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A parabola opening up or down has vertex (0,0)(0,0) and passes through (16,−16)(16,-16). Write its equation in vertex form.\newlineSimplify any fractions.\newline______

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Q. A parabola opening up or down has vertex (0,0)(0,0) and passes through (16,−16)(16,-16). Write its equation in vertex form.\newlineSimplify any fractions.\newline______
  1. Vertex Form Explanation: What is the vertex form of the parabola?\newlineThe vertex form of a parabola is given by the equation y=a(x−h)2+ky = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola.
  2. Vertex at Origin: What is the equation of a parabola with a vertex at (0,0)(0, 0)?\newlineSince the vertex is at the origin (0,0)(0, 0), we substitute h=0h = 0 and k=0k = 0 into the vertex form equation.\newliney=a(x−0)2+0y = a(x - 0)^2 + 0\newliney=ax2y = ax^2
  3. Determine 'a' Value: Determine the value of 'a' using the point (16,−16)(16, -16).\newlineWe know the parabola passes through the point (16,−16)(16, -16). We substitute x=16x = 16 and y=−16y = -16 into the equation y=ax2y = ax^2 to find 'a'.\newline−16=a(16)2-16 = a(16)^2\newline−16=256a-16 = 256a
  4. Solve for 'a': Solve for 'a'.\newlineDivide both sides of the equation by 256256 to solve for 'a'.\newlinea=−16/256a = -16 / 256\newlinea=−1/16a = -1 / 16
  5. Final Equation: Write the final equation of the parabola in vertex form.\newlineNow that we have the value of aa, we can write the equation of the parabola.\newliney=(−116)x2y = \left(-\frac{1}{16}\right)x^2\newlineThis is the equation of the parabola in vertex form.

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