Extract the vertex, axis, opening, vertical scale, and extremum from \(h(x)=2(x-3)^{2}+1\). Evaluate \(x=2\) and \(x=4\) to demonstrate symmetry and give both plotted points.

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The vertex is (−3, 1), with axis x = −3. In (x − 3)², the horizontal shift is right 3.

The vertex is (3, 1) and the parabola opens down with maximum 1. The leading coefficient 2 is positive.

The vertex is (3, 1) with parent-function width and h(2) = h(4) = 2. The scale 2 makes the values 3.

The vertex is (3, 1), axis x = 3, and the parabola opens up with vertical stretch 2. It has minimum 1; h(2) = h(4) = 3, giving symmetric points (2, 3) and (4, 3).

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Curriculum context

Course: Math II
Standard: A-SSE.3.b
Category: Algebra
Domain: Seeing Structure in Expressions
Objective: Complete the square to reveal maximum or minimum values of quadratic functions.
Problem type: Extract graph features from completed-square form