Answer :
### Part 1: Transformations of Point [tex]\( C \)[/tex]
Let's start with the initial coordinates of point [tex]\( C \)[/tex] which are [tex]\( (5, -2) \)[/tex].
#### i. Reflection in the [tex]\( x \)[/tex]-axis
A reflection in the [tex]\( x \)[/tex]-axis changes the sign of the [tex]\( y \)[/tex]-coordinate while keeping the [tex]\( x \)[/tex]-coordinate the same.
So, [tex]\( C^{\prime} \)[/tex] after reflection in the [tex]\( x \)[/tex]-axis will be:
[tex]\[ (5, 2) \][/tex]
#### ii. Reflection in the [tex]\( y \)[/tex]-axis
A reflection in the [tex]\( y \)[/tex]-axis changes the sign of the [tex]\( x \)[/tex]-coordinate while keeping the [tex]\( y \)[/tex]-coordinate the same.
Thus, [tex]\( C^{\prime} \)[/tex] after reflection in the [tex]\( y \)[/tex]-axis will be:
[tex]\[ (-5, -2) \][/tex]
#### iii. Translation 7 units to the left and 4 units up
For translation, you subtract 7 from the [tex]\( x \)[/tex]-coordinate and add 4 to the [tex]\( y \)[/tex]-coordinate.
Hence, [tex]\( C^{\prime} \)[/tex] after the translation will be:
[tex]\[ (5 - 7, -2 + 4) \][/tex]
[tex]\[ (-2, 2) \][/tex]
### Part 2: Transformations of Point [tex]\( A \)[/tex]
The initial coordinates of point [tex]\( A \)[/tex] are [tex]\( (-1, 4) \)[/tex].
#### i. Transformation to [tex]\( A^{\prime}(-5, -1) \)[/tex]
We need to determine the rule that transforms [tex]\( A \)[/tex] to [tex]\( A^{\prime} \)[/tex].
By comparing the coordinates:
[tex]\[ (-5, -1) \][/tex]
with the initial coordinates [tex]\((-1, 4)\)[/tex], we find the changes in [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ \Delta x = -5 - (-1) = -4 \][/tex]
[tex]\[ \Delta y = -1 - 4 = -5 \][/tex]
The transformation rule is a translation by [tex]\((-4, -5)\)[/tex].
#### ii. Transformation to [tex]\( A^{\prime}(1, 4) \)[/tex]
For this transformation, we observe the change from [tex]\((-1, 4)\)[/tex] to [tex]\((1, 4)\)[/tex].
The [tex]\( y \)[/tex]-coordinate remains the same, and the [tex]\( x \)[/tex]-coordinate is the negative of the initial [tex]\( x \)[/tex]-coordinate. Therefore, this is a reflection in the [tex]\( y \)[/tex]-axis.
#### iii. Transformation to [tex]\( A^{\prime}(-1, -4) \)[/tex]
Here, we look at the change from [tex]\((-1, 4)\)[/tex] to [tex]\((-1, -4)\)[/tex].
The [tex]\( x \)[/tex]-coordinate stays the same, and the [tex]\( y \)[/tex]-coordinate is the negative of the initial [tex]\( y \)[/tex]-coordinate. Therefore, this transformation is a reflection in the [tex]\( x \)[/tex]-axis.
### Summary of Results
#### Transformations for Point [tex]\( C \)[/tex]:
1. Reflection in the [tex]\( x \)[/tex]-axis: [tex]\( C^{\prime} = (5, 2) \)[/tex]
2. Reflection in the [tex]\( y \)[/tex]-axis: [tex]\( C^{\prime} = (-5, -2) \)[/tex]
3. Translation 7 units left and 4 units up: [tex]\( C^{\prime} = (-2, 2) \)[/tex]
#### Transformations and Rules for Point [tex]\( A \)[/tex]:
1. To [tex]\( A^{\prime}(-5, -1) \)[/tex]: Translation by [tex]\( (-4, -5) \)[/tex]
2. To [tex]\( A^{\prime}(1, 4) \)[/tex]: Reflection in the [tex]\( y \)[/tex]-axis
3. To [tex]\( A^{\prime}(-1, -4) \)[/tex]: Reflection in the [tex]\( x \)[/tex]-axis
Let's start with the initial coordinates of point [tex]\( C \)[/tex] which are [tex]\( (5, -2) \)[/tex].
