Problem

Analyze \(f(x)=(x-2)^{2}-3\) by stating its vertex, axis, opening, scale, and extremum. Evaluate inputs \(\text{one}~\text{unit}\) to either side of the axis and report the resulting symmetric point pair.

Big Picture

What this problem is really about

Vertex form supplies the center and scale, while symmetry lets one offset predict its matching point. We’ll read the vertex, axis, opening, width, and extremum, then evaluate inputs the same distance from the axis and pair them with their common output.

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Preview only

The vertex is (2, −3) and f(1) = f(3) = 1. Substituting gives (−1)² − 3 = −2.

The vertex is (2, −3), axis x = 2, and the parabola opens up with parent-function width. It has minimum −3; f(1) = f(3) = −2, so (1, −2) and (3, −2) are symmetric points.

The vertex is (−2, −3), with axis x = −2. In (x − 2)², the horizontal shift is right 2, not left 2.

The vertex is (2, −3) and the parabola opens down with maximum −3. The leading coefficient is positive, so it opens up.

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Analyze \(f(x)=(x-2)^{2}-3\) by stating its vertex, axis, opening, scale, and extremum. Evaluate inputs \(\text{one}~\text{unit}\) to either side of the axis and report the resulting symmetric point pair.

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