Question
On this coordinate plane, point A is translated into point A′. Which functions give the two translations?
Answers
GPT-4o mini
To determine the functions that represent the translations of point A to point A′ on a coordinate plane, we need to know the coordinates of both points and the direction and distance of the translation.
Let's denote point A by its coordinates \( A(x_A, y_A) \) and point A′ by its coordinates \( A'(x_{A'}, y_{A'}) \). The translation from point A to point A′ can be mathematically expressed in terms of functions for the x-coordinate and y-coordinate as follows:
1. **Translation of the x-coordinate**:
\[
x_{A'} = x_A + t_x
\]
where \( t_x \) is the horizontal translation distance (could be positive or negative, depending on the direction).
2. **Translation of the y-coordinate**:
\[
y_{A'} = y_A + t_y
\]
where \( t_y \) is the vertical translation distance (also could be positive or negative).
Thus, the functions describing the translation can generally be expressed as:
\[
f(x) = x + t_x
\]
\[
g(y) = y + t_y
\]
Without the specific coordinates of points A and A′, we can't provide the actual values for \( t_x \) and \( t_y \). However, if you can provide the coordinates of points A and A′, I can help calculate the specific translation values.
Let's denote point A by its coordinates \( A(x_A, y_A) \) and point A′ by its coordinates \( A'(x_{A'}, y_{A'}) \). The translation from point A to point A′ can be mathematically expressed in terms of functions for the x-coordinate and y-coordinate as follows:
1. **Translation of the x-coordinate**:
\[
x_{A'} = x_A + t_x
\]
where \( t_x \) is the horizontal translation distance (could be positive or negative, depending on the direction).
2. **Translation of the y-coordinate**:
\[
y_{A'} = y_A + t_y
\]
where \( t_y \) is the vertical translation distance (also could be positive or negative).
Thus, the functions describing the translation can generally be expressed as:
\[
f(x) = x + t_x
\]
\[
g(y) = y + t_y
\]
Without the specific coordinates of points A and A′, we can't provide the actual values for \( t_x \) and \( t_y \). However, if you can provide the coordinates of points A and A′, I can help calculate the specific translation values.
A is on the 3,-3 ans a' is on the 3,5
GPT-4o mini
Given the coordinates:
- Point \( A(3, -3) \)
- Point \( A'(3, 5) \)
We can find the translations in the x and y directions.
1. **Translation of the x-coordinate:**
- The x-coordinate remains the same.
- Thus, the translation for the x-coordinate is:
\[
x_{A'} = x_A + t_x \implies 3 = 3 + t_x \implies t_x = 0
\]
2. **Translation of the y-coordinate:**
- The y-coordinate changes from -3 to 5.
- Thus, the translation for the y-coordinate is:
\[
y_{A'} = y_A + t_y \implies 5 = -3 + t_y \implies t_y = 5 + 3 = 8
\]
Now, we have our translations:
- The function for the x-coordinate is:
\[
f(x) = x + 0 = x
\]
- The function for the y-coordinate is:
\[
g(y) = y + 8
\]
In summary, the translations from point A to point A′ can be represented by the following functions:
- **X translation:** \( f(x) = x \) (no change)
- **Y translation:** \( g(y) = y + 8 \) (an upward translation of 8 units)
- Point \( A(3, -3) \)
- Point \( A'(3, 5) \)
We can find the translations in the x and y directions.
1. **Translation of the x-coordinate:**
- The x-coordinate remains the same.
- Thus, the translation for the x-coordinate is:
\[
x_{A'} = x_A + t_x \implies 3 = 3 + t_x \implies t_x = 0
\]
2. **Translation of the y-coordinate:**
- The y-coordinate changes from -3 to 5.
- Thus, the translation for the y-coordinate is:
\[
y_{A'} = y_A + t_y \implies 5 = -3 + t_y \implies t_y = 5 + 3 = 8
\]
Now, we have our translations:
- The function for the x-coordinate is:
\[
f(x) = x + 0 = x
\]
- The function for the y-coordinate is:
\[
g(y) = y + 8
\]
In summary, the translations from point A to point A′ can be represented by the following functions:
- **X translation:** \( f(x) = x \) (no change)
- **Y translation:** \( g(y) = y + 8 \) (an upward translation of 8 units)