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A parabola opening up or down has vertex (0,2)(0,2) and passes through (12,20)(12,20). Write its equation in vertex form.\newlineSimplify any fractions.

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Full solution

Q. A parabola opening up or down has vertex (0,2)(0,2) and passes through (12,20)(12,20). 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)?\newlineSubstitute 00 for hh and 22 for kk in the vertex form.\newliney=a(x−0)2+2y = a(x - 0)^2 + 2\newliney=ax2+2y = ax^2 + 2
  3. Use Point for 'a': Use the point (12,20)(12, 20) to find the value of 'aa'.\newlineReplace the variables with (12,20)(12, 20) in the equation.\newlineSubstitute 1212 for xx and 2020 for yy.\newline20=a(12)2+220 = a(12)^2 + 2\newline20=144a+220 = 144a + 2
  4. Solve for 'a': Solve for 'a'.\newlineSubtract 22 from both sides of the equation.\newline20−2=144a20 - 2 = 144a\newline18=144a18 = 144a\newlineDivide both sides by 144144.\newlinea=18144a = \frac{18}{144}\newlinea=18a = \frac{1}{8}
  5. Write Equation with 'a': Write the equation of the parabola with the value of 'a'.\newlineSubstitute 18\frac{1}{8} for aa in the equation from Step 22.\newliney=(18)x2+2y = \left(\frac{1}{8}\right)x^2 + 2\newlineVertex form of the parabola: y=(18)x2+2y = \left(\frac{1}{8}\right)x^2 + 2

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