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A parabola opening up or down has vertex (0,0)(0,0) and passes through (−20,20)(-20,20). Write its equation in vertex form.\newlineSimplify any fractions.

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Q. A parabola opening up or down has vertex (0,0)(0,0) and passes through (−20,20)(-20,20). 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 at (00, 00): What is the equation of a parabola with a vertex at (0,0)(0, 0)?\newlineSince the vertex is at the origin (0,0)(0, 0), we substitute h=0h = 0 and k=0k = 0 into the vertex form equation.\newliney=a(x−0)2+0y = a(x - 0)^2 + 0\newliney=ax2y = ax^2
  3. Value of 'a' Calculation: Determine the value of 'a' using the point (−20,20)(-20, 20) that lies on the parabola.\newlineWe substitute x=−20x = -20 and y=20y = 20 into the equation y=ax2y = ax^2 to find the value of 'a'.\newline20=a(−20)220 = a(-20)^2\newline20=400a20 = 400a
  4. Solve for 'a': Solve for 'a'.\newlineDivide both sides of the equation by 400400 to solve for 'a'.\newline20400=a\frac{20}{400} = a\newline120=a\frac{1}{20} = a
  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 as:\newliney=120x2y = \frac{1}{20}x^2

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