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f(x)=-2x^(2)+40 x-104 The function h is defined by h(x)=f(x-3). For what value of x does h(x) obtain its maximum?

f(x)=−2x2+40x−104 f(x)=-2 x^{2}+40 x-104 \newlineThe function h h is defined by h(x)=f(x−3) h(x)=f(x-3) . For what value of x x does h(x) h(x) obtain its maximum?

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Q. f(x)=−2x2+40x−104 f(x)=-2 x^{2}+40 x-104 \newlineThe function h h is defined by h(x)=f(x−3) h(x)=f(x-3) . For what value of x x does h(x) h(x) obtain its maximum?
  1. Substitute and Find h(x): First, let's substitute x−3x-3 into f(x)f(x) to find h(x)h(x):h(x)=f(x−3)=−2(x−3)2+40(x−3)−104.h(x) = f(x-3) = -2(x-3)^2 + 40(x-3) - 104.
  2. Expand and Simplify: Expand and simplify the equation:\newlineh(x)=−2(x2−6x+9)+40x−120−104h(x) = -2(x^2 - 6x + 9) + 40x - 120 - 104\newline=−2x2+12x−18+40x−120−104= -2x^2 + 12x - 18 + 40x - 120 - 104\newline=−2x2+52x−242.= -2x^2 + 52x - 242.
  3. Find Maximum Value: To find the maximum value, we need the vertex of the parabola, which is given by x=−b2ax = -\frac{b}{2a} for a quadratic equation ax2+bx+cax^2 + bx + c: Here, a=−2a = -2 and b=52b = 52. x=−522⋅−2=524=13x = -\frac{52}{2 \cdot -2} = \frac{52}{4} = 13.

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