91Ó°ÊÓ

The vertex of a parabola is a very important point. Imagine it as the tip or the highest/lowest point on the curve. In this case, the parabola's equation is in the vertex form, which is very handy. The vertex form of a parabola is written as \( y = a(x-h)^2 + k \). This format tells you where the vertex is right away. Here, the vertex is at the point \((h, k)\).

For our particular parabola, comparing our equation \(-4\left(x+\frac{1}{2}\right)^{2}=y\) with \( y = a(x-h)^2 + k \), we find the values of \(h\) and \(k\):

• \(h = -\frac{1}{2}\)
• \(k = 0\)

So, the vertex is at \((-\frac{1}{2}, 0)\).

This point is the center of the parabola's symmetry. For any given \(x\), the points on the parabola above and below are mirrored across this vertex point.

The focus of a parabola is crucial for understanding its shape and direction. It is a single point within the parabola where all the parabola's directions point towards.

To find the focus when a parabola is in vertex form, use the formula for vertical parabolas: \( (h, k + \frac{1}{4a}) \). This calculation uses the \(a\) from the equation, which affects how "wide" the parabola looks.

• Here, \(a = -4\), \(h = -\frac{1}{2}\), and \(k = 0\).
• Substitute these values: \( k + \frac{1}{4(-4)} = k - \frac{1}{16} \).
• The focus becomes \((-\frac{1}{2}, 0 - \frac{1}{16})\), or \((-\frac{1}{2}, -\frac{1}{16})\).

This point inside the parabola helps it "focus" light or objects that enter it, directing them to this specific point. In graphical terms, the parabola wraps around this focus.

In contrast to the focus, the directrix of a parabola is a line, not a point. The directrix provides another component of balance in the structure of the parabola.

For a vertical parabola, the equation of the directrix is \( y = k - \frac{1}{4a} \). It works opposite the focus; this line doesn't touch the parabola but helps determine its shape.

• Take \(a = -4\) and \(k = 0\) for our parabola.
• Substitute into the formula: \( y = 0 - \frac{1}{-16} = \frac{1}{16} \).

The directrix, \( y = \frac{1}{16} \), is located above the vertex in this instance. It ensures that each point on the parabola is the same distance from the focus as it is from this line. This balance between the focus and the directrix is what gives the parabola its uniform and predictable shape.