To determine the equation of the parabola given its focus and directrix, follow these steps:
1. Identify the components:
- The focus of the parabola is given as [tex]\((4, 0)\)[/tex].
- The directrix of the parabola is [tex]\(x = -4\)[/tex].
2. Understand the structure of a parabolic equation with a horizontal directrix:
- When a parabola opens sideways (in this case, opens right), its standard form is [tex]\(y^2 = 4ax\)[/tex], where [tex]\(a\)[/tex] is the distance between the vertex and the focus (and also between the vertex and the directrix).
3. Calculate the distance [tex]\(a\)[/tex]:
- The vertex of the parabola is exactly midway between the focus and the directrix.
- The focus is at [tex]\(x = 4\)[/tex] and the directrix at [tex]\(x = -4\)[/tex].
- Distance between the focus and the directrix is [tex]\(4 - (-4) = 8\)[/tex].
- Therefore, the distance [tex]\(a\)[/tex] (from the vertex to the focus or to the directrix) will be half of this distance: [tex]\(a = \frac{8}{2} = 4\)[/tex].
4. Form the equation:
- Substitute [tex]\(a = 4\)[/tex] into the standard form [tex]\(y^2 = 4ax\)[/tex].
- This gives: [tex]\(y^2 = 4 \cdot 4 \cdot x\)[/tex].
- Simplified, it becomes: [tex]\(y^2 = 16x\)[/tex].
Therefore, the equation that represents the parabola is:
[tex]\[ \boxed{y^2 = 16x} \][/tex]
