Question

the team mascot shoots a rolled t-shirt from a special t-shirt cannon to a section of people in the stands at a basketball game. the t-shirt starts at a height of 8 feet when it leaves the cannon and 1 second later reaches a maximum height of 24 feet before coming back down to a lucky winner.

if the path of the t-shirt is represented by a parabola, which function could be used to represent the height of the t-shirt as a function of time, \\(t\\), in seconds?

\\(f(t) = -16(t + 1)^2 - 24\\)

\\(f(t) = -16(t - 1)^2 + 24\\)

\\(f(t) = -16(t + 1)^2 + 24\\)

\\(f(t) = -16(t - 1)^2 - 24\\)

Explanation:

Identify the vertex of the parabola

The problem states that the T-shirt reaches a maximum height of \(24\) feet at \(1\) second. This gives the vertex of the parabolic path as:

$$ (h, k) = (1, 24) $$

Write the vertex form of the quadratic function

The vertex form of a quadratic function is:

$$ f(t) = a(t - h)^2 + k $$

Substituting the vertex \((1, 24)\):

$$ f(t) = a(t - 1)^2 + 24 $$

Determine the leading coefficient using the initial value

The T-shirt starts at a height of \(8\) feet when it leaves the cannon at \(t = 0\), so \(f(0) = 8\):

$$ 8 = a(0 - 1)^2 + 24 $$

$$ 8 = a(1) + 24 \implies a = -16 $$

Substituting \(a = -16\) back into the vertex form yields:

$$ f(t) = -16(t - 1)^2 + 24 $$

Answer:

• \(f(t) = -16(t + 1)^2 - 24\)
• \(f(t) = -16(t - 1)^2 + 24\) (Correct answer)
• \(f(t) = -16(t + 1)^2 + 24\)
• \(f(t) = -16(t - 1)^2 - 24\)