In the xy-plane, the graph of the equation y = −x 2 + 9x − 100 intersects the line y = c at exactly one point. What is the value of c?

Question

Accepted Answer

Choice C is correct. In the xy-plane, the graph of the line y = c is a horizontal line that crosses the y-axis at y = c and the graph of the quadratic equation y = −x 2 + 9x − 100 is a parabola. A parabola can intersect a horizontal line at exactly one point only at its vertex. Therefore, the value of c should be equal to the y-coordinate of the vertex of the graph of the given equation. For a quadratic equation in vertex form, y = a(x − h) 2 + k, the vertex of its graph in the xy-plane is (h, k). The given quadratic equation, y = −x 2 + 9x − 100, can be rewritten as y = −(x 2 − 2( 9 ⁄ 2 )x + ( 9 ⁄ 2 ) 2 ) + ( 9 ⁄ 2 ) 2 − 100, or y = −(x − 9 ⁄ 2 ) 2 + − 319 ⁄ 4 . Thus, the value of c is equal to − 319 ⁄ 4 . Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

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