Problem

Analyze \(y=(x-2)^{2}+3\) from vertex form. Identify the parent function, \(h\), \(k\), vertex, axis of symmetry, opening direction, and width relative to \(y=x^{2}\), then give a complete graph description.

Big Picture

What this problem is really about

Vertex form separates the graph’s location from its shape. We’ll compare the rule with a times the quantity x minus h squared plus k, use h and k for the vertex and axis, use a for opening and width, and then describe the shifts from the parent parabola.

Answer choices

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Preview only

It is an upward-opening parabola with vertex (2, 3) because y=(x-h)²+k has vertex (h, k) and the squared term’s coefficient is positive.

It is an upward-opening parabola with vertex (-2, 3) because the value inside the parentheses gives the x-coordinate directly.

It is a downward-opening parabola with vertex (2, 3) because the +3 moves the graph up and makes it turn downward.

It is an upward-opening parabola with vertex (3, 2) because the numbers 2 and 3 tell how far the graph moves, in that order.

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Analyze \(y=(x-2)^{2}+3\) from vertex form. Identify the parent function, \(h\), \(k\), vertex, axis of symmetry, opening direction, and width relative to \(y=x^{2}\), then give a complete graph description.

What this problem is really about

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