Bot GPT 4-o mini To find the correct equation that matches the given points \((0, -4)\), \((-1, -2)\), \((1, -2)\), \((-2, 4)\), and \((2, 4)\), we need to start with the transformation of the function \(y = 2x^2\).
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Identify the vertex form and its transformations: The standard upward opening parabola is \(y = 2(x - h)^2 + k\), where \((h, k)\) is the vertex of the parabola.
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Determine the vertex: From the points given, particularly \((0, -4)\), it seems the vertex might be at \((0, -4)\).
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Vertical Shift: Since the graph appears to open upwards and the lowest point (vertex) is \(-4\), we can infer that the entire parabola is shifted downwards by \(4\) units.
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Building the equation: The equation then becomes:
\[ y = 2x^2 - 4 \]
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Verification: Let's check if this equation passes through the given points.
- For \((0, -4)\): \[ y = 2(0)^2 - 4 = -4 \quad \text{(correct)} \]
- For \((-1, -2)\): \[ y = 2(-1)^2 - 4 = 2(1) - 4 = -2 \quad \text{(correct)} \]
- For \((1, -2)\): \[ y = 2(1)^2 - 4 = 2(1) - 4 = -2 \quad \text{(correct)} \]
- For \((-2, 4)\): \[ y = 2(-2)^2 - 4 = 2(4) - 4 = 8 - 4 = 4 \quad \text{(correct)} \]
- For \((2, 4)\): \[ y = 2(2)^2 - 4 = 2(4) - 4 = 8 - 4 = 4 \quad \text{(correct)} \]
Since all calculations confirm that the points fit this equation, the correct transformation for the function to match the graph with the specified points is:
\[ \boxed{y = 2x^2 - 4} \]
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