Question
y = ( -x - 2)^2
f (x) = ( x - 2)^2
(3, 1)
The left parabola is a reflection of the right parabola. Enter the coordinate on the left parabola that corresponds to the identified coordinate on the right parabola.
( _, _ )
f (x) = ( x - 2)^2
(3, 1)
The left parabola is a reflection of the right parabola. Enter the coordinate on the left parabola that corresponds to the identified coordinate on the right parabola.
( _, _ )
Answers
GPT-4o mini
To find the corresponding coordinate on the left parabola given the point (3, 1) on the right parabola, we need to analyze the equations of the parabolas.
1. **Identify the right parabola**:
The right parabola is given by \( f(x) = (x - 2)^2 \).
2. **Evaluate \( f(3) \)**:
\[
f(3) = (3 - 2)^2 = (1)^2 = 1
\]
The point (3, 1) is indeed on the right parabola.
3. **Identify the left parabola**:
The left parabola is given by \( y = (-x - 2)^2 \), which can be rewritten as \( y = (-(x + 2))^2 \) or \( y = (-1)^2 \cdot (x + 2)^2 \).
4. **Determine the relation between the parabolas**:
The left parabola is a reflection of the right parabola across the line \( x = -2 \). To find the corresponding x-coordinate for the point (3, 1):
5. **Calculate the reflection**:
The line \( x = -2 \) is the axis of reflection. The distance between the point 3 and this line is calculated as follows:
\[
d = 3 - (-2) = 3 + 2 = 5
\]
To find the reflected x-coordinate, subtract this distance from -2:
\[
x' = -2 - 5 = -7
\]
6. **Determine the y-coordinate on the left parabola \( y = (-x - 2)^2 \)**:
Use \( x' = -7 \) to find the corresponding y-coordinate:
\[
y' = (-(-7) - 2)^2 = (7 - 2)^2 = (5)^2 = 25
\]
Thus, the coordinate on the left parabola that corresponds to the point (3, 1) on the right parabola is:
\[
(-7, 25)
\]
Therefore, the answer is:
\[
(-7, 25)
\]
1. **Identify the right parabola**:
The right parabola is given by \( f(x) = (x - 2)^2 \).
2. **Evaluate \( f(3) \)**:
\[
f(3) = (3 - 2)^2 = (1)^2 = 1
\]
The point (3, 1) is indeed on the right parabola.
3. **Identify the left parabola**:
The left parabola is given by \( y = (-x - 2)^2 \), which can be rewritten as \( y = (-(x + 2))^2 \) or \( y = (-1)^2 \cdot (x + 2)^2 \).
4. **Determine the relation between the parabolas**:
The left parabola is a reflection of the right parabola across the line \( x = -2 \). To find the corresponding x-coordinate for the point (3, 1):
5. **Calculate the reflection**:
The line \( x = -2 \) is the axis of reflection. The distance between the point 3 and this line is calculated as follows:
\[
d = 3 - (-2) = 3 + 2 = 5
\]
To find the reflected x-coordinate, subtract this distance from -2:
\[
x' = -2 - 5 = -7
\]
6. **Determine the y-coordinate on the left parabola \( y = (-x - 2)^2 \)**:
Use \( x' = -7 \) to find the corresponding y-coordinate:
\[
y' = (-(-7) - 2)^2 = (7 - 2)^2 = (5)^2 = 25
\]
Thus, the coordinate on the left parabola that corresponds to the point (3, 1) on the right parabola is:
\[
(-7, 25)
\]
Therefore, the answer is:
\[
(-7, 25)
\]