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Create a function to represent the point [tex](x, y)[/tex] being reflected across the [tex]y[/tex]-axis.

[tex]F(x, y) = (-x, y)[/tex]

Sagot :

GinnyAnswer
Sure! Let's go through the process step by step to create a function that represents the point [tex]\((x, y)\)[/tex] being reflected across the [tex]\(y\)[/tex]-axis.

### Reflection Across the [tex]\(y\)[/tex]-Axis

When you reflect a point across the [tex]\(y\)[/tex]-axis, the [tex]\(x\)[/tex]-coordinate of the point changes sign, while the [tex]\(y\)[/tex]-coordinate remains the same.

For a point [tex]\((x, y)\)[/tex]:
- The reflected point will be [tex]\((-x, y)\)[/tex].

In mathematical terms, this transformation can be described by the function:

[tex]\[ F(x, y) = (-x, y) \][/tex]

### Step-by-Step Solution

1. Define the Input Coordinates:
- Let the original coordinates of the point be [tex]\((x, y)\)[/tex].

2. Reflect Across the [tex]\(y\)[/tex]-Axis:
- To find the reflected point, negate the [tex]\(x\)[/tex]-coordinate: [tex]\(-x\)[/tex].
- The [tex]\(y\)[/tex]-coordinate remains the same: [tex]\(y\)[/tex].

3. Formulate the Reflected Coordinates:
- The new coordinates after reflection will be [tex]\((-x, y)\)[/tex].

### Example

Let's work through an example to illustrate this:

Original Point:
- Suppose the original point is [tex]\((3, 4)\)[/tex].

Reflection Process:
- Negate the [tex]\(x\)[/tex]-coordinate: [tex]\(-3\)[/tex].
- Keep the [tex]\(y\)[/tex]-coordinate unchanged: [tex]\(4\)[/tex].

Reflected Point:
- The reflected point will be [tex]\((-3, 4)\)[/tex].

### General Function

To generalize this process for any point [tex]\((x, y)\)[/tex], we define a function [tex]\(F\)[/tex] as follows:

[tex]\[ F(x, y) = (-x, y) \][/tex]

So, if you are given any point [tex]\((x, y)\)[/tex] and you want to find the coordinates of the point after it has been reflected across the [tex]\(y\)[/tex]-axis, you simply apply the transformation provided by the function [tex]\(F\)[/tex].

### Conclusion
The function [tex]\(F(x, y) = (-x, y)\)[/tex] represents the point [tex]\((x, y)\)[/tex] being reflected across the [tex]\(y\)[/tex]-axis. This transformation changes the sign of the [tex]\(x\)[/tex]-coordinate and keeps the [tex]\(y\)[/tex]-coordinate unchanged.
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