Question
Question
The function f(x)=x2
is transformed 3 times to become f(−x+4)+3
. Place the transformations in the order in which they occurred.
Type 1 for: translate left 4 units
Type 2 for: translate up 3 units
Type 3 for: reflect over the x-axis
(2 points)
First Transformation:
Second Transformation:
Third Transformation:
The function f(x)=x2
is transformed 3 times to become f(−x+4)+3
. Place the transformations in the order in which they occurred.
Type 1 for: translate left 4 units
Type 2 for: translate up 3 units
Type 3 for: reflect over the x-axis
(2 points)
First Transformation:
Second Transformation:
Third Transformation:
Answers
GPT-4o mini
To find out the order of transformations that take the function \( f(x) = x^2 \) to \( f(-x + 4) + 3 \), we analyze the transformations step by step:
1. **Reflecting over the x-axis**: The original function \( f(x) = x^2 \) transforms to \( f(-x) = (-x)^2 = x^2 \) when reflected. However, to reflect over the x-axis, we would write it as \( -f(x) = -x^2 \). In our function, there's no reflection taking place, thus this step is not applicable.
2. **Translating left 4 units**: The expression \( f(-x + 4) \) can be analyzed to find the impact of the function. The inner part of the function gives us a hint:
\[
-x + 4
\]
This indicates a horizontal shift to the left. To get from \( x \) to \( -x + 4 \), we treat it as
\[
f(x) \to f(x - 4)
\]
which means we first translate the function left by 4 units.
3. **Translating up 3 units**: After performing the horizontal shift, we need to translate the entire function up by 3 units as indicated by the \( +3 \) at the end of the function.
Now we can summarize the transformations in the order they occurred:
- **First Transformation**: Translate left 4 units (Type 1)
- **Second Transformation**: Translate up 3 units (Type 2)
Since there is no reflection happening in this transformation, the reflection (Type 3) does not apply here based on the expression \( f(-x + 4) + 3 \).
Thus:
- **First Transformation**: 1
- **Second Transformation**: 2
- **Third Transformation**: (Not applicable; no reflection in this case)
Final answer:
- First Transformation: 1
- Second Transformation: 2
- Third Transformation: (No reflection, so leave blank or N/A)
1. **Reflecting over the x-axis**: The original function \( f(x) = x^2 \) transforms to \( f(-x) = (-x)^2 = x^2 \) when reflected. However, to reflect over the x-axis, we would write it as \( -f(x) = -x^2 \). In our function, there's no reflection taking place, thus this step is not applicable.
2. **Translating left 4 units**: The expression \( f(-x + 4) \) can be analyzed to find the impact of the function. The inner part of the function gives us a hint:
\[
-x + 4
\]
This indicates a horizontal shift to the left. To get from \( x \) to \( -x + 4 \), we treat it as
\[
f(x) \to f(x - 4)
\]
which means we first translate the function left by 4 units.
3. **Translating up 3 units**: After performing the horizontal shift, we need to translate the entire function up by 3 units as indicated by the \( +3 \) at the end of the function.
Now we can summarize the transformations in the order they occurred:
- **First Transformation**: Translate left 4 units (Type 1)
- **Second Transformation**: Translate up 3 units (Type 2)
Since there is no reflection happening in this transformation, the reflection (Type 3) does not apply here based on the expression \( f(-x + 4) + 3 \).
Thus:
- **First Transformation**: 1
- **Second Transformation**: 2
- **Third Transformation**: (Not applicable; no reflection in this case)
Final answer:
- First Transformation: 1
- Second Transformation: 2
- Third Transformation: (No reflection, so leave blank or N/A)