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Answer :
To determine which function represents the path of the T-shirt, we need to examine how each function describes a parabola. In this case, the parabola represents the height of the T-shirt as it moves through the air.
The general form of a parabolic function is:
[tex]\[ f(t) = a(t - h)^2 + k \][/tex]
where [tex]\( (h, k) \)[/tex] is the vertex of the parabola.
Let's analyze the given options one by one:
1. [tex]\( f(t) = -16(t-1)^2 + 24 \)[/tex]
- This function is in the form [tex]\( f(t) = a(t-h)^2 + k \)[/tex].
- Vertex: [tex]\((h, k) = (1, 24)\)[/tex].
- The value of [tex]\( a = -16 \)[/tex] is negative, indicating that the parabola opens downwards.
- This function represents a parabola with a maximum height at [tex]\( t = 1 \)[/tex] seconds, and the peak height is 24 units.
2. [tex]\( f(t) = -16(t+1)^2 + 24 \)[/tex]
- Vertex: [tex]\((h, k) = (-1, 24)\)[/tex].
- The parabola opens downwards with a peak at [tex]\( t = -1 \)[/tex], but in the context of time (which is typically non-negative), this doesn't seem appropriate for modeling the path of a T-shirt.
3. [tex]\( f(t) = -16(t-1)^2 - 24 \)[/tex]
- Vertex: [tex]\((h, k) = (1, -24)\)[/tex].
- The parabola opens downwards, but the peak height is negative, indicating it does not rise above the point considered as the reference level (such as the ground or the initial throw height).
4. [tex]\( f(t) = -16(t+1)^2 - 24 \)[/tex]
- Vertex: [tex]\((h, k) = (-1, -24)\)[/tex].
- The parabola opens downwards and, like option 3, does not effectively model a realistic scenario for a T-shirt moving upwards first.
Based on these analyses, option 1 is the most suitable function. It models the T-shirt reaching a peak or maximum height at [tex]\( t = 1 \)[/tex] second, with a height of 24 units before coming back down. Therefore, the function [tex]\( f(t) = -16(t-1)^2 + 24 \)[/tex] correctly represents the path of the T-shirt.
The general form of a parabolic function is:
[tex]\[ f(t) = a(t - h)^2 + k \][/tex]
where [tex]\( (h, k) \)[/tex] is the vertex of the parabola.
Let's analyze the given options one by one:
1. [tex]\( f(t) = -16(t-1)^2 + 24 \)[/tex]
- This function is in the form [tex]\( f(t) = a(t-h)^2 + k \)[/tex].
- Vertex: [tex]\((h, k) = (1, 24)\)[/tex].
- The value of [tex]\( a = -16 \)[/tex] is negative, indicating that the parabola opens downwards.
- This function represents a parabola with a maximum height at [tex]\( t = 1 \)[/tex] seconds, and the peak height is 24 units.
2. [tex]\( f(t) = -16(t+1)^2 + 24 \)[/tex]
- Vertex: [tex]\((h, k) = (-1, 24)\)[/tex].
- The parabola opens downwards with a peak at [tex]\( t = -1 \)[/tex], but in the context of time (which is typically non-negative), this doesn't seem appropriate for modeling the path of a T-shirt.
3. [tex]\( f(t) = -16(t-1)^2 - 24 \)[/tex]
- Vertex: [tex]\((h, k) = (1, -24)\)[/tex].
- The parabola opens downwards, but the peak height is negative, indicating it does not rise above the point considered as the reference level (such as the ground or the initial throw height).
4. [tex]\( f(t) = -16(t+1)^2 - 24 \)[/tex]
- Vertex: [tex]\((h, k) = (-1, -24)\)[/tex].
- The parabola opens downwards and, like option 3, does not effectively model a realistic scenario for a T-shirt moving upwards first.
Based on these analyses, option 1 is the most suitable function. It models the T-shirt reaching a peak or maximum height at [tex]\( t = 1 \)[/tex] second, with a height of 24 units before coming back down. Therefore, the function [tex]\( f(t) = -16(t-1)^2 + 24 \)[/tex] correctly represents the path of the T-shirt.
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