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A parabola opening up or down has vertex (0,−3)(0,-3) and passes through (−6,−12)(-6,-12). 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 (−6,−12)(-6,-12). Write its equation in vertex form.\newlineSimplify any fractions.
  1. Vertex Form of Parabola: 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)?\newlineSubstitute 00 for hh and −3-3 for kk in the vertex form.\newliney=a(x−0)2+(−3)y = a(x - 0)^2 + (-3)\newliney=ax2−3y = ax^2 - 3
  3. Use Point to Find aa: Use the point (−6,−12)(-6, -12) to find the value of aa.\newlineReplace the variables with (−6,−12)(-6, -12) in the equation.\newlineSubstitute −6-6 for xx and −12-12 for yy.\newline−12=a(−6)2−3-12 = a(-6)^2 - 3\newline−12=36a−3-12 = 36a - 3
  4. Solve for a: Solve for a.\newline−12=36a−3-12 = 36a - 3\newlineAdd 33 to both sides.\newline−9=36a-9 = 36a\newlineDivide both sides by 3636.\newline−936=a-\frac{9}{36} = a\newlineSimplify the fraction.\newline−14=a-\frac{1}{4} = a
  5. Equation with a=−14a=-\frac{1}{4}: What is the equation of the parabola if a=−14a = -\frac{1}{4}?\newlineSubstitute −14-\frac{1}{4} for aa in the equation y=ax2−3y = ax^2 - 3.\newliney=(−14)x2−3y = (-\frac{1}{4})x^2 - 3\newlineVertex form of the parabola: y=−14x2−3y = -\frac{1}{4}x^2 - 3

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