Answer :
To solve the problem of determining which function represents the height of the T-shirt as it follows a parabolic path, let's break down the characteristics of the functions provided in options.
Each given function is in the standard vertex form of a parabola:
[tex]\[ f(t) = a(t-h)^2 + k \][/tex]
Here, [tex]\( (h, k) \)[/tex] is the vertex of the parabola. The value of [tex]\( a \)[/tex] determines the direction of the opening. If [tex]\( a \)[/tex] is negative, the parabola opens downwards, indicating a maximum point, which is typically the highest point of the T-shirt's path.
Let's analyze each option:
1. [tex]\( f(t) = -16(t-1)^2 + 24 \)[/tex]
- Vertex: [tex]\( (1, 24) \)[/tex]
- This parabola opens downward (since [tex]\( a = -16 \)[/tex]) and has a maximum height of 24 at [tex]\( t = 1 \)[/tex].
2. [tex]\( f(t) = -16(t+1)^2 + 24 \)[/tex]
- Vertex: [tex]\( (-1, 24) \)[/tex]
- This parabola also opens downward, with a maximum height of 24 but at [tex]\( t = -1 \)[/tex], which is negative and doesn't make sense for time.
3. [tex]\( f(t) = -16(t-1)^2 - 24 \)[/tex]
- Vertex: [tex]\( (1, -24) \)[/tex]
- This parabola opens downward, but the maximum value at [tex]\( t = 1 \)[/tex] is -24, meaning the height is negative, which is not plausible.
4. [tex]\( f(t) = -16(t+1)^2 - 24 \)[/tex]
- Vertex: [tex]\( (-1, -24) \)[/tex]
- This parabola opens downward with a maximum at [tex]\( t = -1 \)[/tex] and a maximum height of -24, both of which are not plausible.
Considering a logical time progression, the T-shirt should achieve the highest point above ground level at a positive time value with a reasonable height.
From the analysis, the first option [tex]\( f(t) = -16(t-1)^2 + 24 \)[/tex] meets the criteria with the T-shirt reaching a maximum height of 24 units at [tex]\( t = 1 \)[/tex] second. Therefore, this function accurately models the height of the T-shirt as a function of time.
Each given function is in the standard vertex form of a parabola:
[tex]\[ f(t) = a(t-h)^2 + k \][/tex]
Here, [tex]\( (h, k) \)[/tex] is the vertex of the parabola. The value of [tex]\( a \)[/tex] determines the direction of the opening. If [tex]\( a \)[/tex] is negative, the parabola opens downwards, indicating a maximum point, which is typically the highest point of the T-shirt's path.
Let's analyze each option:
1. [tex]\( f(t) = -16(t-1)^2 + 24 \)[/tex]
- Vertex: [tex]\( (1, 24) \)[/tex]
- This parabola opens downward (since [tex]\( a = -16 \)[/tex]) and has a maximum height of 24 at [tex]\( t = 1 \)[/tex].
2. [tex]\( f(t) = -16(t+1)^2 + 24 \)[/tex]
- Vertex: [tex]\( (-1, 24) \)[/tex]
- This parabola also opens downward, with a maximum height of 24 but at [tex]\( t = -1 \)[/tex], which is negative and doesn't make sense for time.
3. [tex]\( f(t) = -16(t-1)^2 - 24 \)[/tex]
- Vertex: [tex]\( (1, -24) \)[/tex]
- This parabola opens downward, but the maximum value at [tex]\( t = 1 \)[/tex] is -24, meaning the height is negative, which is not plausible.
4. [tex]\( f(t) = -16(t+1)^2 - 24 \)[/tex]
- Vertex: [tex]\( (-1, -24) \)[/tex]
- This parabola opens downward with a maximum at [tex]\( t = -1 \)[/tex] and a maximum height of -24, both of which are not plausible.
Considering a logical time progression, the T-shirt should achieve the highest point above ground level at a positive time value with a reasonable height.
From the analysis, the first option [tex]\( f(t) = -16(t-1)^2 + 24 \)[/tex] meets the criteria with the T-shirt reaching a maximum height of 24 units at [tex]\( t = 1 \)[/tex] second. Therefore, this function accurately models the height of the T-shirt as a function of time.
