Final answer:
The equation of the parabola with the given vertex and focus is y² = 8x, which is option a).
Step-by-step explanation:
The equation of a parabola with vertex at the origin and focus on the x-axis can be derived using the standard form of a parabola that is symmetric about the x-axis, which is y² = 4px, where p is the distance from the vertex to the focus.
In this case, the focus is at (8,0), so p is 8. Therefore, by plugging in the value of p, we get the equation y² = 4*8*x, which simplifies to y² = 32x. Among the given options, the one that matches this form is a) y² = 8x.
A parabola is a U-shaped curve that is defined by a specific mathematical equation and has several distinctive properties. It is a type of conic section and is characterized by its symmetry, vertex, and axis.
Parabolas have many real-world applications and are commonly found in physics (such as projectile motion), engineering (designing reflectors or antennas), and mathematics, where they're fundamental in algebra, calculus, and geometry.