- - -_the_graph_of_the - -_equation - ~ In the xy-plane, y = −x^2 +9x−100 y = c intersects the line at exactly one point. What is the value of c? −
Correct answer: C (parsed from the explanation — this source omits an answer key for this item)
Explanation
= 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 =- 2+9 -100 y x x 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 = - 2+ equation in vertex form, y a^x hh k, the vertex of its graph in the xy-plane is =- 2+9 -100 ^h,kh. The given quadratic equation, y x x , can be rewritten as =- 2-2 9 + 9 2 + 9 2 -100 =- - 9 2 + - 319 y dx b 2 lx b 2 l n b 2 l , or y bx 2 l b 4 l. Thus, the value of c is 319 - equal to 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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