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A parabola opening up or down has vertex (0,3)(0,3) and passes through (6,6)(6,6). 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 (6,6)(6,6). 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. Use Point to Find 'a': Use the point (6,6)(6, 6) to find the value of 'a'.\newlineWe know the parabola passes through the point (6,6)(6, 6), so we can substitute x=6x = 6 and y=6y = 6 into the equation to solve for 'a'.\newline6=a(6)2+36 = a(6)^2 + 3\newline6=36a+36 = 36a + 3
  4. Solve for 'a': Solve for 'a'.\newlineSubtract 33 from both sides of the equation to isolate the term with 'a'.\newline6−3=36a6 - 3 = 36a\newline3=36a3 = 36a\newlineDivide both sides by 3636 to solve for 'a'.\newlinea=336a = \frac{3}{36}\newlinea=112a = \frac{1}{12}
  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=112(x−0)2+3y = \frac{1}{12}(x - 0)^2 + 3\newlineSimplify the equation.\newliney=112x2+3y = \frac{1}{12}x^2 + 3

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