To determine the new coordinates of a point after it is reflected across the [tex]\(x\)[/tex]-axis, we use the following rule: When a point [tex]\((x, y)\)[/tex] is reflected across the [tex]\(x\)[/tex]-axis, its new coordinates become [tex]\((x, -y)\)[/tex].
Given the original point [tex]\((-4, -2)\)[/tex]:
1. Identify the original coordinates:
[tex]\[
(x, y) = (-4, -2)
\][/tex]
2. Apply the reflection rule for the [tex]\(x\)[/tex]-axis. This means we change the [tex]\(y\)[/tex]-coordinate to its negative value while the [tex]\(x\)[/tex]-coordinate remains unchanged:
[tex]\[
(x, y) \rightarrow (x, -y)
\][/tex]
Substituting the given values into the rule:
[tex]\[
\text{Original coordinates: } (-4, -2) \rightarrow \text{New coordinates: } (-4, -(-2))
\][/tex]
3. Calculate the new [tex]\(y\)[/tex]-coordinate by negating the original [tex]\(y\)[/tex]-coordinate:
[tex]\[
-(-2) = 2
\][/tex]
4. Therefore, the new coordinates after reflecting [tex]\((-4, -2)\)[/tex] across the [tex]\(x\)[/tex]-axis are:
[tex]\[
(-4, 2)
\][/tex]
Thus, the new coordinates are [tex]\((-4, 2)\)[/tex].
