** Which of the following is an equivalent form of the equation of the graph sho...**

### Question

Which of the following is an equivalent form of the equation of the graph shown in the *xy*-plane above, from which the coordinates of vertex *A *can be identified as constants in the equation?

### Options

The correct answer is D.

### Explanation:

Choice D is correct. Any quadratic function q can be written in the form \(q(x) = a(x - h)^2 + k\), where a, h, and k are constants and (h, k) is the vertex of the parabola when q is graphed in the coordinate plane. This form can be reached by completing the square in the expression that defines q. The equation of the graph is \(y = x^2 - 2x - 15\). Since the coefficient of x is -2, this equation can be written in terms of \((x - 1)^2 = x^2 - 2x + 1\) as follows: \(y = x^2 -2x - 15 = (x^2 - 2x + 1) - 16 = (x - 1)^2 - 16\). From this form of the equation, the coefficients of the vertex can be read as (1,-16).

Choices A and C are incorrect because the coordinates of the vertex A do not appear as constants in these equations. Choice B is incorrect because it is not equivalent to the given equation.

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