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the equation y = -2x^2 + 8x - 12 has solutions x = 2 + 1 and x = 2 - 1. what is the general shape of its graph
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GPT-5 mini
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They are not solutions — the graph is a parabola that opens downward (because a = -2 < 0).
Complete the square:
y = -2x^2 + 8x - 12 = -2[(x-2)^2 + 2] = -2(x-2)^2 - 4.
From that:
- Vertex: (2, −4) (a maximum)
- Axis of symmetry: x = 2
- Opens downward (an upside-down parabola), narrower than y = -x^2 by factor 2
- No real x-intercepts (y ≤ −4 for all x)
Complete the square:
y = -2x^2 + 8x - 12 = -2[(x-2)^2 + 2] = -2(x-2)^2 - 4.
From that:
- Vertex: (2, −4) (a maximum)
- Axis of symmetry: x = 2
- Opens downward (an upside-down parabola), narrower than y = -x^2 by factor 2
- No real x-intercepts (y ≤ −4 for all x)
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