91Ó°ÊÓ

In parabola equations, the vertex form is especially helpful because it simplifies finding the vertex directly from the equation. When you have a parabola and know its vertex, you can write the equation in vertex form: \[ y = a(x-h)^2 + k \]Here, \(h,k\) represents the vertex of the parabola, and \(a\) dictates the 'openness' and direction of the parabola. If \(a \)> 0, the parabola opens upwards; if \(a < 0\), it opens downwards. In our case, since the vertex is (0,0), the equation simplifies to:\[ y = ax^2 \]Understanding vertex form allows you to immediately see the location of the vertex and how the parabola is oriented.

The directrix is a fixed line that helps define a parabola along with its focus. For a parabola with a vertical orientation and a given vertex at \( (0, 0) \), the directrix gives you important positional information.

• Directrix above vertex: Parabola opens downward.
• Directrix below vertex: Parabola opens upward.
• In our exercise, the directrix is \( y = \frac{1}{3} \), indicating the parabola opens downward.

The distance between the vertex and the directrix is key to determining the \(a \) value in the parabola's equation.

The distance formula is key to finding the value of \( p \), which is the distance from the vertex of the parabola to the directrix. A vertical parabola's equation \( y = ax^2 \) depends on this distance.The formula for finding \( p \) when the directrix is known is:\[ p = - \text{(distance from vertex to directrix)} \]Given \((0, 0) \) as the vertex and \( y = \frac{1}{3} \) as the directrix, we apply:\[ p = - \frac{1}{3} \] This negative sign indicates that the parabola opens downward. Use \( p \) to find the value \(a\). Finally, substitute into \( a = -\frac{1}{4p} \): \[ a = -\frac{1}{4 \times -\frac{1}{3}} = \frac{3}{4}\]This gives us the equation: \( y = \frac{3}{4} x^2 \).