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:

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<reasoning>

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:
\[
f(t) = -16(t - 1)^2 + 24
\]
</reasoning>

<answer>
<mcq-option>\(f(t) = -16(t + 1)^2 - 24\)</mcq-option>
<mcq-correct>\(f(t) = -16(t - 1)^2 + 24\)</mcq-correct>
<mcq-option>\(f(t) = -16(t + 1)^2 + 24\)</mcq-option>
<mcq-option>\(f(t) = -16(t - 1)^2 - 24\)</mcq-option>
</answer>

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Answer:

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<reasoning>

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:
\[
f(t) = -16(t - 1)^2 + 24
\]
</reasoning>

<answer>
<mcq-option>\(f(t) = -16(t + 1)^2 - 24\)</mcq-option>
<mcq-correct>\(f(t) = -16(t - 1)^2 + 24\)</mcq-correct>
<mcq-option>\(f(t) = -16(t + 1)^2 + 24\)</mcq-option>
<mcq-option>\(f(t) = -16(t - 1)^2 - 24\)</mcq-option>
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