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If the path of the T-shirt is represented by a parabola, which function could be used to represent the height of the T-shirt as a function of time, [tex]t[/tex], in seconds?

A. [tex]f(t) = -16(t - 1)^2 + 24[/tex]
B. [tex]f(t) = -16(t + 1)^2 + 24[/tex]
C. [tex]f(t) = -16(t - 1)^2 - 24[/tex]
D. [tex]f(t) = -16(t + 1)^2 - 24[/tex]

Answer :

To determine which function represents the height of the T-shirt as a function of time, [tex]t[/tex] in seconds, we need to analyze each of the given functions. Each function is in the form of the vertex form of a quadratic equation, which is typically written as:

[tex]f(t) = a(t - h)^2 + k[/tex]

In this equation, [tex](h, k)[/tex] represents the vertex of the parabola, and [tex]a[/tex] indicates the direction and width of the parabola.

Let's break down each provided equation:


  1. [tex]f(t) = -16(t - 1)^2 + 24[/tex]


    • Vertex: [tex](1, 24)[/tex]

    • This implies the maximum height of the T-shirt is 24 units when [tex]t = 1[/tex] second. The negative sign in front of 16 indicates the parabola opens downward.



  2. [tex]f(t) = -16(t + 1)^2 + 24[/tex]


    • Vertex: [tex](-1, 24)[/tex]

    • Here, the maximum height is again 24, but at [tex]t = -1[/tex] second, which is typically not possible in real scenarios for time unless we consider backward motion.



  3. [tex]f(t) = -16(t - 1)^2 - 24[/tex]


    • Vertex: [tex](1, -24)[/tex]

    • The parabola opens downward with a maximum height of -24 units at [tex]t = 1[/tex] second. Generally, heights can't be negative in real-world scenarios.



  4. [tex]f(t) = -16(t + 1)^2 - 24[/tex]


    • Vertex: [tex](-1, -24)[/tex]

    • This function indicates a maximum height of -24 at [tex]t = -1[/tex] second.




Considering these analyses, the most sensible and realistic function that represents the height of the T-shirt, which opens downwards and has a possible maximum height is:

[tex]f(t) = -16(t - 1)^2 + 24[/tex]

This function describes a scenario where the T-shirt reaches a maximum height of 24 units at [tex]t = 1[/tex] second, a plausible event in real-world motion.

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Rewritten by : Barada