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The vertex form of a parabola is a handy way to express the quadratic equation of a parabola. For a parabola with a vertex at the origin \(0, 0\), this form simplifies things greatly.The vertex form for a parabola that opens horizontally is given by the equation:\[x = a(y - k)^2 + h\]Here, \((h, k)\) is the vertex of the parabola. With the vertex at the origin, \(h = 0\) and \(k = 0\), so the equation becomes:\[x = ay^2\]The parameter \(a\) determines the shape and direction of the parabola.- If \(a\) is positive, the parabola opens to the right.- If \(a\) is negative, it opens to the left.Understanding the vertex form helps in easily identifying key features of the parabola, such as its vertex and the direction it opens.

The focus and directrix are essential geometric entities that help in defining a parabola.### Role of the FocusThe focus of a parabola is a fixed point on its interior. For a horizontal parabola with a vertex at the origin \(0,0\) and a focus at \((-8, 0)\), the parabola opens horizontally and curves around this focus.- The focus is used to determine the parabola's orientation and shape.### Role of the DirectrixOpposite to the focus, the directrix of a parabola is a line.- For a horizontal parabola, it is vertical. The vertex is exactly halfway between the focus and the directrix.- In our case, since the focus is \((-8, 0)\), the directrix is the vertical line \((x = 8)\).The parabola is defined in such a way that for any point on the parabola, the distance to the focus is equal to the distance to the directrix.

Horizontal parabolas are a specific orientation of parabolas. Unlike the more common vertical parabolas that open up or down, these parabolas open left or right.### Characteristics- ** Vertex **: The starting point is typically at the origin \(0, 0\) which simplifies equations.- ** Direction **: The orientation (left or right) depends on the sign of \(a\) in the equation \(x = ay^2\).### Equation of Horizontal ParabolasFor a horizontal parabola with a vertex at the origin, the equation takes the form:- \(x = ay^2\)Where:- \(a = \text{positive}\) means the parabola opens to the right.- \(a = \text{negative}\) means it opens to the left, like in our exercise where \(a = -\frac{1}{32}\).Focusing on these key elements helps simplify the understanding of how horizontal parabolas behave, making it easier to graph and comprehend their properties.