#### i. Reflection in the [tex]\( x \)[/tex]-axis
A reflection in the [tex]\( x \)[/tex]-axis changes the sign of the [tex]\( y \)[/tex]-coordinate while keeping the [tex]\( x \)[/tex]-coordinate the same.
So, [tex]\( C^{\prime} \)[/tex] after reflection in the [tex]\( x \)[/tex]-axis will be:
[tex]\[ (5, 2) \][/tex]
#### ii. Reflection in the [tex]\( y \)[/tex]-axis
A reflection in the [tex]\( y \)[/tex]-axis changes the sign of the [tex]\( x \)[/tex]-coordinate while keeping the [tex]\( y \)[/tex]-coordinate the same.
Thus, [tex]\( C^{\prime} \)[/tex] after reflection in the [tex]\( y \)[/tex]-axis will be:
[tex]\[ (-5, -2) \][/tex]
#### iii. Translation 7 units to the left and 4 units up
For translation, you subtract 7 from the [tex]\( x \)[/tex]-coordinate and add 4 to the [tex]\( y \)[/tex]-coordinate.
Hence, [tex]\( C^{\prime} \)[/tex] after the translation will be:
[tex]\[ (5 - 7, -2 + 4) \][/tex]
[tex]\[ (-2, 2) \][/tex]
### Part 2: Transformations of Point [tex]\( A \)[/tex]
The initial coordinates of point [tex]\( A \)[/tex] are [tex]\( (-1, 4) \)[/tex].
#### i. Transformation to [tex]\( A^{\prime}(-5, -1) \)[/tex]
We need to determine the rule that transforms [tex]\( A \)[/tex] to [tex]\( A^{\prime} \)[/tex].
By comparing the coordinates:
[tex]\[ (-5, -1) \][/tex]
with the initial coordinates [tex]\((-1, 4)\)[/tex], we find the changes in [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ \Delta x = -5 - (-1) = -4 \][/tex]
[tex]\[ \Delta y = -1 - 4 = -5 \][/tex]
The transformation rule is a translation by [tex]\((-4, -5)\)[/tex].
#### ii. Transformation to [tex]\( A^{\prime}(1, 4) \)[/tex]
For this transformation, we observe the change from [tex]\((-1, 4)\)[/tex] to [tex]\((1, 4)\)[/tex].
The [tex]\( y \)[/tex]-coordinate remains the same, and the [tex]\( x \)[/tex]-coordinate is the negative of the initial [tex]\( x \)[/tex]-coordinate. Therefore, this is a reflection in the [tex]\( y \)[/tex]-axis.
#### iii. Transformation to [tex]\( A^{\prime}(-1, -4) \)[/tex]
Here, we look at the change from [tex]\((-1, 4)\)[/tex] to [tex]\((-1, -4)\)[/tex].
The [tex]\( x \)[/tex]-coordinate stays the same, and the [tex]\( y \)[/tex]-coordinate is the negative of the initial [tex]\( y \)[/tex]-coordinate. Therefore, this transformation is a reflection in the [tex]\( x \)[/tex]-axis.
### Summary of Results
#### Transformations for Point [tex]\( C \)[/tex]:
1. Reflection in the [tex]\( x \)[/tex]-axis: [tex]\( C^{\prime} = (5, 2) \)[/tex]
2. Reflection in the [tex]\( y \)[/tex]-axis: [tex]\( C^{\prime} = (-5, -2) \)[/tex]
3. Translation 7 units left and 4 units up: [tex]\( C^{\prime} = (-2, 2) \)[/tex]
#### Transformations and Rules for Point [tex]\( A \)[/tex]:
1. To [tex]\( A^{\prime}(-5, -1) \)[/tex]: Translation by [tex]\( (-4, -5) \)[/tex]
2. To [tex]\( A^{\prime}(1, 4) \)[/tex]: Reflection in the [tex]\( y \)[/tex]-axis
3. To [tex]\( A^{\prime}(-1, -4) \)[/tex]: Reflection in the [tex]\( x \)[/tex]-axis
