Question
The graph of g(x), shown below in pink, has the same shape as the graph of f(x) = x2, shown in gray.
The equation for g(x) is
5
choose your answer...
へ
choose your answer...
8(x) = (x+1)2-2
g(x) = (x+2)-1
g(x) = (x-1)2 -2
g(x) = (x-2)'-1
The equation for g(x) is
5
choose your answer...
へ
choose your answer...
8(x) = (x+1)2-2
g(x) = (x+2)-1
g(x) = (x-1)2 -2
g(x) = (x-2)'-1
Answers
GPT-4o mini
To determine the correct equation for \( g(x) \) given that it has the same shape as \( f(x) = x^2 \), we need to identify the transformations applied to the basic quadratic function.
The transformations include:
1. **Horizontal shifts**: Moving the graph left or right based on the \( (x - h) \) term.
2. **Vertical shifts**: Moving the graph up or down based on the \( k \) term in \( (x - h)^2 + k \).
3. **Vertical stretches/compressions**: Scaling the graph vertically based on a coefficient in front of the quadratic term.
The function \( f(x) = x^2 \) can be transformed to \( g(x) \) as follows:
- If \( g(x) = a(x - h)^2 + k \), then:
- \( a \) affects the vertical stretch/compression,
- \( h \) affects the horizontal shift,
- \( k \) affects the vertical shift.
### Analyzing the Given Options:
1. \( g(x) = (x + 1)^2 - 2 \): Shift left by 1 and down by 2.
2. \( g(x) = (x + 2) - 1 \): This is linear, not a quadratic function.
3. \( g(x) = (x-1)^2 - 2 \): Shift right by 1 and down by 2.
4. \( g(x) = (x-2)^2 - 1 \): Shift right by 2 and down by 1.
### Conclusion:
Without seeing the graph and knowing the exact transformations, it's difficult to decide which function correctly describes \( g(x) \). If you know the specific transformations based on the context, you can select one accordingly. However, it seems likely that options 1, 3, and 4 are all valid transformations of the parabola, but choice 2 is not.
If you are able to provide additional information about the location of the vertex or the transformations observed in the graph of \( g(x) \), I can help you identify the most appropriate function. Otherwise, based on common transformations:
- If the graph of \( g(x) \) is opened upwards and has been shifted to the left and down (like option 1), that is a suitable choice.
- If it has been shifted right and down (like option 3), then that would be appropriate instead.
So, based on this analysis, you might want to check the graph again for the vertex' position to solidify your choice.
The transformations include:
1. **Horizontal shifts**: Moving the graph left or right based on the \( (x - h) \) term.
2. **Vertical shifts**: Moving the graph up or down based on the \( k \) term in \( (x - h)^2 + k \).
3. **Vertical stretches/compressions**: Scaling the graph vertically based on a coefficient in front of the quadratic term.
The function \( f(x) = x^2 \) can be transformed to \( g(x) \) as follows:
- If \( g(x) = a(x - h)^2 + k \), then:
- \( a \) affects the vertical stretch/compression,
- \( h \) affects the horizontal shift,
- \( k \) affects the vertical shift.
### Analyzing the Given Options:
1. \( g(x) = (x + 1)^2 - 2 \): Shift left by 1 and down by 2.
2. \( g(x) = (x + 2) - 1 \): This is linear, not a quadratic function.
3. \( g(x) = (x-1)^2 - 2 \): Shift right by 1 and down by 2.
4. \( g(x) = (x-2)^2 - 1 \): Shift right by 2 and down by 1.
### Conclusion:
Without seeing the graph and knowing the exact transformations, it's difficult to decide which function correctly describes \( g(x) \). If you know the specific transformations based on the context, you can select one accordingly. However, it seems likely that options 1, 3, and 4 are all valid transformations of the parabola, but choice 2 is not.
If you are able to provide additional information about the location of the vertex or the transformations observed in the graph of \( g(x) \), I can help you identify the most appropriate function. Otherwise, based on common transformations:
- If the graph of \( g(x) \) is opened upwards and has been shifted to the left and down (like option 1), that is a suitable choice.
- If it has been shifted right and down (like option 3), then that would be appropriate instead.
So, based on this analysis, you might want to check the graph again for the vertex' position to solidify your choice.