The given equation is \( y = -1(x + 3)^2 \). This is a transformation of the parent function \( y = x^2 \).
To analyze the transformations:
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Horizontal Shift: The expression \( (x + 3) \) indicates a horizontal shift to the left by 3 units. This is because for a general transformation \( y = a(x - h)^2 \), the term \( h \) shifts the graph. Here, since we have \( x + 3 \), it implies a shift left (negative direction).
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Vertical Reflection: The coefficient \( a = -1 \) reflects the graph over the x-axis. This means every \( y \) value will be inverted; for instance, if the parent function \( y = x^2 \) has a value of 4 for \( x = 2 \), the transformed function will have a value of -4 at the same \( x \).
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Vertical Stretch/Compression: In this case, \( a = -1 \) does not indicate a vertical stretch or compression since the absolute value of \( a \) is equal to 1. Thus, there is no vertical stretching or compressing.
In summary, the transformations on the parent function \( y = x^2 \) are:
- Shift left by 3 units.
- Reflect over the x-axis.
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