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Difficulty: HardQuadratic Functions and Graphs

A parabola defined by the equation y=ax2+bx+cy = ax^2 + bx + c, where aa, bb, and cc are constants, has its vertex in the second quadrant of the coordinate plane. If the parabola passes through the point (0,0)(0,0), which of the following inequalities must be true?

  1. ab>0ab > 0Answer
  2. B
    ab<0ab < 0
  3. C
    a>0a > 0
  4. D
    b>0b > 0

Answer

The inequality ab>0ab > 0 must be true.
Since the parabola passes through the point (0,0)(0,0) and its vertex (h,k)(h, k) lies in the second quadrant where k>0k > 0, the vertex must represent a maximum value. Therefore, the parabola opens downward, so the leading coefficient aa is negative. The x-coordinate of the vertex, hh, is also negative because it lies in the second quadrant. The vertex x-coordinate is defined by h=−b2ah = -\frac{b}{2a}, which means b=−2ahb = -2ah. Because both aa and hh are negative, their product ahah is positive, which makes bb negative when multiplied by −2-2. Finally, since both aa and bb are negative, their product abab must be positive.

Step-by-Step Solution

1
Determine the signs of the vertex coordinates from its quadrant.
The vertex (h,k)(h, k) lies in the second quadrant, which means h<0h < 0 and k>0k > 0.
Points in the second quadrant have negative x-coordinates and positive y-coordinates.
2
Determine the value of the constant cc.
c=0c = 0
Since the parabola passes through (0,0)(0,0), substituting x=0x = 0 must result in y=0y = 0.
3
Determine the sign of the leading coefficient aa.
a<0a < 0
The vertex is at (h,k)(h, k) with k>0k > 0, and the graph passes through (0,0)(0, 0) where the y-value is 00. Since the maximum value must be at least the value at any other point, the vertex (h,k)(h,k) is a maximum. A parabola with a maximum opens downward, so a<0a < 0.
4
Determine the sign of the coefficient bb using the vertex formula.
b<0b < 0
The x-coordinate of the vertex is h=−b2ah = -\frac{b}{2a}, which can be rewritten as b=−2ahb = -2ah. Since a<0a < 0 and h<0h < 0, their product ahah is positive. Multiplying this positive product by −2-2 yields a negative value for bb.
5
Find the sign of the product abab.
ab>0ab > 0
Since both aa and bb are negative, their product abab must be positive.

Key Concept

Analyzing quadratic coefficients and vertex properties in the coordinate plane.
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Topics

Quadratic Functions and Graphs

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