91Ó°ÊÓ

In the context of a parabola, the vertex is one of its most essential points. For a horizontally opening parabola represented in the form \((y-k)^2 = 4p(x-h)\), the vertex is located at the point

• \((h, k)\)

The vertex acts as the "turning point" of the parabola, marking the spot where the curve changes its direction. In our exercise, converting the equation \(x = 36y^2\) into standard form revealed that the vertex \((h, k)\) is \((0, 0)\). This means the parabola's minimum curvature occurs right at the origin. It serves as a way to clearly describe the position of the parabola in a coordinate plane, making it a critical component in analyzing its properties. The vertex is fundamental because any transformations applied to the parabola will impact this point, relocating the entire graph while maintaining its shape.

The focus is another crucial feature of a parabola, playing an important role in its geometric properties. In a parabola opening horizontally like \((y-k)^2=4p(x-h)\), the focus is located \(p\) units from the vertex along the x-axis. This means it sits inside the parabola. The equation gives us \(p = \frac{1}{144}\), allowing us to determine the precise coordinates of the focus:

• \((h + p, k)\)

With \((h, k) = (0, 0)\), we find the focus at \((\frac{1}{144}, 0)\). Think of the focus as a focal point for guiding the curve's path; it's where light rays entering parallel to the axis of symmetry converge. Such properties highlight the parabolic shapes' utility in satellite dishes and flashlights, where collecting or emitting light is essential.

The directrix of a parabola is a theoretical line that works with the focus to define the parabola's shape mathematically. For our horizontally opening parabola, the equation \((y-k)^2 = 4p(x-h)\) tells us that the directrix is a vertical line positioned \(p\) units left of the vertex.

• The equation for the directrix is \(x = h - p\)

Given the vertex \((0, 0)\) and \(p = \frac{1}{144}\), we calculate the directrix as \(x = -\frac{1}{144}\). Unlike the focus, which is an internal point, the directrix does not physically touch the parabola. Instead, it serves as a guide, ensuring that any point \((x, y)\) on the parabola is equidistant from the focus and the directrix. This exact balance is what mathematically maintains the curve's shape, underscoring its symmetry and elegance in geometric contexts.