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Advanced Math / Nonlinear equations in one variable and systems of equations in two variables Difficulty: Hard

$y equals 18$

$y = - 3 {(x - 18)}^{2} + 15$

If the given equations are graphed in the xy-plane, at how many points do the graphs of the equations intersect?

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Explanation

Choice D is correct. A point $(x, y)$ is a solution to a system of equations if it lies on the graphs of both equations in the xy-plane. In other words, a solution to a system of equations is a point $(x, y)$ at which the graphs intersect. It’s given that the first equation is $y equals 18$ . Substituting $18$ for $y$ in the second equation yields $18 = - 3 {(x - 18)}^{2} + 15$ . Subtracting $15$ from each side of this equation yields $3 = - 3 {(x - 18)}^{2}$ . Dividing each side of this equation by $negative 3$ yields $- 1 = {(x - 18)}^{2}$ . Since the square of a real number is at least $0$ , this equation can't have any real solutions. Therefore, the graphs of the equations intersect at zero points.

Alternate approach: The graph of the second equation is a parabola that opens downward and has a vertex at $left parenthesis 18 comma 15 right parenthesis$ . Therefore, the maximum value of this parabola occurs when $y equals 15$ . The graph of the first equation is a horizontal line at $18$ on the y-axis, or $y equals 18$ . Since $18$ is greater than $15$ , or the horizontal line is above the vertex of the parabola, the graphs of these equations intersect at zero points.

Choice A is incorrect. The graph of $y equals 15$ , not $y equals 18$ , and the graph of the second equation intersect at exactly one point.

Choice B is incorrect. The graph of any horizontal line such that the value of $y$ is less than $15$ , not greater than $15$ , and the graph of the second equation intersect at exactly two points.

Choice C is incorrect and may result from conceptual or calculation errors.