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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.


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