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A parabola opening up or down has vertex (0,0)(0,0) and passes through (4,2)(4,2). Write its equation in vertex form.\newlineSimplify any fractions.\newline______

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Q. A parabola opening up or down has vertex (0,0)(0,0) and passes through (4,2)(4,2). Write its equation in vertex form.\newlineSimplify any fractions.\newline______
  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 at Vertex: 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. Point Substitution: Determine the value of aa using the point (4,2)(4, 2) that lies on the parabola.\newlineWe know that when x=4x = 4, y=2y = 2 for the point on the parabola. We substitute these values into the equation y=ax2y = ax^2 to find aa.\newline2=a(4)22 = a(4)^2\newline2=16a2 = 16a
  4. Solving for 'a': Solve for 'a'.\newlineDivide both sides of the equation by 1616 to solve for 'a'.\newlinea=216a = \frac{2}{16}\newlinea=18a = \frac{1}{8}
  5. 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=18x2y = \frac{1}{8}x^2\newlineThis is the equation of the parabola in vertex form that has a vertex at (0,0)(0, 0) and passes through the point (4,2)(4, 2).

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