From a Verbal Description to a Function
A verbal description may tell you two important facts: the amount at the start, and how much the amount changes for each unit of input. When the per-unit change stays constant, those facts determine a simple function rule.
For example, a service might charge a fixed starting fee plus a fixed amount for every mile travelled. A container might begin with a certain quantity and lose the same amount each minute. The situations differ, but the structure is the same: begin with a fixed amount, then add the accumulated change.
Identify the Parts Before Writing the Rule
Begin by deciding what the input represents and what the output represents. Then identify the starting amount and the constant change for each unit of input. Giving each variable and quantity a clear meaning makes it easier to form the rule correctly.
- Input: the quantity that can vary, such as miles travelled or minutes elapsed.
- Output: the quantity determined by the input, such as total cost or remaining volume.
- Starting amount: the output when the input is \(0\).
- Rate of change: the amount the output changes for each one-unit increase in the input.
The rate can be positive or negative. If the output increases, use a positive rate. If it decreases, use a negative rate. The units of the rate combine output-units with input-units: for example, dollars per mile or liters per minute.
This rule describes the total output, not just the amount of change. The term \(rx\) is the accumulated change after an input of \(x\); the term \(a\) accounts for what was already present at the start.
Why the Rule Has This Form
Suppose the output begins at \(a\), and each input-unit contributes the same change \(r\). After one input-unit, the output is \(a+r\). After two, it is \(a+r+r=a+2r\). After \(x\) input-units, the accumulated change is \(r\) repeated \(x\) times, or \(rx\), giving \(a+rx\).
The same reasoning can be stated as a general justification. Assume that the change in output over any allowable input increase \(h\) is \(rh\), where \(r\) is constant. Taking the initial input to be \(0\), the change from \(0\) to \(x\) is \(rx\). Since the output at \(0\) is \(a\), the output at \(x\) must be \(a+rx\).
This argument also shows why the constant-rate condition matters. If the change per input-unit varies, then multiplying one fixed rate by \(x\) will not generally give the total change. The rule \(a+rx\) is appropriate when the description supports a constant rate over the inputs being modeled.
A Reliable Translation Process
State what the input measures and what the function value measures.
Use the output at input \(0\), not the amount added or removed per unit.
Use a positive rate for an increase and a negative rate for a decrease, with units attached.
Write starting amount plus rate times input, then test an input value against the verbal description.
Restrict the inputs if the real situation makes the rule invalid outside a certain range.
Worked Examples
Worked Example: A Taxi Fare with a Starting Charge
A taxi charges a fixed fee of \(4.50\) dollars, plus \(2.25\) dollars for each mile. Let \(m\) be the number of miles travelled and let \(C(m)\) be the fare in dollars. Find a function rule and the fare for a \(6\)-mile trip.
The input is \(m\), measured in miles. The output is \(C(m)\), measured in dollars. At \(m=0\), the fare is the fixed fee, so the starting amount is \(4.50\) dollars. The fare increases by \(2.25\) dollars per mile, so the rate is \(2.25\) dollars per mile.
For a \(6\)-mile trip, substitute \(m=6\):
The model gives a fare of \(18.00\) dollars. Check the calculation against the description: six miles at \(2.25\) dollars per mile costs \(13.50\) dollars, and adding the \(4.50\)-dollar starting fee gives \(18.00\) dollars. The input should satisfy \(m\geq 0\), since a trip cannot have a negative distance.
Worked Example: A Container Losing Liquid at a Constant Rate
A container holds \(120\) liters of liquid initially and loses \(7\) liters each minute. Let \(t\) be the elapsed time in minutes and \(V(t)\) the volume in liters. Find the rule and determine the volume after \(8\) minutes.
