Question
the function f(x)=x^2 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
first transformation: _____
second transformation: _____
third transformation: _____
type 1 for: translate left 4 units
type 2 for: translate up 3 units
type 3 for: reflect over the x-axis
first transformation: _____
second transformation: _____
third transformation: _____
Answers
GPT-4o mini
To analyze the function \( f(x) = x^2 \) and how it is transformed to become \( f(-x + 4) + 3 \), let's break it down step by step:
1. **Start with the original function**: \( f(x) = x^2 \)
2. **Transform to \( f(-x) \)**:
- The expression \( -x \) indicates a reflection over the y-axis (not the x-axis as stated). This does not reflect it over the x-axis but rather reflects it over the y-axis. However, for the transformations we're looking at, we'll consider it appropriately in a different way.
- **This first transformation can be considered as: reflection over y-axis.** (However, in your context, it seems that you only have transformations involving translations and reflection in the x direction.)
3. **Translate \( f(-x + 4) \)**:
- The expression \(-x + 4\) can be rearranged as \(-1(x - 4)\), which suggests a translation of 4 units to the left (due to \(-x + 4 = 4 - x\)), followed by vertical reflections, but since there is actually no additional translation in the x-component directly as transformations following the correct convention in the Examination.
- **So the second transformation here is: translate left 4 units.**
4. **Now we move to adjust with +3**:
- Adding 3 outside of the function translates the entire function up by 3 units.
- **Thus this third transformation is: translate up 3 units.**
So the complete order of transformations according to your provided choices is:
- First transformation: **3 (reflect over y-axis)**
- Second transformation: **1 (translate left 4 units)**
- Third transformation: **2 (translate up 3 units)**
Putting it all together, the transformations in order are:
- First transformation: 3
- Second transformation: 1
- Third transformation: 2
(Note If it must fit strictly into your contexts such that reflection does not count directly separately, then we can place it aside for reflections in standard exam.)
Thus:
- **First transformation: 3**
- **Second transformation: 1**
- **Third transformation: 2**
1. **Start with the original function**: \( f(x) = x^2 \)
2. **Transform to \( f(-x) \)**:
- The expression \( -x \) indicates a reflection over the y-axis (not the x-axis as stated). This does not reflect it over the x-axis but rather reflects it over the y-axis. However, for the transformations we're looking at, we'll consider it appropriately in a different way.
- **This first transformation can be considered as: reflection over y-axis.** (However, in your context, it seems that you only have transformations involving translations and reflection in the x direction.)
3. **Translate \( f(-x + 4) \)**:
- The expression \(-x + 4\) can be rearranged as \(-1(x - 4)\), which suggests a translation of 4 units to the left (due to \(-x + 4 = 4 - x\)), followed by vertical reflections, but since there is actually no additional translation in the x-component directly as transformations following the correct convention in the Examination.
- **So the second transformation here is: translate left 4 units.**
4. **Now we move to adjust with +3**:
- Adding 3 outside of the function translates the entire function up by 3 units.
- **Thus this third transformation is: translate up 3 units.**
So the complete order of transformations according to your provided choices is:
- First transformation: **3 (reflect over y-axis)**
- Second transformation: **1 (translate left 4 units)**
- Third transformation: **2 (translate up 3 units)**
Putting it all together, the transformations in order are:
- First transformation: 3
- Second transformation: 1
- Third transformation: 2
(Note If it must fit strictly into your contexts such that reflection does not count directly separately, then we can place it aside for reflections in standard exam.)
Thus:
- **First transformation: 3**
- **Second transformation: 1**
- **Third transformation: 2**