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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. 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,−2)(0, -2)?\newlineSince the vertex is (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 (4,2)(4, 2): Use the point (4,2)(4, 2) to find the value of aa. We know the parabola passes through the point (4,2)(4, 2), so we substitute x=4x = 4 and y=2y = 2 into the equation to solve for aa. 2=a(4)2−22 = a(4)^2 - 2 2=16a−22 = 16a - 2
  4. Solve for 'a': Solve for 'a'.\newlineAdd 22 to both sides of the equation to isolate the term with 'a'.\newline2+2=16a2 + 2 = 16a\newline4=16a4 = 16a\newlineDivide both sides by 1616 to solve for 'a'.\newline416=a\frac{4}{16} = a\newline14=a\frac{1}{4} = a
  5. Write Equation in Vertex Form: Write the equation of the parabola in vertex form using the value of 'a'.\newlineNow that we have found a=14a = \frac{1}{4}, we substitute it back into the equation y=ax2−2y = ax^2 - 2.\newliney=(14)x2−2y = \left(\frac{1}{4}\right)x^2 - 2\newlineThis is the equation of the parabola in vertex form.

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