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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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Full solution

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 Parabola Orientation: Since the directrix is y=3y = 3, which is a horizontal line, the parabola is vertical and opens either up or down.
  2. Locate Vertex and Directrix: The vertex is at (0,0)(0,0), and the directrix is above it at y=3y = 3, so the parabola opens downward.
  3. Calculate Distance to Directrix: The distance between the vertex and the directrix is the absolute value of the difference in their y-coordinates, which is ∣0−3∣=3|0 - 3| = 3.
  4. Use Formula to Find 'a': The value of 'a' in the vertex form equation y=a(x−h)2+ky = a(x-h)^2+k is related to the distance from the vertex to the directrix by the formula 4a=−distance4a = -\text{distance}, since the parabola opens downward.
  5. Substitute Distance to Solve for 'a': Substitute the distance into the formula to find 'a': 4a=−34a = -3, so a=−34a = -\frac{3}{4}.

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