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 1, negative 3), (1, 1), (4, negative 4). A blue triangle has the following vertices: (negative 3, 1), (negative 1, 5), (2, 0).
Question content area bottom
Part 1
The translation rule is (x,y)right arrow(xplus(
enter your response here),yplus(
enter your response here)).
-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 1, negative 3), (1, 1), (4, negative 4). A blue triangle has the following vertices: (negative 3, 1), (negative 1, 5), (2, 0).
Question content area bottom
Part 1
The translation rule is (x,y)right arrow(xplus(
enter your response here),yplus(
enter your response here)).
Answers
GPT-4o mini
To find the translation rule for the black triangle with vertices at \((-1, -3)\), \((1, 1)\), and \((4, -4)\) to the blue triangle with vertices at \((-3, 1)\), \((-1, 5)\), and \((2, 0)\), we need to determine the changes in the \(x\) and \(y\) coordinates.
Let's analyze the movement from the first vertex of the black triangle to the first vertex of the blue triangle:
1. From \((-1, -3)\) to \((-3, 1)\):
- Change in \(x\): \(-3 - (-1) = -3 + 1 = -2\)
- Change in \(y\): \(1 - (-3) = 1 + 3 = 4\)
Next, we can verify the same changes against the second and third vertices to ensure consistency.
2. From \((1, 1)\) to \((-1, 5)\):
- Change in \(x\): \(-1 - 1 = -2\)
- Change in \(y\): \(5 - 1 = 4\)
3. From \((4, -4)\) to \((2, 0)\):
- Change in \(x\): \(2 - 4 = -2\)
- Change in \(y\): \(0 - (-4) = 4\)
All the vertex translations confirm that the change in \(x\) is \(-2\) and the change in \(y\) is \(4\).
Therefore, the translation rule is:
\[
(x, y) \rightarrow (x - 2, y + 4)
\]
So, you should enter \(-2\) for the \(x\) change and \(4\) for the \(y\) change:
The translation rule is \((x,y) \rightarrow (x - 2,y + 4)\).
Let's analyze the movement from the first vertex of the black triangle to the first vertex of the blue triangle:
1. From \((-1, -3)\) to \((-3, 1)\):
- Change in \(x\): \(-3 - (-1) = -3 + 1 = -2\)
- Change in \(y\): \(1 - (-3) = 1 + 3 = 4\)
Next, we can verify the same changes against the second and third vertices to ensure consistency.
2. From \((1, 1)\) to \((-1, 5)\):
- Change in \(x\): \(-1 - 1 = -2\)
- Change in \(y\): \(5 - 1 = 4\)
3. From \((4, -4)\) to \((2, 0)\):
- Change in \(x\): \(2 - 4 = -2\)
- Change in \(y\): \(0 - (-4) = 4\)
All the vertex translations confirm that the change in \(x\) is \(-2\) and the change in \(y\) is \(4\).
Therefore, the translation rule is:
\[
(x, y) \rightarrow (x - 2, y + 4)
\]
So, you should enter \(-2\) for the \(x\) change and \(4\) for the \(y\) change:
The translation rule is \((x,y) \rightarrow (x - 2,y + 4)\).