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A parabola opening up or down has vertex (0,−3)(0,-3) and passes through (8,−7)(8,-7). Write its equation in vertex form.\newlineSimplify any fractions.

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Q. A parabola opening up or down has vertex (0,−3)(0,-3) and passes through (8,−7)(8,-7). Write its equation in vertex form.\newlineSimplify any fractions.
  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. Equation with Vertex: What is the equation of a parabola with a vertex at (0,−3)(0, -3)?\newlineSince the vertex is (0,−3)(0, -3), we substitute h=0h = 0 and k=−3k = -3 into the vertex form equation.\newliney=a(x−0)2−3y = a(x - 0)^2 - 3\newliney=ax2−3y = ax^2 - 3
  3. Value of 'a' Calculation: Determine the value of 'a' using the point (8,−7)(8, -7).\newlineWe know the parabola passes through the point (8,−7)(8, -7), so we substitute x=8x = 8 and y=−7y = -7 into the equation to find 'a'.\newline−7=a(8)2−3-7 = a(8)^2 - 3\newline−7=64a−3-7 = 64a - 3
  4. Solving for 'a': Solve for 'a'.\newlineAdd 33 to both sides of the equation to isolate the term with 'a'.\newline−7+3=64a−3+3-7 + 3 = 64a - 3 + 3\newline−4=64a-4 = 64a\newlineNow, divide both sides by 6464 to solve for 'a'.\newline−4/64=a-4 / 64 = a\newline−1/16=a-1 / 16 = a
  5. Final Equation in Vertex Form: 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)(x−0)2−3y = \left(-\frac{1}{16}\right)(x - 0)^2 - 3\newlineSimplify the equation by removing the 00 inside the parentheses.\newliney=(−116)x2−3y = \left(-\frac{1}{16}\right)x^2 - 3

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