Check out the answer to yesterday’s Challenge Problem below!
Question
Line k lies in a coordinate plane. Is the slope of line k positive?
(1) Line k and the graph of the function f(x) = x^2 – bx, where b is positive, intersect on the x-axis.
(2) Line k and the graph of the function g(x) = –x^2 – c, where c is positive, intersect on the y-axis.
(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.
Answer
The given information simply sets the stage: a coordinate plane.
Statement (1) states that the graph of the function f(x) = x^2 – bx, where b is positive, and line k intersect on the x-axis. Since the function is a quadratic, we know that the graph of the function is a parabola. Let’s find the places that this function intersects the x-axis. This is equivalent to finding the values of x for which the function equals zero.
f(x) = x^2 – bx = x(x – b) = 0. Thus, x = 0 or x = b, a positive number. Therefore, line k touches the x-axis at one (or possibly both) of the points (0,0) and (b, 0). However, we do not know another point guaranteed to be on the line. Thus, we do not know the slope of line k. INSUFFICIENT.
Statement (2) states that the graph of the function g(x) and line k intersect on the y-axis. Again, we know that the graph of the function is a parabola, since the function itself is quadratic. To find where the parabola intersects the y-axis, we find the value of the function at x = 0.
g(x) = –x^2 – c
g(0) = –0^2 – c = –c
Since c is positive, –c is negative. Thus, the line k touches the y-axis at the point (0, –c). However, we do not know another point guaranteed to be on the line. Thus, we do not know the slope of line k. INSUFFICIENT.
Statements (1) and (2) together: Line k goes through (0, –c) AND either (0,0) or (b, 0). If line k goes through (0, –c) and (b, 0), where both b and c are positive, then we know that the slope of line k is positive (both the rise and the run are positive). However, line k could go through (0, –c) and (0, 0), which would mean that line k is vertical (it actually would coincide with the y-axis). A vertical line has an undefined slope. Since the slope could either be positive or undefined, the two statements together are INSUFFICIENT.
The correct answer is E: Statements (1) and (2) TOGETHER are INSUFFICIENT to answer the question.
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