m misunderstanding.

Looking again at the original text:

In A left, it says:

> B (-3,0) → B'(-6,-5)

But mathematically, with rule (x+3, y-5), that’s impossible.

Unless... is the point B actually (-9, 0)? Then -9 + 3 = -6, 0 -5 = -5 → yes!

Ah! Perhaps there was a misreading of the coordinate.

Maybe in the graph, point B is at (-9, 0), not (-3, 0)?

Because otherwise, it doesn’t make sense.

Similarly, in B left: B (-2,0) → if rule is (x+2, y-3), then (-2+2=0, 0-3=-3) → (0,-3)

But student wrote (-4,-3) — which would require x - 2.

But A and C are correct with addition.

This suggests either:

1. There’s a typo in the problem’s listed coordinates for B in both cases.

OR

2. The student made errors in calculating B’ in both cases.

Given that A and C are consistent with the rules, and only B is off, and in both cases the student got B’ wrong by doing subtraction instead of addition for x, it’s likely the student made calculation errors.

Therefore, the correct answers are:

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Final Correct Answers:

Part A – Left:

A (-2, 1) → A’ (1, -4) ✔️
B (-3, 0) → B’ (0, -5)
C (1, 2) → C’ (4, -3) ✔️

Part A – Right:

C (2, 0) → C’ (-4, 0) ✔️
F (5, 4) → F’ (-1, 4) ✔️
G (1, 4) → G’ (-5, 4) ✔️

Part B – Left:

A (-3, 1) → A’ (-1, -2) ✔️
B (-2, 0) → B’ (0, -3)
C (2, 2) → C’ (4, -1) ✔️

Part B – Right:

C (2, 0) → C’ (3, 0) ✔️
F (3, 5) → F’ (4, 5) ✔️
G (-1, 4) → G’ (0, 4) ✔️

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Final Answer:
A left: B’ = (0, -5)
A right: all correct
B left: B’ = (0, -3)
B right: all correct