Question 116·Medium·Linear Functions
A cell-phone company bills its customers each month according to the formula
where is the total charge, in dollars, and is the number of minutes used that month.
What is the best interpretation of the number in this formula?
For questions about interpreting parts of a linear model, first identify which variable is the input (like minutes) and which is the output (like total cost). Recognize that the coefficient of the variable is the rate of change (e.g., cost per unit), while the constant term is the output when the input is zero. Quickly plug in 0 for the input to see what the constant means in context, then choose the option that matches that real-world interpretation exactly, watching out for distractors that confuse rate, maximum, or a specific computed value.
Hints
Focus on the structure of the formula
Look at the formula . One part has in it, and one part does not. What does the part without usually represent in a linear cost formula?
Think about using 0 minutes
What would the total charge be if the customer used minutes? How does that value relate to the number 18?
Separate usage costs from non-usage costs
The term changes when changes. The number 18 does not change. In a real-world billing situation, what kind of charge stays the same no matter how much you use?
Desmos Guide
Enter an equivalent equation in Desmos
In Desmos, type an equation using instead of , such as y = 0.12x + 18. This represents the same relationship between minutes (x) and total charge (y).
Find the value when no minutes are used
Either click on the graph where or type x = 0 as a second line to see the point where the graph crosses that vertical line. Look at the corresponding -value when and think about what that -value means in terms of the phone bill.
Connect the graph back to the context
Notice that the -intercept (where the graph crosses the -axis) is the same as the constant term in the equation. Interpret that -intercept in words: it is the total charge when the number of minutes (x) is 0.
Step-by-step Explanation
Identify what each symbol represents
The formula is .
- is the total charge in dollars.
- is the number of minutes used.
- is multiplied by .
- is a constant added on at the end.
We are asked specifically about the meaning of the number in this billing context.
Recall what a constant term means in a linear cost formula
In a linear formula of the form
the constant term is the value of the total cost when the amount is . It does not change when the amount changes; it is a fixed part of the cost.
Find the total charge when no minutes are used
To see what represents here, set (no minutes used) and calculate :
So even if a customer uses minutes, they still pay that month.
Match this meaning to the correct answer choice
Because the customer is charged even when they use minutes, is a fixed monthly charge that does not depend on usage. That matches the choice “The fixed monthly fee before any usage charges.”