Multiple choice

g(x) = bx2 – 6x + d In the given quadratic function, b and d are constants. The graph of y = g(x) in the xy-plane is a parabola that opens upward and has a vertex at the point (m, n) where m and n are constants. If n > 0 and g(-4) = g(10), which of the following must be true? I. d > 0 II. b > 0

  1. I only

  2. II only

  3. I and II

  4. Neither I nor II
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Parabola opens upward, so b > 0. Vertex x-coordinate m = -(-6)/(2b) = 3/b. Since g(-4) = g(10), the axis of symmetry is x = (-4+10)/2 = 3. Thus 3/b = 3, so b = 1. g(x) = x^2 - 6x + d. Vertex y-coordinate n = g(3) = 9 - 18 + d = d - 9. Given n > 0, d - 9 > 0, so d > 9. Both I (d > 0) and II (b > 0) are true.

AI explanation

Because g(-4) = g(10), the vertical line of symmetry for the parabola is the average of the x-coordinates, giving x = (-4 + 10) / 2 = 3, so m = 3. The function value at this axis of symmetry equals the minimum value n, so n = g(3) = b(9) - 6(3) + d = 9b - 18 + d; because the parabola opens upward, b > 0. Since n > 0 and 9b - 18 is strictly greater than -18 (because b > 0), it follows that d must be greater than 18 - 9b, and because -18 + d > -9b > 0, d is strictly greater than 18 - 9b which forces d > 0. Thus, both b > 0 and d > 0 must be true.

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