Problem

Interpret \(3(x-2)^{2}\) within \(f(x)=3(x-2)^{2}+1\). Identify the center and squared distance, explain the positive scale and vertical contribution from the vertex level, and infer opening and relative width.

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What this problem is really about

The squared binomial measures squared distance from the center, and its coefficient scales vertical change from the vertex level. We’ll isolate that output difference, use nonnegativity and the coefficient’s sign to determine the opening, and use the coefficient’s magnitude to compare the parabola’s width with the parent graph.

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Center 2; squared distance (x − 2)²; a = 3; contribution +3(x − 2)²; f(x) − 1 ≥ 0; opens up and is wider than y = x².

Center 6; squared distance (x − 6)²; a = 3; contribution +3(x − 6)²; f(x) − 1 = 3(x − 6)² ≥ 0; opens up and is narrower than y = x².

Center 2; squared distance −(x − 2)²; a = −3; contribution −3(x − 2)²; f(x) − 1 ≤ 0; opens down and is narrower than y = x².

Center 2; squared distance (x − 2)²; a = 3; contribution +3(x − 2)²; f(x) − 1 = 3(x − 2)² ≥ 0; opens up and is narrower than y = x².

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Interpret \(3(x-2)^{2}\) within \(f(x)=3(x-2)^{2}+1\). Identify the center and squared distance, explain the positive scale and vertical contribution from the vertex level, and infer opening and relative width.

What this problem is really about

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