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Mathematics>Standard Multiple Choice

Read the following SAT test question and then click on a button to select your answer.

In the xy -plane, the graph of the line with equation y=a intersects the graph of the quadratic function f(x)=x^2 -(6x) +8 in exactly one point. What is the value of a?

(A) -3
(B) -1
(C) 1
(D) 3
(E) 4

答案和解析

答案：B

解析：

The graph in the xy -plane of the quadratic function fx = (x^2)-(6x)+8 is a parabola. If the graph of the line with equation y=a intersects the graph of this parabola in exactly one point, that point must be the vertex of the parabola, and the y -coordinate of the point must be a. The graph of (fx)=(x^2)-(6x)+8=((x-2)(x-4)) intersects the x -axis at x=2 and x=4, so the x -coordinate of the vertex of the parabola is halfway between x=2 and x=4 on the x -axis at x=3. Thus the y -coordinate of the vertex is (f3)=(3^2)-(63)+8=-1. Therefore, if the graph of the line with equation y=a intersects the graph of the quadratic function (fx)=(x^2)-(6x)+8 in exactly one point, the value of a must be -1.