Sure, let's go through this step-by-step.
1. Vertex of the Parabola:
- Given the parabola: [tex]\( y = \frac{x^2}{16} + x - 2 \)[/tex]
- The vertex of this parabola is at the coordinates [tex]\((-8, -6)\)[/tex].
2. Focus of the Parabola:
- The coordinates of the focus are given as [tex]\((-8, -2.0)\)[/tex].
3. Directrix of the Parabola:
- The directrix can be described as a line with the equation [tex]\( y = -10.0 \)[/tex].
So, here are the answers for each part of the question:
1. The coordinates of the building's entrance, located at the parabola's focus, are: [tex]\((-8, -2.0)\)[/tex].
2. The coordinates of the lawn's gate, located at the parabola's vertex, are: [tex]\((-8, -6)\)[/tex].
3. The building's front wall, located along the directrix of the parabolic lawn area, is given by the line [tex]\( y = -10.0 \)[/tex].
To summarize, fill in the blanks as follows:
"The building's entrance is located at the parabola's focus, which has the coordinates [tex]\(\boxed{(-8, -2.0)}\)[/tex]. The lawn's gate is located at the parabola's vertex, which has the coordinates [tex]\(\boxed{(-8, -6)}\)[/tex]. The building's front wall is located along the directrix of the parabolic lawn area. The directrix of the parabola is [tex]\(\boxed{y = -10.0}\)[/tex]."
