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Mathematics>Standard Multiple Choice

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

If the graph of the function function f in the x y-plane contains the points (0,-9), (1,-4), and (3,0), which of the following CANNOT be true?

(A) The graph of function f has a maximum value.
(B) y less than or equal to 0 for all points (x,y) on the graph of function f.
(C) The graph of function f is symmetric with respect to a line.
(D) The graph of function f is a line.
(E) The graph of function f is a parabola.

答案和解析

答案：D

解析：

If the graph of the function function f, which contains the points (0,-9), (1,-4), and (3,0), were a line, then the slope of the segment connecting the points (0,-9) and (1,-4) would be the same as the slope of the segment connecting the points (1,-4) and (3,0). However, the slope of the segment connecting the points (0,-9) and (1,-4) is 5, and the slope of the segment connecting the points (1,-4) and (3,0) is 2. Therefore, the graph of function f cannot be a line.

The statements in the other four options could be true. For example, if the equation of function f were y = -(x-3)^2, then the graph of function f would contain the points (0,-9), (1,-4), and (3,0). The graph of function f would be a parabola symmetric with respect to the line with equation x = 3. The maximum value of function f would occur at the point (3,0), and y less than or equal to 0 would be true for all points (x,y) on the graph of function f.