The zeros of a parabola are −2 and −8. The maximum value of the function is 18. The parabola is drawn with a solid line, and the i

Question

The zeros of a parabola are −2 and −8. The maximum value of the function is 18. The parabola is drawn with a solid line, and the inside of the parabola is shaded. What quadratic inequality is represented by this description?

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Thu Thảo 1 month 2022-12-24T13:16:16+00:00 1 Answer 0 views 0

Answer ( 1 )

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    2022-12-24T13:17:42+00:00
    The quadratic inequality is represented by this description is y > 9x²/8 + 45x/4 + 369/8

    How to find the quadratic inequality represented by the description?

    Since the function is a parabola, we represent is as y = f(x) = ax² + bx + c
    Now, given that the zeros of a parabola are −2 and −8, we have that
    f(-2) = 0
    a(-2)² + b(-2) + c = 0
    4a – 2b + c = 0
    4a – 2b = -c     (1)
    Also, f(-8) = 0
    a(-8)² + b(-8) + c = 0
    64a – 8b + c = 0
    64a – 8b = -c    (2)
    Also, dy/dx = d(ax² + bx + c)/dx = 2ax + b
    At maximum value, dy/dx = 0.
    So, 2ax + b = 0
    2ax = -b
    x = -b/2a
    Substituting this into y, we have
    f(-b/2a) = a(-b/2a)² + b(-b/2a) + c
    f(-b/2a) = ab²/4a² – b²/2a + c
    f(-b/2a) = b²/4a – b²/2a + c
    f(-b/2a) = – b²/4a + c
    Since the maximum value of y is 18. So,
    f(-b/2a) = – b²/4a + c = 18
    – b²/4a + c = 18
    c = 18 + b²/4a
    Substituting c into equation (1) and (2), we have
    So, in equation (1)
    4a – 2b = -c     (1) × 16
    4a – 2b = -(18 + b²/4a)     (1)
    4a – 2b = -18 – b²/4a  
    16a² – 8ab = -72a  (4)
    Also, in equation (2)
    64a – 8b = -c    (2)       ×  1
    64a – 8b = -(18 + b²/4a)    (2)
    64a – 8b = -18 – b²/4a)
    256a² – 32ab = -72a  (5)
    Subtracting equations (4) and (5),we have
    16a² – 8ab = -72a  (4)
    256a² – 32ab = -72a  (5)
    -240a² + 24ab = 0
    -24a(10a – b) = 0
    ⇒ 24a = 0 or 10a – b = 0
    ⇒ a = 0 or 10a = b
    ⇒ a = 0 or b = 10a
    Substituting b = 10a into equation (4), we have
    16a² – 8ab = -72a  (4)
    16a² – 8a(10a) = -72a  (4)
    16a² – 80a² = -72a  (4)
    -64a² = -72a
    -64a² + 72a = 0
    -8a(8a – 9) = 0
    ⇒ -8a = 0 or 8a – 9 = 0
    ⇒ a = 0 or 8a = 9
    ⇒ a = 0 or a = 9/8
    Substituting a into b, we have
    b = 10a
    b = 10 × 9/8
    b = 5 × 9/4
    b = 45/4
    Substituting a nand b into c, we have
    c = 18 + b²/4a
    c = 18 + (10a)²/4a
    c = 18 + 100a²/4a
    c = 18 + 25a
    c = 18 + 25 × 9/8
    c = 18 + 225/8
    c = (144 + 225)/8
    c = 369/8
    So, susbtituting the values of a, b and c into y, we have
    y = ax² + bx + c
    y =  9x²/8 + 45x/4 + 369/8

    The region represented by the shaded region

    Since the equation of the parabola is y =  9x²/8 + 45x/4 + 369/8 and the region shaded is the inside of the parabola, we use the inequality sign > since the region is greater than y.
    Also, since there is a solid line bounding the region, the line is not included in the inequality. so, the greater than sign > is used.
    So, the shaded region is y > 9x²/8 + 45x/4 + 369/8
    So, the quadratic inequality is represented by this description is y >  9x²/8 + 45x/4 + 369/8
    Learn more about quadratic inequality here:
    brainly.com/question/27914608
    #SPJ1

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