Question

The blue figure is a translation image of the black figure. Write a rule to describe the translation.
-6
-3
3
6
-6
-3
3
6
x
y



A coordinate system has a horizontal x-axis and a vertical y-axis each labeled from negative 6 to 6 in increments of 1. A black triangle has the following vertices: (negative 4, negative 2), (negative 1, 0), (2, negative 4). A blue triangle has the following vertices: (negative 5, 2), (negative 2, 4), (1, 0).
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Part 1
The translation rule is ​(x,y)right arrow​(xplus​(
  
enter your response here​),yplus​(
  
enter your response here​)).

Answer 1

To find the translation rule that describes the transformation from the black triangle to the blue triangle, we can analyze the coordinates of the corresponding vertices.

The vertices of the black triangle are:
- \( A(-4, -2) \)
- \( B(-1, 0) \)
- \( C(2, -4) \)

The vertices of the blue triangle are:
- \( A'(-5, 2) \)
- \( B'(-2, 4) \)
- \( C'(1, 0) \)

Now, let's determine the translation by finding the changes in the x-coordinates and y-coordinates from the black triangle to the blue triangle.

1. From \( A(-4, -2) \) to \( A'(-5, 2) \):
- Change in x: \(-5 - (-4) = -1\)
- Change in y: \(2 - (-2) = 4\)

2. From \( B(-1, 0) \) to \( B'(-2, 4) \):
- Change in x: \(-2 - (-1) = -1\)
- Change in y: \(4 - 0 = 4\)

3. From \( C(2, -4) \) to \( C'(1, 0) \):
- Change in x: \(1 - 2 = -1\)
- Change in y: \(0 - (-4) = 4\)

In each case, we see that the x-coordinates decrease by 1 and the y-coordinates increase by 4.

Therefore, the translation rule is:

\[
(x, y) \rightarrow (x - 1, y + 4)
\]

Thus, your response for the translation rule is:
- For the x-component: \(-1\)
- For the y-component: \(4\)

Put together, the translation rule is:

\[
(x, y) \rightarrow (x - 1, y + 4)
\]