Question

The number of students riding the bus in one district can be represented by an exponential function, which is modeled by the equation

y=3,765e
−0.25x
. In this equation, x represents the time in years since 2010, and y represents the number of students riding the bus. Complete the sentences interpreting the exponential equation.

The initial value of the function is _______.
The initial value represents the number of _______.
Since the value of
b is _______.
This means there is _______.

Answer 1

For the complete sentences interpreting the exponential equation:

• The initial value of the function is 3,765.
• The initial value represents the number of students riding the bus in 2010.
• Since the value of b is -0.25.
• This means there is a decrease in the number of students riding the bus each year.

How to interpret function?

1. To interpret the exponential function
`\( y = 3,765e^(-0.25x) \)`, break down its components:

Initial Value: The initial value of an exponential function is the value of y when x = 0.

In this case, when x = 0 (the year 2010), the equation becomes y = 3,765e⁰.

Since e⁰ = 1, the initial value is y = 3,765.

So, the initial value of the function is 3,765.

2. Interpretation of the initial value: The initial value represents the number of students riding the bus at the start of the observation period, which is the year 2010.

Therefore, the initial value represents the number of students riding the bus in 2010.

3. Value of b: In this equation, b is the coefficient of x in the exponent, which is -0.25. The value of b is -0.25.

4. Meaning of the value of b: Since b is negative, the function is a decreasing exponential function.

This indicates that the number of students riding the bus is decreasing over time.

Then, there is a decrease in the number of students riding the bus each year.