Answer :
To determine which function could represent the height of the T-shirt as a function of time, we need to analyze the given options, which are in the vertex form of a quadratic function. The vertex form is:
[tex]\[ f(t) = a(t-h)^2 + k \][/tex]
Here, [tex]\( (h, k) \)[/tex] is the vertex of the parabola. The parabola can open upwards or downwards. If [tex]\( a \)[/tex] is negative, the parabola opens downwards, and if positive, it opens upwards. For trajectories, such as the path of a T-shirt, we are typically dealing with a downwards opening parabola due to gravity.
Let's analyze each option:
1. Option 1: [tex]\( f(t) = -16(t-1)^2 + 24 \)[/tex]
- Vertex: [tex]\( (1, 24) \)[/tex]
- The coefficient [tex]\(-16\)[/tex] is negative, indicating it opens downwards.
- The vertex [tex]\( (1, 24) \)[/tex] suggests that the peak height of the T-shirt is 24 units high, reached at [tex]\( t = 1 \)[/tex] second.
2. Option 2: [tex]\( f(t) = -16(t+1)^2 + 24 \)[/tex]
- Vertex: [tex]\( (-1, 24) \)[/tex]
- Similar to option 1, it opens downwards.
- However, the vertex at [tex]\( (-1, 24) \)[/tex] suggests a peak height at a negative time, which is not practical for this scenario.
3. Option 3: [tex]\( f(t) = -16(t-1)^2 - 24 \)[/tex]
- Vertex: [tex]\( (1, -24) \)[/tex]
- Opens downwards, but the vertex is at [tex]\( (1, -24) \)[/tex].
- This suggests the maximum height is -24, which doesn’t make sense physically as a height value.
4. Option 4: [tex]\( f(t) = -16(t+1)^2 - 24 \)[/tex]
- Vertex: [tex]\( (-1, -24) \)[/tex]
- The shape is a downward parabola, vertex at negative time and negative height, which is not suitable.
Among these options, Option 1: [tex]\( f(t) = -16(t-1)^2 + 24 \)[/tex] is the most reasonable as it provides a realistic scenario where the T-shirt reaches its maximum height of 24 units at 1 second within the time frame of the event. Therefore, this is the function that could accurately represent the height of the T-shirt as a function of time.
[tex]\[ f(t) = a(t-h)^2 + k \][/tex]
Here, [tex]\( (h, k) \)[/tex] is the vertex of the parabola. The parabola can open upwards or downwards. If [tex]\( a \)[/tex] is negative, the parabola opens downwards, and if positive, it opens upwards. For trajectories, such as the path of a T-shirt, we are typically dealing with a downwards opening parabola due to gravity.
Let's analyze each option:
1. Option 1: [tex]\( f(t) = -16(t-1)^2 + 24 \)[/tex]
- Vertex: [tex]\( (1, 24) \)[/tex]
- The coefficient [tex]\(-16\)[/tex] is negative, indicating it opens downwards.
- The vertex [tex]\( (1, 24) \)[/tex] suggests that the peak height of the T-shirt is 24 units high, reached at [tex]\( t = 1 \)[/tex] second.
2. Option 2: [tex]\( f(t) = -16(t+1)^2 + 24 \)[/tex]
- Vertex: [tex]\( (-1, 24) \)[/tex]
- Similar to option 1, it opens downwards.
- However, the vertex at [tex]\( (-1, 24) \)[/tex] suggests a peak height at a negative time, which is not practical for this scenario.
3. Option 3: [tex]\( f(t) = -16(t-1)^2 - 24 \)[/tex]
- Vertex: [tex]\( (1, -24) \)[/tex]
- Opens downwards, but the vertex is at [tex]\( (1, -24) \)[/tex].
- This suggests the maximum height is -24, which doesn’t make sense physically as a height value.
4. Option 4: [tex]\( f(t) = -16(t+1)^2 - 24 \)[/tex]
- Vertex: [tex]\( (-1, -24) \)[/tex]
- The shape is a downward parabola, vertex at negative time and negative height, which is not suitable.
Among these options, Option 1: [tex]\( f(t) = -16(t-1)^2 + 24 \)[/tex] is the most reasonable as it provides a realistic scenario where the T-shirt reaches its maximum height of 24 units at 1 second within the time frame of the event. Therefore, this is the function that could accurately represent the height of the T-shirt as a function of time.
