Question
The function created from transformations of apparent function
Answers
GPT-4o mini
It seems like you are asking about the process of creating a new function by applying transformations to an "apparent function" (possibly a basic or a parent function). In mathematics, function transformations involve changing the position, size, or shape of a function's graph. Here are some common transformations:
1. **Vertical Shifts**:
- **Addition**: If you have a function \( f(x) \) and you create a new function \( g(x) = f(x) + k \), the graph of \( g(x) \) will be the graph of \( f(x) \) shifted vertically by \( k \) units. If \( k \) is positive, it shifts up; if negative, it shifts down.
2. **Horizontal Shifts**:
- **Addition**: If \( g(x) = f(x - h) \), the graph of \( g(x) \) will be shifted horizontally. If \( h \) is positive, it shifts to the right; if negative, it shifts to the left.
3. **Vertical Stretch/Compression**:
- **Multiplication**: If \( g(x) = af(x) \), where \( a > 1 \), the graph of \( g(x) \) is a vertical stretch of \( f(x) \). If \( 0 < a < 1 \), it compresses the graph vertically.
4. **Horizontal Stretch/Compression**:
- If \( g(x) = f(bx) \), where \( b > 1 \), the graph of \( g(x) \) is a horizontal compression of \( f(x) \). If \( 0 < b < 1 \), it stretches the graph horizontally.
5. **Reflections**:
- **About the x-axis**: If \( g(x) = -f(x) \), this reflects the graph of \( f(x) \) across the x-axis.
- **About the y-axis**: If \( g(x) = f(-x) \), this reflects the graph of \( f(x) \) across the y-axis.
### Example:
Let’s consider the function \( f(x) = x^2 \) (a parabola).
1. **Vertical Shift**: \( g(x) = x^2 + 3 \) shifts the parabola up by 3 units.
2. **Horizontal Shift**: \( g(x) = (x - 2)^2 \) shifts it to the right by 2 units.
3. **Vertical Stretch**: \( g(x) = 2x^2 \) stretches the parabola vertically by a factor of 2.
4. **Horizontal Compression**: \( g(x) = (2x)^2 \) compresses it horizontally by a factor of 2.
5. **Reflection**: \( g(x) = -x^2 \) reflects the parabola across the x-axis.
By combining these transformations, you can create a wide variety of new functions based on the transformations of a base function. If you have a specific apparent function and specific transformations in mind, please provide those details!
1. **Vertical Shifts**:
- **Addition**: If you have a function \( f(x) \) and you create a new function \( g(x) = f(x) + k \), the graph of \( g(x) \) will be the graph of \( f(x) \) shifted vertically by \( k \) units. If \( k \) is positive, it shifts up; if negative, it shifts down.
2. **Horizontal Shifts**:
- **Addition**: If \( g(x) = f(x - h) \), the graph of \( g(x) \) will be shifted horizontally. If \( h \) is positive, it shifts to the right; if negative, it shifts to the left.
3. **Vertical Stretch/Compression**:
- **Multiplication**: If \( g(x) = af(x) \), where \( a > 1 \), the graph of \( g(x) \) is a vertical stretch of \( f(x) \). If \( 0 < a < 1 \), it compresses the graph vertically.
4. **Horizontal Stretch/Compression**:
- If \( g(x) = f(bx) \), where \( b > 1 \), the graph of \( g(x) \) is a horizontal compression of \( f(x) \). If \( 0 < b < 1 \), it stretches the graph horizontally.
5. **Reflections**:
- **About the x-axis**: If \( g(x) = -f(x) \), this reflects the graph of \( f(x) \) across the x-axis.
- **About the y-axis**: If \( g(x) = f(-x) \), this reflects the graph of \( f(x) \) across the y-axis.
### Example:
Let’s consider the function \( f(x) = x^2 \) (a parabola).
1. **Vertical Shift**: \( g(x) = x^2 + 3 \) shifts the parabola up by 3 units.
2. **Horizontal Shift**: \( g(x) = (x - 2)^2 \) shifts it to the right by 2 units.
3. **Vertical Stretch**: \( g(x) = 2x^2 \) stretches the parabola vertically by a factor of 2.
4. **Horizontal Compression**: \( g(x) = (2x)^2 \) compresses it horizontally by a factor of 2.
5. **Reflection**: \( g(x) = -x^2 \) reflects the parabola across the x-axis.
By combining these transformations, you can create a wide variety of new functions based on the transformations of a base function. If you have a specific apparent function and specific transformations in mind, please provide those details!