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Given the ordered pair [tex]\((1, -3)\)[/tex], where will the transformed image be after the following composition: [tex]\(D_2 \circ R_{y-axis}\)[/tex]?

Sagot :

GinnyAnswer
Certainly! Let's transform the given ordered pair [tex]\((1, -3)\)[/tex] through the described transformations step-by-step.

### Step 1: Reflection over the y-axis

To reflect a point over the y-axis, we change the sign of the x-coordinate while keeping the y-coordinate the same:
- Original Point: [tex]\((1, -3)\)[/tex]
- Reflected Point: [tex]\((-1, -3)\)[/tex]

### Step 2: Scaling by a factor of 2

Next, we scale the reflected coordinates by a factor of 2. This means we multiply both the x and y coordinates by 2:
- Reflected Point: [tex]\((-1, -3)\)[/tex]
- Transformed Point: [tex]\((-2 \cdot (-1), 2 \cdot (-3))\)[/tex]

So, after we multiply:
- Transformed Coordinates: [tex]\((-2, -6)\)[/tex]

### Summary

1. Reflection over the y-axis:
- From [tex]\((1, -3)\)[/tex] to [tex]\((-1, -3)\)[/tex]

2. Scaling by a factor of 2:
- From [tex]\((-1, -3)\)[/tex] to [tex]\((-2, -6)\)[/tex]

Therefore, the final transformed image of the original point [tex]\((1, -3)\)[/tex] through the composition [tex]\(D_2 \circ R_{\text{y-axis}}\)[/tex] will be:
[tex]\[ (-2, -6) \][/tex]

Additionally, the intermediate reflected coordinates are [tex]\((-1, -3)\)[/tex].
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