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A parabola opening up or down has vertex (0,4)(0,4) 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 (0,4)(0,4) 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 with Vertex: What is the equation of a parabola with a vertex at (0,4)(0, 4)?\newlineSince the vertex is (0,4)(0, 4), we substitute h=0h = 0 and k=4k = 4 into the vertex form equation.\newliney=a(x−0)2+4y = a(x - 0)^2 + 4\newliney=ax2+4y = ax^2 + 4
  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 substitute x=4x = 4 and y=2y = 2 into the equation to solve for 'a'.\newline2=a(4)2+42 = a(4)^2 + 4\newline2=16a+42 = 16a + 4
  4. Solve for 'a': Solve for 'a'.\newlineSubtract 44 from both sides of the equation to isolate the term with 'a'.\newline2−4=16a2 - 4 = 16a\newline−2=16a-2 = 16a\newlineDivide both sides by 1616 to solve for 'a'.\newline−216=a-\frac{2}{16} = a\newline−18=a-\frac{1}{8} = 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.\newliney=(−18)x2+4y = (-\frac{1}{8})x^2 + 4\newlineThis is the equation of the parabola in vertex form.

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