1998 AHSME Problem 14

Geometry Level 2

A parabola has vertex of (4,-5) and has two x-intercepts, one positive, and one negative. If this parabola is the graph of y = a x 2 + b x + c y = ax^{2} + bx + c , which of a,b, and c must be positive?

a and b only only b only a only c

Daniel Hirschberg by Daniel Hirschberg , 39, USA

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2 solutions

Amanda Lim
Apr 1, 2014

Formula for parabola: y=a(x-h)^2 + k; if a>0, parabola opens upward. Since the parabola hits the x-axis while its vertex is below the x-axis, the parabola opens upward or a is positive.

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The vertex is in the 4 4 th quadrant. If it doesn't opens upwards, then it will not intercept the x x -axis, and thus a a must be positive.

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