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# 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

A)
$$y = (x + 3)(x - 5)$$
B)
$$y = (x - 3)(x + 5)$$
C)
$$y = x(x - 2) - 15$$
D)
$$y = (x - 1)^2 - 16$$

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.