Answer :
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].
### 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].
