The correct reasoning is:

The amount of money you pay in a month is a linear function because the amount paid per month has a constant growth rate based on the number of movies ordered.

This is because the total cost increases by a fixed amount ($3.99) for each additional movie ordered. The flat rate ($8) is a constant that doesn’t change, and the charge per movie is a constant increase. This makes the total cost a linear function of the number of movies ordered.

Growth rate and initial value:

• Growth rate: The cost per additional movie ordered, which is $3.99. This is the amount the total cost increases for each additional movie, so the growth rate is 3.99 per movie.
• Initial value: The flat rate charge of $8. This is the cost when no movies are ordered, so the initial value is 8 dollars.

Formula for C:

Using $m$ for the number of movies ordered, and $C$ for the total cost in dollars, the formula for the total monthly cost is:

$C = 8 + 3.99m$

Where:

• $8$ is the initial value (the flat rate),
• $3.99$ is the growth rate (the cost per movie),
• $m$ is the number of movies ordered.

Would you like further explanation on how the formula is derived or have any other questions about linear functions?

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Here are 5 related questions to expand on this:

1. What does it mean for a function to have a "constant growth rate"?
2. How can you graph a linear function like $C = 8 + 3.99m$?
3. How would you modify the formula if the flat rate increased to $10 per month?
4. How can you calculate the total cost for 5 on-demand movies using the formula?
5. What happens to the total cost if you order 0 movies?

Tip: When dealing with linear functions, identifying the "initial value" and "growth rate" helps in quickly formulating the equation.