Question
Error Analysis The blue figure is the image of the black figure after a translation. Your friend says that a rule that describes the translation is (x,y)right arrow(xnegative 3,yplus 1). What is the correct rule? What mistake did your friend likely make?
-6
-3
3
6
-6
-3
3
6
x
y
A horizontal x-axis is labeled from negative 6 to 6. A vertical y-axis is labeled from negative 6 to 6. A triangle has the following vertices in clockwise order: (negative 3, 1); (0, 4); (1, negative 1). A second triangle has the following vertices in clockwise order: (0, 0), (3, 3), (4, negative 2).
Question content area bottom
Part 1
Which is the correct rule for the translation?
A.
(x,y)right arrow(xplus 3,ynegative 1)
B.
(x,y)right arrow(x,ynegative 1)
C.
(x,y)right arrow(xplus 3,y)
D.
(x,y)right arrow(xnegative 1,yplus 3)
-6
-3
3
6
-6
-3
3
6
x
y
A horizontal x-axis is labeled from negative 6 to 6. A vertical y-axis is labeled from negative 6 to 6. A triangle has the following vertices in clockwise order: (negative 3, 1); (0, 4); (1, negative 1). A second triangle has the following vertices in clockwise order: (0, 0), (3, 3), (4, negative 2).
Question content area bottom
Part 1
Which is the correct rule for the translation?
A.
(x,y)right arrow(xplus 3,ynegative 1)
B.
(x,y)right arrow(x,ynegative 1)
C.
(x,y)right arrow(xplus 3,y)
D.
(x,y)right arrow(xnegative 1,yplus 3)
Answers
GPT-4o mini
To find the correct translation rule from the black triangle to the blue triangle, we need to determine how the coordinates of the vertices of the black triangle change to form the coordinates of the vertices of the blue triangle.
### Black Triangle Vertices:
1. A(-3, 1)
2. B(0, 4)
3. C(1, -1)
### Blue Triangle Vertices:
1. D(0, 0)
2. E(3, 3)
3. F(4, -2)
### Translation Analysis:
- **Vertex A(-3, 1) to D(0, 0)**:
- From x: \(-3\) to \(0\) → \(0 = -3 + 3\) (translated to the right by 3 units)
- From y: \(1\) to \(0\) → \(0 = 1 - 1\) (translated down by 1 unit)
- **Vertex B(0, 4) to E(3, 3)**:
- From x: \(0\) to \(3\) → \(3 = 0 + 3\) (translated to the right by 3 units)
- From y: \(4\) to \(3\) → \(3 = 4 - 1\) (translated down by 1 unit)
- **Vertex C(1, -1) to F(4, -2)**:
- From x: \(1\) to \(4\) → \(4 = 1 + 3\) (translated to the right by 3 units)
- From y: \(-1\) to \(-2\) → \(-2 = -1 - 1\) (translated down by 1 unit)
### Rule Determination:
From the analysis above, we can conclude that all three vertices undergo the same translation:
- The x-coordinates increase by \(3\) (translating to the right).
- The y-coordinates decrease by \(1\) (translating down).
### Translation Rule:
Thus, the correct translation rule is:
\[
(x, y) \rightarrow (x + 3, y - 1)
\]
### Friend's Mistake:
Your friend’s translation rule was \((x, y) \rightarrow (x - 3, y + 1)\). The likely mistake was in the direction of the translations:
- The friend mistakenly translated the x-coordinates to the left instead of to the right (using -3 instead of +3).
- The friend also mistakenly translated the y-coordinates up instead of down (using +1 instead of -1).
### Final Answer:
The correct rule is **A.** \((x,y) \rightarrow (x+3, y-1)\).
### Black Triangle Vertices:
1. A(-3, 1)
2. B(0, 4)
3. C(1, -1)
### Blue Triangle Vertices:
1. D(0, 0)
2. E(3, 3)
3. F(4, -2)
### Translation Analysis:
- **Vertex A(-3, 1) to D(0, 0)**:
- From x: \(-3\) to \(0\) → \(0 = -3 + 3\) (translated to the right by 3 units)
- From y: \(1\) to \(0\) → \(0 = 1 - 1\) (translated down by 1 unit)
- **Vertex B(0, 4) to E(3, 3)**:
- From x: \(0\) to \(3\) → \(3 = 0 + 3\) (translated to the right by 3 units)
- From y: \(4\) to \(3\) → \(3 = 4 - 1\) (translated down by 1 unit)
- **Vertex C(1, -1) to F(4, -2)**:
- From x: \(1\) to \(4\) → \(4 = 1 + 3\) (translated to the right by 3 units)
- From y: \(-1\) to \(-2\) → \(-2 = -1 - 1\) (translated down by 1 unit)
### Rule Determination:
From the analysis above, we can conclude that all three vertices undergo the same translation:
- The x-coordinates increase by \(3\) (translating to the right).
- The y-coordinates decrease by \(1\) (translating down).
### Translation Rule:
Thus, the correct translation rule is:
\[
(x, y) \rightarrow (x + 3, y - 1)
\]
### Friend's Mistake:
Your friend’s translation rule was \((x, y) \rightarrow (x - 3, y + 1)\). The likely mistake was in the direction of the translations:
- The friend mistakenly translated the x-coordinates to the left instead of to the right (using -3 instead of +3).
- The friend also mistakenly translated the y-coordinates up instead of down (using +1 instead of -1).
### Final Answer:
The correct rule is **A.** \((x,y) \rightarrow (x+3, y-1)\).