To create a function that represents the reflection of a point [tex]\((x, y)\)[/tex] across the [tex]\( y \)[/tex]-axis, follow these steps:
1. Understand the Reflection Rule: When reflecting a point [tex]\((x, y)\)[/tex] across the [tex]\( y \)[/tex]-axis, the [tex]\( x \)[/tex]-coordinate changes sign (i.e., it is negated), while the [tex]\( y \)[/tex]-coordinate remains the same.
So, if you have a point [tex]\((x, y)\)[/tex], its reflection across the [tex]\( y \)[/tex]-axis will be [tex]\((-x, y)\)[/tex].
2. Define the Function: We need to define a function [tex]\( F \)[/tex] that takes a point [tex]\((x, y)\)[/tex] and outputs the reflected point [tex]\((-x, y)\)[/tex].
[tex]\[
F(x, y) = (-x, y)
\][/tex]
Let's walk through the process with a specific example to make it clearer:
### Example
Given a point [tex]\((3, 4)\)[/tex]:
1. Identify the coordinates: [tex]\( x = 3 \)[/tex] and [tex]\( y = 4 \)[/tex].
2. Apply the reflection rule:
- The [tex]\( x \)[/tex]-coordinate is negated: [tex]\( 3 \)[/tex] becomes [tex]\(-3\)[/tex].
- The [tex]\( y \)[/tex]-coordinate remains the same: [tex]\( 4 \)[/tex] stays as [tex]\( 4 \)[/tex].
Therefore, the reflected point is [tex]\((-3, 4)\)[/tex].
### Formal Function Definition
To generalize, the function [tex]\( F(x, y) \)[/tex] can be written as:
[tex]\[
F(x, y) = (-x, y)
\][/tex]
So, using the function:
- For a general point [tex]\((x, y)\)[/tex]:
[tex]\[
F(x, y) = (-x, y)
\][/tex]
- For the specific point [tex]\((3, 4)\)[/tex]:
[tex]\[
F(3, 4) = (-3, 4)
\][/tex]
Thus, the reflected point of [tex]\((3, 4)\)[/tex] across the [tex]\( y \)[/tex]-axis is [tex]\((-3, 4)\)[/tex].
