Answer :
To determine which function represents the height of the T-shirt as a function of time, we need to analyze the given options:
1. [tex]$f(t) = -16(t-1)^2 + 24$[/tex]
- This is a quadratic function describing a parabola that opens downwards because the coefficient of the squared term, [tex]\(-16\)[/tex], is negative.
- The vertex form of the parabola is [tex]\((h, k)\)[/tex], where in this function [tex]\(h = 1\)[/tex] and [tex]\(k = 24\)[/tex]. This means the vertex is at [tex]\((1, 24)\)[/tex].
- This vertex indicates the highest point of the trajectory, where [tex]\(t = 1\)[/tex] and the height is 24 units.
2. [tex]$f(t) = -16(t+1)^2 + 24$[/tex]
- This quadratic also opens downwards with a vertex at [tex]\((-1, 24)\)[/tex].
3. [tex]$f(t) = -16(t-1)^2 - 24$[/tex]
- Opens downwards with a vertex at [tex]\((1, -24)\)[/tex].
4. [tex]$f(t) = -16(t+1)^2 - 24$[/tex]
- Opens downwards with a vertex at [tex]\((-1, -24)\)[/tex].
Considering these characteristics:
- For a realistic projectile motion (like throwing a T-shirt), the vertex should represent the highest point, and this height should be positive (above ground level).
- Function [tex]$f(t) = -16(t-1)^2 + 24$[/tex] has a positive vertex height of 24, which makes it realistic for depicting a physical object's trajectory. The vertex [tex]\((1, 24)\)[/tex] implies that at [tex]\(t = 1\)[/tex] second, the T-shirt reaches its maximum height of 24 units.
Thus, the function [tex]$f(t) = -16(t-1)^2 + 24$[/tex] is the one that fits a typical projectile motion with a positive peak height, making it the correct option to represent the height of the T-shirt over time.
1. [tex]$f(t) = -16(t-1)^2 + 24$[/tex]
- This is a quadratic function describing a parabola that opens downwards because the coefficient of the squared term, [tex]\(-16\)[/tex], is negative.
- The vertex form of the parabola is [tex]\((h, k)\)[/tex], where in this function [tex]\(h = 1\)[/tex] and [tex]\(k = 24\)[/tex]. This means the vertex is at [tex]\((1, 24)\)[/tex].
- This vertex indicates the highest point of the trajectory, where [tex]\(t = 1\)[/tex] and the height is 24 units.
2. [tex]$f(t) = -16(t+1)^2 + 24$[/tex]
- This quadratic also opens downwards with a vertex at [tex]\((-1, 24)\)[/tex].
3. [tex]$f(t) = -16(t-1)^2 - 24$[/tex]
- Opens downwards with a vertex at [tex]\((1, -24)\)[/tex].
4. [tex]$f(t) = -16(t+1)^2 - 24$[/tex]
- Opens downwards with a vertex at [tex]\((-1, -24)\)[/tex].
Considering these characteristics:
- For a realistic projectile motion (like throwing a T-shirt), the vertex should represent the highest point, and this height should be positive (above ground level).
- Function [tex]$f(t) = -16(t-1)^2 + 24$[/tex] has a positive vertex height of 24, which makes it realistic for depicting a physical object's trajectory. The vertex [tex]\((1, 24)\)[/tex] implies that at [tex]\(t = 1\)[/tex] second, the T-shirt reaches its maximum height of 24 units.
Thus, the function [tex]$f(t) = -16(t-1)^2 + 24$[/tex] is the one that fits a typical projectile motion with a positive peak height, making it the correct option to represent the height of the T-shirt over time.