The initial volume is \(120\) liters, so \(V(0)=120\). The volume decreases by \(7\) liters per minute. Because this is a decrease, the signed rate is \(-7\) liters per minute. Therefore,
Substitute \(t=8\):
After \(8\) minutes, the model gives \(64\) liters. Check: the container loses \(7\cdot 8=56\) liters in \(8\) minutes, and \(120-56=64\) liters remain. The rule describes the draining process only until the container is empty. Solve \(120-7t\geq 0\): this gives \(t\leq 120/7\). Thus the relevant time domain is \(0\leq t\leq 120/7\). Continuing the rule to larger times would give negative volumes, which do not fit this situation.
Worked Example: A Fundraiser with an Amount Already Collected
A club has already collected \(35\) dollars. It then collects \(12\) dollars for every ticket sold. Let \(n\) be the number of additional tickets sold and \(R(n)\) be the total dollars collected. Find the rule and the total after selling \(9\) additional tickets.
When \(n=0\), the club still has the \(35\) dollars already collected. This is the starting amount. Each additional ticket adds \(12\) dollars, so the constant rate is \(12\) dollars per ticket. The function is
For \(9\) additional tickets:
The total is \(143\) dollars. Check the parts: the nine tickets bring in \(108\) dollars, and adding that to the original \(35\) dollars gives \(143\) dollars. Here \(n\) must be a nonnegative whole number, because it counts tickets. The formula can be evaluated for other real inputs algebraically, but those inputs would not represent a possible number of tickets in this context.
Check the Meaning, Units, and Domain
A correctly written expression should match the verbal relationship in more than one way. Its value at input \(0\) should equal the stated starting amount. Its rate should have the correct size and direction. Its units should fit the output, and its inputs should make sense in the situation.
| Verbal information | Role in the model | Example |
|---|---|---|
| Amount present at the start | Starting amount \(a\) | \(35\) dollars collected already |
| Amount added per unit | Positive rate \(r\) | \(12\) dollars per ticket |
| Amount removed per unit | Negative rate \(r\) | \(-7\) liters per minute |
| Number of units elapsed or used | Input \(x\) | Miles, minutes, or tickets |
Pay particular attention to what the input counts. “Already collected” describes the amount at input \(0\); “for each additional ticket” describes the rate. If the description uses a different reference point rather than an input of \(0\), first express the input relative to the stated start. The basic form \(a+rx\) uses \(x=0\) at the start of the modeled change.
Common Mistake: Multiplying the Rate but Forgetting the Start
Suppose a repair costs a \(30\)-dollar call-out fee plus \(20\) dollars per hour. A common incorrect rule is \(C(h)=20h\). For \(3\) hours, that gives \(C(3)=20(3)=60\) dollars. But this calculation counts only the hourly charge and leaves out the call-out fee.
The correct rule includes both parts:
Check the verbal description: three hours cost \(60\) dollars at the hourly rate, and the fixed fee adds \(30\) dollars, for a total of \(90\) dollars. A quick way to catch the mistake is to set \(h=0\). The incorrect rule gives \(C(0)=0\), but the description says the call-out fee is charged even when no repair time has elapsed, so the model must give \(C(0)=30\).
Why Building the Model Carefully Matters
Turning words into a rule requires preserving the meaning of each quantity. A fixed starting amount is not interchangeable with a per-unit rate, and a decrease must be represented with the right sign. Once these pieces are identified, the function makes predictions for any appropriate input and gives a precise way to check whether a proposed answer fits the situation.
Check Your Understanding
For each situation, identify the input, output, starting amount, rate, and an appropriate function rule. Check that the sign and units make sense.
- A parking garage charges a \(6\)-dollar entry fee and \(3\) dollars for each hour parked. What is the charge after \(5\) hours?
- A \(90\)-liter tank loses \(4\) liters per minute. Write a rule for the volume after \(t\) minutes, and state when the model first reaches zero.
- A baker has \(18\) muffins ready and bakes \(12\) more each hour. How many muffins are ready after \(3\) hours?
- A subscription account starts with \(75\) dollars and is charged \(9\) dollars each month. Write a rule for the balance after \(m\) months. Why is the rate negative?
- In a model \(P(x)=11+5x\), what do \(11\) and \(5\) represent if \(P\) is measured in dollars and \(x\) counts items?
