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Find the minimum value of the function f(x)=0.7x^(2)+8x+25 to the nearest hundredth. Answer:

Find the minimum value of the function f(x)=0.7x2+8x+25 f(x)=0.7 x^{2}+8 x+25 to the nearest hundredth.\newlineAnswer:

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Full solution

Q. Find the minimum value of the function f(x)=0.7x2+8x+25 f(x)=0.7 x^{2}+8 x+25 to the nearest hundredth.\newlineAnswer:
  1. Calculate Vertex: To find the minimum value of the quadratic function f(x)=0.7x2+8x+25f(x) = 0.7x^2 + 8x + 25, we can use the vertex formula. The vertex of a parabola given by f(x)=ax2+bx+cf(x) = ax^2 + bx + c is at the point (h,k)(h, k), where h=−b2ah = -\frac{b}{2a} and kk is the value of the function at x=hx = h.
  2. Calculate x-coordinate: First, we calculate the x-coordinate of the vertex, hh, using the formula h=−b2ah = -\frac{b}{2a}. Here, a=0.7a = 0.7 and b=8b = 8.\newlineh=−82×0.7=−81.4=−5.71428571…h = -\frac{8}{2 \times 0.7} = -\frac{8}{1.4} = -5.71428571\ldots
  3. Round x-coordinate: Now we round hh to the nearest hundredth to get h≈−5.71h \approx -5.71.
  4. Calculate y-coordinate: Next, we calculate the y-coordinate of the vertex, kk, by substituting hh back into the function f(x)f(x).k=f(−5.71)=0.7(−5.71)2+8(−5.71)+25k = f(-5.71) = 0.7(-5.71)^2 + 8(-5.71) + 25
  5. Perform Calculations: Perform the calculations:\newlinek=0.7×32.6041+8×−5.71+25k = 0.7 \times 32.6041 + 8 \times -5.71 + 25\newlinek=22.82287−45.68+25k = 22.82287 - 45.68 + 25\newlinek=−0.85713k = -0.85713
  6. Round y-coordinate: Now we round kk to the nearest hundredth to get k≈−0.86k \approx -0.86.

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