Question

the graph shows the population of a bacteria in an experiment, measured every hour.

which function represents the population of the bacteria after \\(t\\) hours?

\\(f(t) = 10(1.4)^t\\)
\\(f(t) = 10(2.0)^t\\)
\\(f(t) = 14(1.4)^t\\)
\\(f(t) = 14(2.0)^t\\)

Explanation:

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<reasoning>

Identify the initial value from the graph

Using the Y-Intercept and Graph Interpretation knowledge points
\[
f(0) = 10
\]

Determine the growth factor

Using the Exponential Functions and Exponential Modeling knowledge points
\[

$$\begin{aligned} f(t) &= a \cdot b^t \\ f(0) &= a = 10 \\ f(1) &\approx 14 \\ 10 \cdot b^1 &= 14 \\ b &= 1.4 \end{aligned}$$

\]

Formulate the final function

Using the Exponential Functions knowledge point
\[
f(t) = 10(1.4)^t
\]
</reasoning>

<answer>
<mcq-correct>(A) \(f(t) = 10(1.4)^t\)</mcq-correct>
<mcq-option>(B) \(f(t) = 10(2.0)^t\)</mcq-option>
<mcq-option>(C) \(f(t) = 14(1.4)^t\)</mcq-option>
<mcq-option>(D) \(f(t) = 14(2.0)^t\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the initial value from the graph

Using the Y-Intercept and Graph Interpretation knowledge points
\[
f(0) = 10
\]

Determine the growth factor

Using the Exponential Functions and Exponential Modeling knowledge points
\[

$$\begin{aligned} f(t) &= a \cdot b^t \\ f(0) &= a = 10 \\ f(1) &\approx 14 \\ 10 \cdot b^1 &= 14 \\ b &= 1.4 \end{aligned}$$

\]

Formulate the final function

Using the Exponential Functions knowledge point
\[
f(t) = 10(1.4)^t
\]
</reasoning>

<answer>
<mcq-correct>(A) \(f(t) = 10(1.4)^t\)</mcq-correct>
<mcq-option>(B) \(f(t) = 10(2.0)^t\)</mcq-option>
<mcq-option>(C) \(f(t) = 14(1.4)^t\)</mcq-option>
<mcq-option>(D) \(f(t) = 14(2.0)^t\)</mcq-option>
</answer>

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