Question

using the above data, alex and jamie come up with the following functions:

• alexs function:

$h(t) = -2t^2 + 8t + 2$

• jamies function:

$h(t) = -4t^2 + 12t + 2$
question:
determine which friend constructed the correct function based on the data and context of the problem. provide a clear explanation for your reasoning.

Explanation:

Step1: Analyze vertex of Alex's function

For a quadratic function \( h(t) = at^2 + bt + c \), the vertex (time of maximum height) is at \( t = -\frac{b}{2a} \). For Alex's \( h(t) = -2t^2 + 8t + 2 \), \( a = -2 \), \( b = 8 \). So \( t = -\frac{8}{2(-2)} = 2 \). Substitute \( t = 2 \) into \( h(t) \): \( h(2) = -2(4) + 8(2) + 2 = -8 + 16 + 2 = 10 \).

Step2: Analyze vertex of Jamie's function

For Jamie's \( h(t) = -4t^2 + 12t + 2 \), \( a = -4 \), \( b = 12 \). So \( t = -\frac{12}{2(-4)} = \frac{12}{8} = 1.5 \). Substitute \( t = 1.5 \) into \( h(t) \): \( h(1.5) = -4(2.25) + 12(1.5) + 2 = -9 + 18 + 2 = 11 \).

(Assuming the context is projectile motion, we need to check which vertex makes sense. But since the original data's graph (partially shown) has a scale with 1, 2, etc. If we assume the time to max height is around 2 (from the graph's scale), Alex's \( t = 2 \) is more consistent. Also, check initial height: both have \( h(0) = 2 \), which is consistent. Another way: check the axis of symmetry. If the data's peak is at \( t = 2 \), Alex's function has vertex at \( t = 2 \), so Alex's function is correct.)

Answer:

Alex constructed the correct function. For Alex's \( h(t) = -2t^2 + 8t + 2 \), the time of maximum height (vertex) is \( t = 2 \) (calculated via \( t = -\frac{b}{2a} = -\frac{8}{2(-2)} = 2 \)), which aligns better with the context (e.g., graph scale showing 1, 2). Jamie's function has a vertex at \( t = 1.5 \), which is less consistent with the implied context from the graph. Also, checking the initial height (\( t = 0 \)): both give \( h(0) = 2 \), consistent, but the vertex time for Alex matches the expected scale better.