Question

julia finds that the data in the table can be modeled by the function \\(y = 5(4)^x\\).

which statement about julias finding is true?

julia is correct because the distance starts at 5 feet and increases by a factor of 4.
julia is correct because the function is true for \\((0, 5)\\) and \\((1, 20)\\).
julia is not correct because the function is not true for the point \\((2, 80)\\).
julia is not correct because the distance does not increase by a constant factor each minute.

Explanation:

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<reasoning>

Evaluate the function at given table values

\[

$$\begin{aligned} &y = 5(4)^x\\ &x = 0 \implies y = 5(4)^0 = 5\\ &x = 1 \implies y = 5(4)^1 = 20\\ &x = 2 \implies y = 5(4)^2 = 80\\ &x = 4 \implies y = 5(4)^4 = 1280 e 320 \end{aligned}$$

\]

Analyze the rate of change in the table

\[

$$\begin{aligned} &\text{From } x=0 \text{ to } x=1: \Delta x = 1, \text{ factor } = \frac{20}{5} = 4\\ &\text{From } x=1 \text{ to } x=2: \Delta x = 1, \text{ factor } = \frac{80}{20} = 4\\ &\text{From } x=2 \text{ to } x=4: \Delta x = 2, \text{ factor } = \frac{320}{80} = 4\\ &\text{From } x=4 \text{ to } x=8: \Delta x = 4, \text{ factor } = \frac{640}{320} = 2 \end{aligned}$$

\]

Determine the correct statement

\[

$$\begin{aligned} &\text{The factor per minute is not constant:}\\ &\text{For } x \in [2, 4], \text{ factor per minute is } \sqrt{4} = 2\\ &\text{For } x \in [4, 8], \text{ factor per minute is } \sqrt[4]{2} \approx 1.19\\ &\implies \text{Julia is not correct because the distance does not increase by a constant factor each minute.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>Julia is correct because the distance starts at 5 feet and increases by a factor of 4.</mcq-option>
<mcq-option>Julia is correct because the function is true for (0, 5) and (1, 20).</mcq-option>
<mcq-option>Julia is not correct because the function is not true for the point (2, 80).</mcq-option>
<mcq-correct>Julia is not correct because the distance does not increase by a constant factor each minute.</mcq-correct>
</answer>

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Answer:

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<reasoning>

Evaluate the function at given table values

\[

$$\begin{aligned} &y = 5(4)^x\\ &x = 0 \implies y = 5(4)^0 = 5\\ &x = 1 \implies y = 5(4)^1 = 20\\ &x = 2 \implies y = 5(4)^2 = 80\\ &x = 4 \implies y = 5(4)^4 = 1280 e 320 \end{aligned}$$

\]

Analyze the rate of change in the table

\[

$$\begin{aligned} &\text{From } x=0 \text{ to } x=1: \Delta x = 1, \text{ factor } = \frac{20}{5} = 4\\ &\text{From } x=1 \text{ to } x=2: \Delta x = 1, \text{ factor } = \frac{80}{20} = 4\\ &\text{From } x=2 \text{ to } x=4: \Delta x = 2, \text{ factor } = \frac{320}{80} = 4\\ &\text{From } x=4 \text{ to } x=8: \Delta x = 4, \text{ factor } = \frac{640}{320} = 2 \end{aligned}$$

\]

Determine the correct statement

\[

$$\begin{aligned} &\text{The factor per minute is not constant:}\\ &\text{For } x \in [2, 4], \text{ factor per minute is } \sqrt{4} = 2\\ &\text{For } x \in [4, 8], \text{ factor per minute is } \sqrt[4]{2} \approx 1.19\\ &\implies \text{Julia is not correct because the distance does not increase by a constant factor each minute.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>Julia is correct because the distance starts at 5 feet and increases by a factor of 4.</mcq-option>
<mcq-option>Julia is correct because the function is true for (0, 5) and (1, 20).</mcq-option>
<mcq-option>Julia is not correct because the function is not true for the point (2, 80).</mcq-option>
<mcq-correct>Julia is not correct because the distance does not increase by a constant factor each minute.</mcq-correct>
</answer>

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