Bunuel wrote:
In the xy-plane, does y = a(x - k)^2 + p intersect with x-axis?
(1) a < 0
(2) p > 0
y can either be upward or a downward parabola.
If y is an upward parabola, the minima of the parabola lies at (k,p)
If y is an downward parabola, the maxima of the parabola lies at (k,p)
Also, y = a(x - k)^2 + p represents a parabola shifted to the right by k units if the value of k is positive or shifted to the left by k units if the value of k is negative.
Similarly, if the value of p is positive, the parabola is shifted upwards by p units. If the value of p is negative, the parabola is shifted downwards by k units.
Statement 1 a < 0
This statement tells us that the parabola is a downward parabola. However the information is not sufficient.
If the parabola is shifted downward (depicted using green), the parabola will not intersect with x-axis, however if the parabola is shifted upward (depicted using blue), the parabola will intersect with x axis.
Attachment:
Downward Parabola.jpg [ 37.24 KiB | Viewed 739 times ]
The statement is thus not sufficient, and we can eliminate A and D.
Statement 2 p > 0This statement tells us the vertical shift was made in the upward direction. The statement is not sufficient as we don't know whether the original parabola was upward or was it downward.
If the parabola was upward, then a vertical shift will result the parabola to not intersect with x-axis. This is depicted by the blue and red pair.
However if the parabola was a downward parabola, then a vertical shift will result the parabola to intersect with x-axis This is depicted by the blue and red pair.
Attachment:
Upward Parabola.jpg [ 21.12 KiB | Viewed 729 times ]
Attachment:
Downward Parabola-2.jpg [ 20.82 KiB | Viewed 715 times ]
CombinedThe statements combined tells us that the parabola is downward and has been shifted upward. Hence the possible parabola is
Attachment:
Downward Parabola-2.jpg [ 20.82 KiB | Viewed 715 times ]
Hence, the statements combined is sufficient.
Option C