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Write the equation in vertex form for the parabola with vertex (0,0)(0,0) and directrix y=−3y = -3.\newlineSimplify any fractions.\newline______

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Q. Write the equation in vertex form for the parabola with vertex (0,0)(0,0) and directrix y=−3y = -3.\newlineSimplify any fractions.\newline______
  1. Identify Orientation: Identify the orientation of the parabola based on the directrix.\newlineSince the directrix is y=−3y = -3, which is a horizontal line, the parabola is vertical.
  2. Determine Opening Direction: Determine the direction the parabola opens. The vertex (0,0)(0,0) is above the directrix y=−3y = -3, so the parabola opens upwards.
  3. Calculate Distance: Calculate the distance between the vertex and the directrix.\newlineThe distance is the absolute value of the difference in y-coordinates, which is ∣0−(−3)∣=3|0 - (-3)| = 3.
  4. Find Value of 'a': Find the value of 'a' using the distance from the vertex to the directrix.\newlineThe distance is equal to 14a\frac{1}{4a}, so 3=14a3 = \frac{1}{4a}. Solving for 'a' gives a=14×3=112a = \frac{1}{4\times 3} = \frac{1}{12}.
  5. Write Equation in Vertex Form: Write the equation in vertex form using the vertex (h,k)=(0,0)(h,k) = (0,0) and the value of ′a′'a'. The vertex form is y=a(x−h)2+ky = a(x-h)^2 + k. Substituting the values gives y=112(x−0)2+0y = \frac{1}{12}(x-0)^2 + 0.

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