Exponential Functions from Situations Quick Check

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Question
Use the image to answer the question.

An increasing curve with an arrow on the upper end is plotted on a coordinate plane. The x-axis is labeled people and ranges from 0 to 5 in increments of 1. The y-axis is labeled money and ranges from 0 to 500 in increments of 100.

Mason is completing a study for his psychology course. For the study, he begins with $20 and asks individuals on the street if they want the money or if they would like him to double it and give it to the next person. He continues to do this until someone takes the money. This situation can be modeled by the exponential equation m(p)=20(2)p , where m is the money and p is the number of people who opt to pass it on to the next person. Which of the following statements is true about the equation?

(1 point)
Responses

The range (m-values) includes the values (m≥20).
The range ( m -values) includes the values left parenthesis m greater than or equal to 20 right parenthesis .

As p increases, m decreases slowly at first and then quickly.
As p increases, m decreases slowly at first and then quickly.

As p increases, m increases quickly at first and then slowly.
As p increases, m increases quickly at first and then slowly.

The domain (p-values) include the values greater than or equal to 20 (p≥20).
The domain ( p -values) include the values greater than or equal to 20 left parenthesis p greater than or equal to 20 right parenthesis .
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Answer 1

To analyze the exponential equation \( m(p) = 20(2)^p \), let's break down the components:

1. Understanding the context:

   • \( m \) represents the money available after passing it on to the next person.
   • \( p \) represents the number of people who choose to pass the money on.

2. Examining the equation:

   • The equation starts with an initial value of $20 when \( p = 0 \).
   • As \( p \) increases, the value of \( m \) increases because the term \( (2)^p \) grows exponentially.

3. Range of the function:

   • When \( p = 0 \), \( m(0) = 20(2^0) = 20 \).
   • As \( p \) increases, \( m \) continues to increase without bound, so the range of \( m \) is \( m \geq 20 \). Therefore, the statement "The range (m-values) includes the values \( m \geq 20 \)" is true.

4. Behavior of the function:

   • As \( p \) increases, \( m(p) \) increases quickly at first due to the exponential nature of the function and then continues to grow quickly but at a larger scale. This means it does not decrease; it only increases.

5. Domain of the function:

   • The domain of the function is not restricted to \( p \geq 20 \); instead, \( p \) can take on any non-negative integer value (typically \( p \geq 0 \) in this context).

Conclusion:

The only true statement regarding the exponential equation \( m(p) = 20(2)^p \) is:

The range (m-values) includes the values \( m \geq 20 \).