Problem

Use the displayed graph to interpret \((x-4)^{2}\) in \(f(x)=(x-4)^{2}+7\). Identify the center, signed difference, absolute distance, squared-distance term, and why inputs equally far on opposite sides have equal outputs.

Unannotated graph of f(x)=(x−4)²+7; the parabola is shown without naming its center, symmetry axis, or equal-output inputs.

Big Picture

What this problem is really about

The binomial inside vertex form measures signed horizontal displacement from its center, while absolute value turns that displacement into distance. We’ll identify the center from x minus h, square the displacement to remove its sign, and compare inputs the same distance to the left and right to explain their equal outputs and the graph’s symmetry.

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Center 4; signed difference x − 4; distance |x − 4|; squared term is interpreted as the linear distance x − 4; inputs 4 − d and 4 + d are claimed to have different outputs.

Center 4; signed difference x − 4; distance x − 4 without absolute value; squared term (x − 4)²; inputs 4 − d and 4 + d are claimed to have different outputs.

Center 4; signed difference x − 4; distance |x − 4|; squared-distance term (x − 4)²; inputs 4 − d and 4 + d have equal outputs.

Center −4; signed difference x − (−4) = x + 4; distance |x + 4|; squared term (x + 4)²; inputs −4 − d and −4 + d have equal outputs.

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Use the displayed graph to interpret \((x-4)^{2}\) in \(f(x)=(x-4)^{2}+7\). Identify the center, signed difference, absolute distance, squared-distance term, and why inputs equally far on opposite sides have equal outputs.

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