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Difficulty: HardQuadratic Functions and Graphs

In the xyxy-plane, the vertex of the parabola with equation y=−2x2+bx+cy = -2x^2 + bx + c, where bb and cc are constants, lies on the line y=3x+5y = 3x + 5. If the parabola has its maximum value at x=−2x = -2, what is the value of cc?

  1. A
    3
  2. -9Answer
  3. C
    -7
  4. D
    -1

Answer

-9
The maximum value of the quadratic function y=−2x2+bx+cy = -2x^2 + bx + c occurs at its vertex. Since the maximum occurs at x=−2x = -2, the xx-coordinate of the vertex is −2-2. The vertex lies on the line y=3x+5y = 3x + 5, so substituting x=−2x = -2 into this equation gives the yy-coordinate of the vertex: y=3(−2)+5=−1y = 3(-2) + 5 = -1. Thus, the vertex of the parabola is (−2,−1)(-2, -1). The vertex form of a quadratic function is y=a(x−h)2+ky = a(x - h)^2 + k, where (h,k)(h, k) is the vertex. Substituting a=−2a = -2, h=−2h = -2, and k=−1k = -1 gives y=−2(x+2)2−1y = -2(x + 2)^2 - 1. Expanding this equation yields y=−2(x2+4x+4)−1=−2x2−8x−9y = -2(x^2 + 4x + 4) - 1 = -2x^2 - 8x - 9. Comparing this to the standard form y=−2x2+bx+cy = -2x^2 + bx + c shows that c=−9c = -9.

Step-by-Step Solution

1
Identify the x-coordinate of the vertex of the parabola
The x-coordinate of the vertex is h=−2h = -2.
For a quadratic function in standard form with a negative leading coefficient, the maximum value occurs at the vertex. Thus, the x-coordinate of the vertex is −2-2.
2
Determine the y-coordinate of the vertex using the equation of the line
The y-coordinate of the vertex is k=−1k = -1.
Since the vertex (h,k)(h, k) lies on the line y=3x+5y = 3x + 5 and h=−2h = -2, substituting x=−2x = -2 yields y=3(−2)+5=−1y = 3(-2) + 5 = -1.
3
Write the quadratic equation in vertex form and expand it to find c
y=−2x2−8x−9y = -2x^2 - 8x - 9, which gives c=−9c = -9.
Using the vertex form y=a(x−h)2+ky = a(x - h)^2 + k with a=−2a = -2, h=−2h = -2, and k=−1k = -1, the equation is y=−2(x+2)2−1y = -2(x + 2)^2 - 1. Expanding this yields y=−2(x2+4x+4)−1=−2x2−8x−9y = -2(x^2 + 4x + 4) - 1 = -2x^2 - 8x - 9. Comparing this with y=−2x2+bx+cy = -2x^2 + bx + c shows that c=−9c = -9.

Key Concept

Vertex form of a quadratic function and coordinate geometry.
Estimated Time:2m 0s
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Topics

Quadratic Functions and Graphs

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