IDNLearn.com provides a seamless experience for finding accurate answers. Our Q&A platform offers detailed and trustworthy answers to ensure you have the information you need.
Sagot :
### 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
Thank you for using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. Your search for answers ends at IDNLearn.com. Thanks for visiting, and we look forward to helping you again soon.
