Final answer:
The equation in vertex form out of the given options is A. f(x) = (x - 3)² - 4. To prove a projectile's trajectory is parabolic, solve the horizontal motion equation for time and substitute into the vertical motion equation. Quadratic equations are solved using the quadratic formula which provides the roots of the equation.
Step-by-step explanation:
The student is asking to write the equation of a parabola in vertex form. The vertex form of a parabola's equation is generally expressed as f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. Looking at the given options, A. f(x) = (x - 3)² - 4 is already in vertex form, reflecting a parabola with a vertex at (3, -4).
To prove that the trajectory of a projectile is parabolic, you can start by solving the equation x = Vo*cos(θ)t for t, which will give you t = x / (Vo*cos(θ)). Then, substitute this expression for t into the equation for y, which is y = Vo*sin(θ)t - (1/2)gt². After substitution, you will get the equation in the form of y = ax + bx², which shows that the projectile's trajectory is a parabola.
For a quadratic equation of the form ax² + bx + c = 0, the solutions can be found using the quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), which yields the roots of the equation.