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A parabola opening up or down has vertex (0,−1)(0,-1) and passes through (−12,17)(-12,17). Write its equation in vertex form.\newlineSimplify any fractions.

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Q. A parabola opening up or down has vertex (0,−1)(0,-1) and passes through (−12,17)(-12,17). 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: What is the equation of a parabola with a vertex at (0,−1)(0, -1)?\newlineSubstitute 00 for hh and −1-1 for kk in the vertex form.\newliney=a(x−0)2+(−1)y = a(x - 0)^2 + (-1)\newliney=ax2−1y = ax^2 - 1
  3. Use Point to Find aa: Use the point (−12,17)(-12, 17) to find the value of aa. Replace the variables with (−12,17)(-12, 17) in the equation. Substitute −12-12 for xx and 1717 for yy. 17=a(−12)2−117 = a(-12)^2 - 1 17=144a−117 = 144a - 1
  4. Solve for a: Solve for a.\newlineAdd 11 to both sides of the equation.\newline17+1=144a17 + 1 = 144a\newline18=144a18 = 144a\newlineDivide both sides by 144144.\newline18144=a\frac{18}{144} = a\newlineSimplify the fraction.\newline18=a\frac{1}{8} = a
  5. Write Equation with aa: Write the equation of the parabola using the value of aa. Substitute 18\frac{1}{8} for aa in the equation y=ax2−1y = ax^2 - 1. y=(18)x2−1y = \left(\frac{1}{8}\right)x^2 - 1 Vertex form of the parabola: y=(18)x2−1y = \left(\frac{1}{8}\right)x^2 - 1

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