Answer :
To solve this problem, we consider the characteristics of a parabolic motion, such as the T-shirt's path.
We're given a set of quadratic functions that describe the height of the T-shirt as a function of time [tex]\( t \)[/tex], and we want to identify the correct function.
The general form of a quadratic function for this type of motion is:
[tex]\[ f(t) = a(t-h)^2 + k \][/tex]
Here, [tex]\( (h, k) \)[/tex] represents the vertex of the parabola. Since the path of the T-shirt is influenced by gravity, the coefficient [tex]\( a \)[/tex] is typically negative, reflecting the downward opening of the parabola.
Let's analyze each function option given:
1. [tex]\( f(t) = -16(t-1)^2 + 24 \)[/tex]
- The vertex form, [tex]\(-16(t-1)^2 + 24\)[/tex], indicates that the vertex [tex]\((h, k)\)[/tex] is at [tex]\((1, 24)\)[/tex]. This suggests that the maximum height of the T-shirt is 24 units at [tex]\( t = 1 \)[/tex] second.
2. [tex]\( f(t) = -16(t+1)^2 + 24 \)[/tex]
- Here, the vertex [tex]\((h, k)\)[/tex] is [tex]\((-1, 24)\)[/tex], indicating the maximum height is also 24 units, but at [tex]\( t = -1 \)[/tex], which doesn't make sense since time can't be negative in this context.
3. [tex]\( f(t) = -16(t-1)^2 - 24 \)[/tex]
- This function's vertex is [tex]\((1, -24)\)[/tex], suggesting that the maximum height is a negative value. This is physically unrealistic for the height of a T-shirt thrown upwards.
4. [tex]\( f(t) = -16(t+1)^2 - 24 \)[/tex]
- The vertex [tex]\((-1, -24)\)[/tex] suggests both a negative maximum height and an unrealistic time.
Considering the physical context and realistic values, the function [tex]\( f(t) = -16(t-1)^2 + 24 \)[/tex] is suitable. It has a maximum height of 24 at time [tex]\( t = 1 \)[/tex], which is reasonable and realistic.
Therefore, the correct function to describe the height of the T-shirt as a function of time is:
[tex]\[ f(t) = -16(t-1)^2 + 24 \][/tex]
We're given a set of quadratic functions that describe the height of the T-shirt as a function of time [tex]\( t \)[/tex], and we want to identify the correct function.
The general form of a quadratic function for this type of motion is:
[tex]\[ f(t) = a(t-h)^2 + k \][/tex]
Here, [tex]\( (h, k) \)[/tex] represents the vertex of the parabola. Since the path of the T-shirt is influenced by gravity, the coefficient [tex]\( a \)[/tex] is typically negative, reflecting the downward opening of the parabola.
Let's analyze each function option given:
1. [tex]\( f(t) = -16(t-1)^2 + 24 \)[/tex]
- The vertex form, [tex]\(-16(t-1)^2 + 24\)[/tex], indicates that the vertex [tex]\((h, k)\)[/tex] is at [tex]\((1, 24)\)[/tex]. This suggests that the maximum height of the T-shirt is 24 units at [tex]\( t = 1 \)[/tex] second.
2. [tex]\( f(t) = -16(t+1)^2 + 24 \)[/tex]
- Here, the vertex [tex]\((h, k)\)[/tex] is [tex]\((-1, 24)\)[/tex], indicating the maximum height is also 24 units, but at [tex]\( t = -1 \)[/tex], which doesn't make sense since time can't be negative in this context.
3. [tex]\( f(t) = -16(t-1)^2 - 24 \)[/tex]
- This function's vertex is [tex]\((1, -24)\)[/tex], suggesting that the maximum height is a negative value. This is physically unrealistic for the height of a T-shirt thrown upwards.
4. [tex]\( f(t) = -16(t+1)^2 - 24 \)[/tex]
- The vertex [tex]\((-1, -24)\)[/tex] suggests both a negative maximum height and an unrealistic time.
Considering the physical context and realistic values, the function [tex]\( f(t) = -16(t-1)^2 + 24 \)[/tex] is suitable. It has a maximum height of 24 at time [tex]\( t = 1 \)[/tex], which is reasonable and realistic.
Therefore, the correct function to describe the height of the T-shirt as a function of time is:
[tex]\[ f(t) = -16(t-1)^2 + 24 \][/tex]
