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A parabola opening up or down has vertex (0,−2)(0,-2) and passes through (8,−10)(8,-10). Write its equation in vertex form.\newlineSimplify any fractions.

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Q. A parabola opening up or down has vertex (0,−2)(0,-2) and passes through (8,−10)(8,-10). Write its equation in vertex form.\newlineSimplify any fractions.
  1. Vertex Form Explanation: What is the vertex form of the parabola?\newlineVertex 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,−2)(0, -2)?\newlineSubstitute 00 for hh and −2-2 for kk in the vertex form.\newliney=a(x−0)2+(−2)y = a(x - 0)^2 + (-2)\newliney=ax2−2y = ax^2 - 2
  3. Use Point for Value: Use the point (8,−10)(8, -10) to find the value of aa. Replace the variables with (8,−10)(8, -10) in the equation. Substitute 88 for xx and −10-10 for yy. −10=a(8)2−2-10 = a(8)^2 - 2 −10=64a−2-10 = 64a - 2
  4. Solve for a: Solve for a.\newline−10=64a−2-10 = 64a - 2\newlineAdd 22 to both sides of the equation.\newline−8=64a-8 = 64a\newlineDivide both sides by 6464.\newline−864=a-\frac{8}{64} = a\newlineSimplify the fraction.\newline−18=a-\frac{1}{8} = a
  5. Equation with a=−18a=-\frac{1}{8}: What is the equation of the parabola if a=−18a = -\frac{1}{8}?\newlineSubstitute −18-\frac{1}{8} for aa in the equation y=ax2−2y = ax^2 - 2.\newliney=(−18)x2−2y = (-\frac{1}{8})x^2 - 2\newlineVertex form of the parabola: y=−18x2−2y = -\frac{1}{8}x^2 - 2

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