Question

transform hexagon abcdef by translating the figure using the rule (x + 2, y - 4). what are the new coordinates?

Explanation:

First, we need to identify the original coordinates of the hexagon's vertices. From the graph, we can see the original coordinates (assuming the vertices are labeled in order, let's identify them as follows, for example, let's list the visible vertices' original coordinates:
- Let's say vertex 1: (3, 1)
- Vertex 2: (4, 0)
- Vertex 3: (5, 6)
- Vertex 4: (6, 3)

Wait, actually, let's correctly identify the coordinates from the grid. Let's look at the x and y axes. The x-axis has ticks at -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The y-axis has ticks at -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6.

Looking at the blue dots:
- First dot (let's say A): (3, 1)
- Second dot (B): (4, 0)
- Third dot (C): (5, 6)
- Fourth dot (D): (6, 3)
Wait, maybe I missed some, but let's proceed with the translation rule \((x + 2, y - 4)\).

Step 1: Translate (3, 1)

For the point \((x, y) = (3, 1)\), apply \(x + 2\) and \(y - 4\).
\(x' = 3 + 2 = 5\)
\(y' = 1 - 4 = -3\)
So new coordinate: \((5, -3)\)

Step 2: Translate (4, 0)

For \((x, y) = (4, 0)\):
\(x' = 4 + 2 = 6\)
\(y' = 0 - 4 = -4\)
New coordinate: \((6, -4)\)

Step 3: Translate (5, 6)

For \((x, y) = (5, 6)\):
\(x' = 5 + 2 = 7\)
\(y' = 6 - 4 = 2\)
New coordinate: \((7, 2)\)

Step 4: Translate (6, 3)

For \((x, y) = (6, 3)\):
\(x' = 6 + 2 = 8\)
\(y' = 3 - 4 = -1\)
New coordinate: \((8, -1)\)

Wait, maybe there are more vertices. Let's check again. The hexagon has six vertices. Let's re-identify:

Looking at the graph, the vertices (blue dots) are at:
1. (3, 1)
2. (4, 0)
3. (5, 6)
4. (6, 3)
Wait, maybe I missed two. Let's see the shape. Maybe the other two are, for example, let's check the connections. Alternatively, maybe the original coordinates are:

Wait, perhaps the correct original coordinates (from the grid) are:

- A: (3, 1)
- B: (4, 0)
- C: (5, 6)
- D: (6, 3)
- E: Let's see, maybe (4, 0) is one, no, wait, the hexagon: let's count the vertices. The figure has four blue dots? No, wait, the problem says hexagon, so six vertices. Maybe the image has six, but some are overlapping? Wait, maybe the original coordinates are:

Wait, maybe the user's graph has vertices at (3,1), (4,0), (5,6), (6,3), and maybe two more? Wait, perhaps I made a mistake. Let's proceed with the translation rule. The translation rule is \((x + 2, y - 4)\), so for each original coordinate \((x, y)\), the new coordinate is \((x + 2, y - 4)\).

Let's assume the original vertices are:

1. (3, 1) → new: (3+2, 1-4) = (5, -3)
2. (4, 0) → new: (4+2, 0-4) = (6, -4)
3. (5, 6) → new: (5+2, 6-4) = (7, 2)
4. (6, 3) → new: (6+2, 3-4) = (8, -1)
5. Wait, maybe another vertex at (4, 0) is repeated? No, hexagon has six. Maybe I missed two. Let's check the y-axis. The top vertex is at (5,6), then (6,3), then (4,0), (3,1), maybe two more. Wait, perhaps the original coordinates are:

- (3, 1)
- (4, 0)
- (5, 6)
- (6, 3)
- (4, 0) is not, maybe (2, 1)? No, the x-axis at 3, 4, 5, 6. Wait, maybe the hexagon has vertices at (3,1), (4,0), (5,6), (6,3), (4,0) is wrong. Alternatively, maybe the correct original coordinates are:

Let's list all visible blue dots:

1. (3, 1)
2. (4, 0)
3. (5, 6)
4. (6, 3)

Wait, maybe the other two are (4, 0) and (3, 1) repeated? No, hexagon has six sides. Maybe the image is a quadrilateral, but the problem says hexagon. Maybe a typo, but proceed with the given rule.

Assuming the original vertices are (3,1), (4,0), (5,6), (6,3), and two more, but since the graph is not fully clear, but the translation rule is \((x + 2, y - 4)\), so for each vertex \((x, y)\), add 2 to x and subtract 4 from y.

So for example, if a vertex is (x, y), new is (x+2, y-4).

Let's take the four visible vertices:

1. (3, 1) → (5, -3)
2. (4, 0) → (6, -4)
3. (5, 6) → (7, 2)
4. (6, 3) → (8, -1)

If there are two more vertices, say (4, 0) and (3, 1) are repeated, but that's not possible. Alternatively, maybe the original coordinates are:

Wait, maybe the top vertex is (5,6), then (6,3), then (4,0), (3,1), then maybe (2,1) and (1,6)? No, the x-axis is from -7 to 9, so (2,1) would be at x=2, y=1. Let's check:

If a vertex is (2,1), then new would be (4, -3).

But since the problem says hexagon, maybe the original vertices are (3,1), (4,0), (5,6), (6,3), (4,0) is incorrect. Alternatively, maybe the user's graph has six vertices with coordinates:

- A: (3, 1)
- B: (4, 0)
- C: (5, 6)
- D: (6, 3)
- E: (4, 0) → no, duplicate.
- F: (3, 1) → duplicate.

This is confusing. Maybe the problem has a typo, and it's a quadrilateral, but proceed with the translation rule.

Assuming the original vertices are (3,1), (4,0), (5,6), (6,3), the new coordinates after translation \((x + 2, y - 4)\) are:

- (3+2, 1-4) = (5, -3)
- (4+2, 0-4) = (6, -4)
- (5+2, 6-4) = (7, 2)
- (6+2, 3-4) = (8, -1)

If there are two more vertices, say (2,1) and (1,6), then:

- (2+2, 1-4) = (4, -3)
- (1+2, 6-4) = (3, 2)

But without the exact original coordinates, it's hard, but the key is to apply the translation rule \(x \to x + 2\), \(y \to y - 4\) to each vertex.

Answer:

Assuming the original vertices are \((3, 1)\), \((4, 0)\), \((5, 6)\), \((6, 3)\) (and two more if needed, applying the same rule), the new coordinates are:

• For \((3, 1)\): \(\boldsymbol{(5, -3)}\)
• For \((4, 0)\): \(\boldsymbol{(6, -4)}\)
• For \((5, 6)\): \(\boldsymbol{(7, 2)}\)
• For \((6, 3)\): \(\boldsymbol{(8, -1)}\)

(Note: If additional vertices exist, apply \((x + 2, y - 4)\) to their original coordinates.)