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what kind of transformation converts the graph of f(x) = -(x+1)^2 into the graph of g(x) = -9 (x+1) ^2
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what kind of transformation converts the graph of f(x) = -(x+1)^2 into the graph of g(x) = -9 (x+1) ^2
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GPT-4o mini
To determine the transformation that converts the graph of \( f(x) = -(x+1)^2 \) into the graph of \( g(x) = -9(x+1)^2 \), we should analyze the modifications made to the original function.
1. **Vertical Stretch/Compression**: The main difference between \( f(x) \) and \( g(x) \) is the coefficient of the squared term. The function \( g(x) \) has a factor of -9 instead of -1.
- The factor of -9 indicates that the graph of \( f(x) \) is not only reflected across the x-axis (the negative sign) but also vertically stretched by a factor of 9.
- In general, multiplying a function by a value greater than 1 (in absolute terms) will stretch it vertically, while multiplying by a value between 0 and 1 compresses it.
Thus, the transformation from \( f(x) = -(x+1)^2 \) to \( g(x) = -9(x+1)^2 \) involves:
- A vertical stretch by a factor of 9.
In summary, the transformation is a vertical stretch by a factor of 9.
1. **Vertical Stretch/Compression**: The main difference between \( f(x) \) and \( g(x) \) is the coefficient of the squared term. The function \( g(x) \) has a factor of -9 instead of -1.
- The factor of -9 indicates that the graph of \( f(x) \) is not only reflected across the x-axis (the negative sign) but also vertically stretched by a factor of 9.
- In general, multiplying a function by a value greater than 1 (in absolute terms) will stretch it vertically, while multiplying by a value between 0 and 1 compresses it.
Thus, the transformation from \( f(x) = -(x+1)^2 \) to \( g(x) = -9(x+1)^2 \) involves:
- A vertical stretch by a factor of 9.
In summary, the transformation is a vertical stretch by a factor of 9.