To determine the various components of the parabola given by the equation [tex]\( y = \frac{x^2}{16} + x - 2 \)[/tex], we need to identify the focus, vertex, and directrix.
1. Vertex: The vertex of a parabola in standard form [tex]\(( y = ax^2 + bx + c )\)[/tex] can be found by completing the square. For the given equation:
[tex]\[
y = \frac{x^2}{16} + x - 2
\][/tex]
The vertex form is derived to be in the form [tex]\( y = a(x - h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] are the coordinates of the vertex.
The calculation shows that the vertex is located at [tex]\((-8, -6)\)[/tex].
2. Focus: The focus of the parabola can be found using the relation between the vertex and the focal length, which is determined by the coefficient of the [tex]\( x^2 \)[/tex] term in the equation. Given the equation’s form and focal length calculations, the focus coordinates are found to be [tex]\((-8, -2)\)[/tex].
3. Directrix: The directrix is a line equidistant from the vertex, located opposite to the focus with respect to the vertex. Based on the previous calculations, the directrix is found to be [tex]\( y = -10 \)[/tex].
Thus, the answers to the dropdown questions are:
- The building's entrance is located at the parabola's focus, which has the coordinates [tex]\((-8, -2)\)[/tex]
- The lawn's gate is located at the parabola's vertex, which has the coordinates [tex]\((-8, -6)\)[/tex]
- The building's front wall is located along the directrix of the parabolic lawn area. The directrix of the parabola is [tex]\( y = -10 \)[/tex]
