Problem

Analyze \(y=2(x+3)^{2}-5\) relative to \(y=x^{2}\). Identify \(a\), \(h\), and \(k\), give the vertex and axis of symmetry, and explain the shifts, opening direction, vertical stretch, and resulting width.

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It is a wide upward-opening parabola with vertex (-3, -5) because the graph shifts left 3 and down 5, and multiplying by 2 stretches it outward.

It is a narrow upward-opening parabola with vertex (3, -5) because the +3 inside the parentheses gives the x-coordinate directly, and the 2 makes the parabola narrower.

It is a narrow downward-opening parabola with vertex (-3, -5) because the graph shifts left 3 and down 5, and the -5 means the parabola opens downward.

It is a narrow upward-opening parabola with vertex (-3, -5) because y=2(x+3)²-5 is in vertex form, so x+3 means left 3, down 5, and the 2 makes the parabola narrower while still opening upward.

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Analyze \(y=2(x+3)^{2}-5\) relative to \(y=x^{2}\). Identify \(a\), \(h\), and \(k\), give the vertex and axis of symmetry, and explain the shifts, opening direction, vertical stretch, and resulting width.

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