Answer :
To find the image of the point [tex]\((0,0)\)[/tex] after two reflections, we will follow each step of the reflection process carefully.
### Step-by-Step Solution:
#### Step 1: Reflection across Line 1, [tex]\(x = -2\)[/tex]
When reflecting a point [tex]\((x, y)\)[/tex] across a vertical line [tex]\(x = c\)[/tex], the formula to find the reflected point is [tex]\((2c - x, y)\)[/tex]. Here, Line 1 is [tex]\(x = -2\)[/tex]. Our original point is [tex]\((0, 0)\)[/tex].
1. Apply the reflection formula:
[tex]\[ x' = 2c - x = 2(-2) - 0 = -4 \][/tex]
[tex]\[ y' = y = 0 \][/tex]
So, the coordinates of the point after the first reflection are:
[tex]\[ (-4, 0) \][/tex]
#### Intermediate Point:
The coordinate after the first reflection is [tex]\((-4,0)\)[/tex].
#### Step 2: Reflection across Line 2 (the [tex]\(y\)[/tex]-axis)
When reflecting a point [tex]\((x, y)\)[/tex] across the [tex]\(y\)[/tex]-axis, the formula to find the reflected point is [tex]\((-x, y)\)[/tex]. Here, the intermediate point is [tex]\((-4, 0)\)[/tex].
1. Apply the reflection formula:
[tex]\[ x'' = -x = -(-4) = 4 \][/tex]
[tex]\[ y'' = y = 0 \][/tex]
So, the coordinates of the point after the second reflection are:
[tex]\[ (4, 0) \][/tex]
### Conclusion:
After performing these two reflections sequentially, the final image of the point [tex]\((0, 0)\)[/tex] is [tex]\((4, 0)\)[/tex].
The correct answer is:
[tex]\[ \boxed{(4, 0)} \][/tex]
### Step-by-Step Solution:
#### Step 1: Reflection across Line 1, [tex]\(x = -2\)[/tex]
When reflecting a point [tex]\((x, y)\)[/tex] across a vertical line [tex]\(x = c\)[/tex], the formula to find the reflected point is [tex]\((2c - x, y)\)[/tex]. Here, Line 1 is [tex]\(x = -2\)[/tex]. Our original point is [tex]\((0, 0)\)[/tex].
1. Apply the reflection formula:
[tex]\[ x' = 2c - x = 2(-2) - 0 = -4 \][/tex]
[tex]\[ y' = y = 0 \][/tex]
So, the coordinates of the point after the first reflection are:
[tex]\[ (-4, 0) \][/tex]
#### Intermediate Point:
The coordinate after the first reflection is [tex]\((-4,0)\)[/tex].
#### Step 2: Reflection across Line 2 (the [tex]\(y\)[/tex]-axis)
When reflecting a point [tex]\((x, y)\)[/tex] across the [tex]\(y\)[/tex]-axis, the formula to find the reflected point is [tex]\((-x, y)\)[/tex]. Here, the intermediate point is [tex]\((-4, 0)\)[/tex].
1. Apply the reflection formula:
[tex]\[ x'' = -x = -(-4) = 4 \][/tex]
[tex]\[ y'' = y = 0 \][/tex]
So, the coordinates of the point after the second reflection are:
[tex]\[ (4, 0) \][/tex]
### Conclusion:
After performing these two reflections sequentially, the final image of the point [tex]\((0, 0)\)[/tex] is [tex]\((4, 0)\)[/tex].
The correct answer is:
[tex]\[ \boxed{(4, 0)} \][/tex]
