The dilation of a triangle on a coordinate grid, illustrating a transformation with a scale factor.

Let’s solve this step by step.

We are given a dilation on a coordinate grid. Dilation means the figure is scaled — made bigger or smaller — from a center point (usually the origin unless stated otherwise). We need to find the rule that tells us how each point (x, y) changes to its new position after the dilation.

Looking at the graph:

- The original triangle has points like M, N, P.
- The dilated image has corresponding points M', N', P'.

Let’s pick one easy point to compare — let’s use point N and N'.

From the grid:

- Point N is at (6, -3) → x = 6, y = -3
- Point N' is at (2, -1) → x = 2, y = -1

Now, let’s see what happened to the coordinates:

Original x: 6 → New x: 2
What do we multiply 6 by to get 2?
→ 6 × ? = 2 → ? = 2/6 = 1/3

Original y: -3 → New y: -1
What do we multiply -3 by to get -1?
→ -3 × ? = -1 → ? = (-1)/(-3) = 1/3

So both x and y are multiplied by 1/3.

Let’s check another point to be sure — let’s try point P and P'.

Point P is at (4, -3)
Point P' is at (4/3, -1)? Wait — looking at the grid again...

Actually, let me double-check the coordinates carefully from the grid.

Looking again:

Assume the grid lines are every 1 unit.

Original triangle vertex N: looks like it’s at (6, -3)
Dilated image N’: looks like it’s at (2, -1)

Yes — 6 → 2 is ÷3 or ×1/3
-3 → -1 is also ÷3 or ×1/3

Another point: Let’s take point M.

Original M: looks like (-3, 6)
Dilated M’: looks like (-1, 2)

Check:
-3 × 1/3 = -1 ✔
6 × 1/3 = 2 ✔

Perfect.

So the rule is:
(x, y) → (1/3 x, 1/3 y)

That matches Option 4.

Wait — let’s look at the options again as written in the problem:

Option 1: (x, y) → (6x, 6y) → too big
Option 2: (x, y) → (1/3 x, 1/3 y) → this is what we found
Option 3: (x, y) → (3x, 3y) → would make it bigger
Option 4: (x, y) → (1/3 x, 1/3 y) — wait, same as option 2?

Hold on — looking back at your text:

You wrote:

> (x, y) → (6x, 6y) (x, y) → (⅓x, ⅓y)
> Option 1 Option 2
>
> (x, y) → (3x, 3y) (x, y) → (⅓x, ⅓y)
> Option 4

Wait — Option 2 and Option 4 are identical? That can’t be right. Probably a typo in how you copied it.

But based on standard multiple choice, and our calculation, the correct rule is multiplying both x and y by 1/3.

In your list, Option 2 says: (x, y) → (⅓x, ⅓y)
And Option 4 also says the same? That must be an error.

But since we calculated correctly, and the only logical answer is scaling by 1/3, then whichever option says (x, y) → (1/3 x, 1/3 y) is correct.

Given that, and assuming Option 2 is meant to be that, and Option 4 might be a duplicate or misprint — but in your original message, Option 4 is listed as (x, y) → (⅓x, ⅓y), same as Option 2.

Wait — perhaps you meant:

Looking again at your input:

“Which of the following algebraic rules describes the dilation shown on the graph above?”

Then four options:

Top left: (x,y) → (6x,6y) — Option 1
Top right: (x,y) → (⅓x, ⅓y) — Option 2
Bottom left: (x,y) → (3x,3y) — no label?
Bottom right: (x,y) → (⅓x, ⅓y) — Option 4

This seems like a formatting issue. But logically, only one should be correct.

Since our math shows scaling by 1/3, and both Option 2 and Option 4 say that, but likely in the actual problem, only one is labeled correctly.

But in your text, you have “Option 2” under top right, and “Option 4” under bottom right — both with same formula.

Perhaps it’s a copy-paste error.

But based on standard problems, and our verification with two points, the correct transformation is:

(x, y) → (1/3 x, 1/3 y)

So if Option 2 is that, then Option 2 is correct.

Alternatively, maybe Option 4 was meant to be different — but as written, both 2 and 4 are same.

However, in many such problems, the correct answer is often listed once.

Given that, and since we verified with coordinates, I’ll go with the rule: multiply both x and y by 1/3.

Final Answer:
Option 2