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A parabola opening up or down has vertex (0,2)(0,2) and passes through (4,−2)(4,-2). 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 (4,−2)(4,-2). Write its equation in vertex form.\newlineSimplify any fractions.
  1. Identify vertex form: Identify the vertex form of a 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. Plug vertex coordinates: Plug the vertex coordinates into the vertex form.\newlineSince the vertex is given as (0,2)(0, 2), we substitute h=0h = 0 and k=2k = 2 into the vertex form equation.\newliney=a(x−0)2+2y = a(x - 0)^2 + 2\newliney=ax2+2y = ax^2 + 2
  3. Use point to find 'a': Use the point (4,−2)(4, -2) to find the value of 'a'.\newlineWe know the parabola passes through the point (4,−2)(4, -2), so we can substitute x=4x = 4 and y=−2y = -2 into the equation to solve for 'a'.\newline−2=a(4)2+2-2 = a(4)^2 + 2\newline−2=16a+2-2 = 16a + 2
  4. Solve for 'a': Solve for 'a'.\newlineSubtract 22 from both sides of the equation to isolate the term with 'a'.\newline−2−2=16a-2 - 2 = 16a\newline−4=16a-4 = 16a\newlineDivide both sides by 1616 to solve for 'a'.\newlinea=−4/16a = -4 / 16\newlinea=−1/4a = -1 / 4
  5. Write 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=(−14)x2+2y = (-\frac{1}{4})x^2 + 2\newlineThis is the equation of the parabola in vertex form.

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