Problem

For \(f(x)=3(x-2)^{2}+5\), identify \(a\), \(h\), and \(k\) in \(a(x-h)^{2}+k\). Use them to state the vertex, symmetry axis, opening direction, and vertical scale, then verify the vertex value by substitution.

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What this problem is really about

Vertex form exposes the horizontal shift, vertical shift, and scale in separate parameters. We’ll match the expression with a times the quantity x minus h squared plus k, use h and k for the vertex and axis, interpret a for opening and width, and substitute the vertex input as a check.

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a = 3, h = 2, and k = 5, so the vertex is (2, 5), the axis is x = 2, and the parabola opens up with a vertical stretch by 3. Also f(2) = 5.

The vertex is (5, 2) and the axis is x = 5. The coordinates h and k have been swapped.

The vertex is (−2, 5) and the axis is x = −2. In (x − h)², h = 2, not −2.

The vertex is (2, 5) and it opens up with the parent-function width. The factor 3 makes the parabola narrower.

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For \(f(x)=3(x-2)^{2}+5\), identify \(a\), \(h\), and \(k\) in \(a(x-h)^{2}+k\). Use them to state the vertex, symmetry axis, opening direction, and vertical scale, then verify the vertex value by substitution.

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