Final answer:
The vertex of the quadratic equation y=4x²+8x+1 is found by computing the x-coordinate using -b/(2a) and substituting it back into the equation to determine the y-coordinate. The quadratic formula is used to solve quadratic equations, and the parabolic trajectory of a projectile is derived by substituting time from the horizontal motion equation into the vertical motion equation.
Step-by-step explanation:
The student is dealing with a quadratic equation of the form y = ax² + bx + c, where 'a', 'b', and 'c' are constants, and 'x' is the variable. The vertex of a parabola represented by a quadratic equation can be found using the formula -b/(2a) for the x-coordinate and substituting this back into the equation to find the corresponding y-coordinate. To find the vertex of the given quadratic equation y = 4x² + 8x + 1, we first calculate the x-coordinate of the vertex using -b/(2a) = -8/(2*4), which simplifies to -1. Substituting x = -1 back into the equation gives us the y-coordinate, resulting in a vertex at (-1, y).
To solve the other provided quadratic equations, the quadratic formula, which is x = (-b ± √(b² - 4ac))/(2a), would be used. In the context of physics, the trajectory of a projectile is proved to be parabolic by solving for 't' in the horizontal motion equation and substituting into the vertical motion equation, resulting in a quadratic equation in the form of y = ax + bx².