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A parabola opening up or down has vertex (0,−2)(0,-2) and passes through (−4,−4)(-4,-4). 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,−4)(-4,-4). 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. Substitute Vertex: Substitute the vertex into the vertex form.\newlineGiven the vertex (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,−4)(-4, -4) to find the value of 'a'.\newlineThe parabola passes through the point (−4,−4)(-4, -4), so we substitute x=−4x = -4 and y=−4y = -4 into the equation to solve for 'a'.\newline−4=a(−4)2−2-4 = a(-4)^2 - 2\newline−4=16a−2-4 = 16a - 2
  4. Solve for 'a': Solve for 'a'.\newlineAdd 22 to both sides of the equation to isolate the term with 'a'.\newline−4+2=16a-4 + 2 = 16a\newline−2=16a-2 = 16a\newlineDivide both sides by 1616 to solve for 'a'.\newline−2/16=a-2 / 16 = a\newline−1/8=a-1 / 8 = a
  5. Write Final Equation: Write the final equation of the parabola in vertex form.\newlineNow that we have found aa to be −18-\frac{1}{8}, we substitute it back into the vertex form equation.\newliney=(−18)(x−0)2−2y = \left(-\frac{1}{8}\right)(x - 0)^2 - 2\newliney=−(18)x2−2y = -\left(\frac{1}{8}\right)x^2 - 2\newlineThis is the equation of the parabola in vertex form.

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