Question
The function f(x)=x2 is transformed 3 times to becomef(−x+4)+3 . Select the correct transformations below. The function was translated 4 units and 3 units , and reflected over the axis.
Answers
Answered by
GPT-4o mini
Let's break down the transformations applied to the function \( f(x) = x^2 \) to reach the final function \( f(-x + 4) + 3 \).
1. **Start with the original function**:
\[
f(x) = x^2
\]
2. **Reflect over the y-axis**:
The transformation that reflects the function over the y-axis is done by replacing \( x \) with \( -x \):
\[
f(-x) = (-x)^2 = x^2
\]
So, reflecting \( f \) does not change its appearance because \( x^2 \) is symmetric with respect to the y-axis.
3. **Horizontal translation (4 units to the right)**:
To translate the function 4 units to the right, we replace \( x \) with \( x - 4 \):
\[
f(x - 4) = (x - 4)^2
\]
4. **Vertical translation (3 units up)**:
To translate the function 3 units up, we add \( 3 \) to the function:
\[
f(x - 4) + 3 = (x - 4)^2 + 3
\]
Putting it all together, our transformations transform \( f(x) = x^2 \) into \( f(-x + 4) + 3 \):
- **Reflection over the y-axis**: This occurs first, creating \( f(-x) \).
- **Horizontal translation**: Moving right 4 units results in \( f(-x + 4) \).
- **Vertical translation**: Moving up 3 units gives \( f(-x + 4) + 3 \).
In conclusion, the correct transformations are:
- The function is reflected over the y-axis, but for \( x^2 \) this does not visibly change it.
- The function is translated 4 units to the right.
- The function is translated 3 units up.
So the statement about its transformations is somewhat misleading since it does not reflect visibly, but to check transformations mentioned:
1. Translated 4 units (to the right)
2. Translated 3 units (up)
3. Reflected (over the y-axis)
Thus, the correct interpretation of your transformations is:
- The function experiences a reflection, a translation right by 4 units, and a translation up by 3 units.
**Please note**: While the final destination function indicates transformations, the reflection part does not alter the shape of the graph, but it is included in the transformation steps.
1. **Start with the original function**:
\[
f(x) = x^2
\]
2. **Reflect over the y-axis**:
The transformation that reflects the function over the y-axis is done by replacing \( x \) with \( -x \):
\[
f(-x) = (-x)^2 = x^2
\]
So, reflecting \( f \) does not change its appearance because \( x^2 \) is symmetric with respect to the y-axis.
3. **Horizontal translation (4 units to the right)**:
To translate the function 4 units to the right, we replace \( x \) with \( x - 4 \):
\[
f(x - 4) = (x - 4)^2
\]
4. **Vertical translation (3 units up)**:
To translate the function 3 units up, we add \( 3 \) to the function:
\[
f(x - 4) + 3 = (x - 4)^2 + 3
\]
Putting it all together, our transformations transform \( f(x) = x^2 \) into \( f(-x + 4) + 3 \):
- **Reflection over the y-axis**: This occurs first, creating \( f(-x) \).
- **Horizontal translation**: Moving right 4 units results in \( f(-x + 4) \).
- **Vertical translation**: Moving up 3 units gives \( f(-x + 4) + 3 \).
In conclusion, the correct transformations are:
- The function is reflected over the y-axis, but for \( x^2 \) this does not visibly change it.
- The function is translated 4 units to the right.
- The function is translated 3 units up.
So the statement about its transformations is somewhat misleading since it does not reflect visibly, but to check transformations mentioned:
1. Translated 4 units (to the right)
2. Translated 3 units (up)
3. Reflected (over the y-axis)
Thus, the correct interpretation of your transformations is:
- The function experiences a reflection, a translation right by 4 units, and a translation up by 3 units.
**Please note**: While the final destination function indicates transformations, the reflection part does not alter the shape of the graph, but it is included in the transformation steps.