Question

The paths a baseball takes after it has been hit are modeled by a function. The baseball's height above the ground is given by a function of the form f(t) = a(t - h)^2 + k, where t is the time in seconds since the baseball was hit. Find an equation for F(t), where F(t) represents the baseball's height above the ground.

A) F(t) = k(t - h)^2 + a
B) F(t) = a(t - h)^2 + k
C) F(t) = h(t^2 - a) + k
D) F(t) = a(t^2 - h) + k

Answer 1

The correct form of the function F(t) representing the height of the baseball in a parabolic motion over time is F(t) = a(t - h)^2 + k, which corresponds to Option B in the list provided.

Step-by-step explanation:

The equation modeling the height of the baseball above the ground as a function of time is given in quadratic form. Remembering that functions are typically represented as f(t), we want the equation to be organized with the function on the left side and the equation on the right side. The correct form of the equation, based on the typical representation of a parabolic motion, is F(t) = a(t - h)^2 + k. Hence, the correct answer is Option B).

In physics, when analyzing projectile motion, the vertical position oftentimes follows a parabolic path represented by a quadratic function. The constants in our equation have specific meanings: 'a' influences the curvature (related to acceleration), 'h' represents the time at which the projectile reaches its maximum height (the vertex of the parabola), and 'k' is the maximum height (the y-coordinate of the vertex).

The student's initial confusion may have stemmed from incorrect arrangement of the variables; however, the correct quadratic equation representing the baseball's height over time follows the vertex form of a parabola, which is mirrored in Option B.