Question

Use the translation left parenthesis x comma y right parenthesis right arrow left parenthesis x plus 3 comma y minus 1 right parenthesis to graph the image of triangle symbol Upper D prime Upper E prime Upper F prime .
-5
5
-5
5
x
y
D
E
F



A horizontal x-axis is labeled from negative 5 to 5. A vertical y-axis is labeled from negative 5 to 5. A triangle has the following vertices in clockwise order: D, (negative 4, 1); E, (negative 2, 5); F (0, negative 1).
Question content area bottom
Part 1
Graph triangle symbol DEF and triangle symbol Upper D prime Upper E prime Upper F prime . Choose the correct graph below.
A.
-5
5
-5
5
x
y



A horizontal x-axis is labeled from negative 5 to 5. A vertical y-axis is labeled from negative 5 to 5. A triangle has the following vertices in clockwise order: (negative 4, 1); (negative 2, 5); (0, negative 1). A second triangle has the following vertices in a clockwise order: (negative 4, 0); (negative 2, 4); (0, negative 2).
B.
-5
5
-5
5
x
y



A horizontal x-axis is labeled from negative 5 to 5. A vertical y-axis is labeled from negative 5 to 5. A triangle has the following vertices in clockwise order: (negative 4, 1); (negative 2, 5); (0, negative 1). A second triangle has the following vertices in a clockwise order: (negative 1, 1); (1, 5); (3, negative 1).
C.
-5
5
-5
5
x
y

Answer 1

To translate the triangle \( DEF \), we will apply the transformation given by the rule \( (x, y) \rightarrow (x + 3, y - 1) \) to each vertex of triangle \( DEF \):

1. Vertex \( D(-4, 1) \):
\[
D' = (-4 + 3, 1 - 1) = (-1, 0)
\]

2. Vertex \( E(-2, 5) \):
\[
E' = (-2 + 3, 5 - 1) = (1, 4)
\]

3. Vertex \( F(0, -1) \):
\[
F' = (0 + 3, -1 - 1) = (3, -2)
\]

Now, we have the vertices of triangle \( D'E'F' \) as:
- \( D'(-1, 0) \)
- \( E'(1, 4) \)
- \( F'(3, -2) \)

Now, let's compare the results with the options provided:

- **Option A:** Triangle vertices are \((-4, 0)\), \((-2, 4)\), \((0, -2)\) - Incorrect
- **Option B:** Triangle vertices are \((-1, 1)\), \((1, 5)\), \((3, -1)\) - Incorrect
- **Option C:** Triangle vertices are \((-1, 0)\), \((1, 4)\), \((3, -2)\) - Correct

Therefore, the correct graph is **Option C**, which accurately reflects the translation of triangle \( DEF \) to triangle \( D'E'F' \).