To determine the parent function and the equation of the transformed function for the parabola, we first need to identify the standard form of the parent function and how the transformation has been applied.
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Identify the Parent Function: The parent function for a parabola is typically \( y = x^2 \).
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Identify Transformations: The transformed function typically takes the form \( y = a(x-h)^2 + k \), where:
- \( h \) represents a horizontal shift,
- \( k \) represents a vertical shift,
- \( a \) determines vertical stretch or compression.
Since I cannot see the graph, let's hypothesize based on common transformations:
- If the parabola is shifted horizontally to the right by 2 units, then the equation would be \( y = (x - 2)^2 \).
- If instead it is shifted horizontally to the left by 2 units, it would be \( y = (x + 2)^2 \).
Based on this understanding, let's evaluate the options:
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The parent function is \( y = x^2 \). The equation of the transformed function is \( y = (x - 2)^2 \).
- This implies a right shift.
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The parent function is \( y = (x + 2)^2 \). The equation of the transformed function is \( y = x^2 \).
- This implies shifting the parent function to the left, which is not a standard transformed equation.
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The parent function is \( y = x^2 \). The equation of the transformed function is \( y = (x + 2)^2 \).
- This implies left shifting the parabola.
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The parent function is \( y = (x - 2)^2 \). The equation of the transformed function is \( y = x^2 \).
- Similar to option 2, this is not a typical form for transformations.
Conclusion
Based on the common transformations, the correct answer should match the transformation applied correctly. If the parabola is indeed shifted to the right, the first option is correct:
The parent function is \( y = x^2 \). The equation of the transformed function is \( y = (x - 2)^2 \).
If it is shifted on the left, then the third option would be applicable. Check what meaning the graph has for clarity.