Let's analyze the transformation needed to go from \( f(x) = x^2 \) to \( g(x) = 3(x + 2)^2 - 1 \).
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Identifying Transformations:
- The term \( (x + 2) \) indicates a horizontal shift. Since it’s \( x + 2 \), the graph shifts to the left by 2 units.
- The coefficient \( 3 \) in front of \( (x + 2)^2 \) stretches the graph vertically. Because it is greater than 1, the graph of \( g(x) \) is narrower than \( f(x) \).
- The term \( -1 \) at the end indicates a vertical shift down by 1 unit.
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Combining Transformations:
- Start with \( f(x) = x^2 \).
- Shift left 2 units, yielding \( f(x + 2) = (x + 2)^2 \).
- Then, apply the vertical stretch by 3: \( g(x) = 3(x + 2)^2 \).
- Finally, shift down 1 unit: \( g(x) = 3(x + 2)^2 - 1 \).
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Conclusion:
- So the transformations needed to form \( g(x) \) from \( f(x) \) are:
- Shift to the left 2 units.
- Vertical stretch: narrower than \( f(x) \).
- Shift down 1 unit.
- So the transformations needed to form \( g(x) \) from \( f(x) \) are:
Therefore, the correct description of the transformation needed is: The graph of g(x) is narrower than f(x) and is shifted to the left 2 units and down 1 unit.
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