Turns a described situation into a linear rule: choosing which quantity is the input, reading the rate and the initial value out of the words, and — the step most often missed — stating the domain the situation allows rather than the one the formula would accept. Distinguishes interpolation from extrapolation and names model breakdown as the reason the distinction matters.
Subject: Precalculus · 65 slides · symbolic lesson
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Title
Precalculus · Chapter 2 — Linear Functions
§2.3 Modeling with Linear Functions, pp. 233-246
Objectives
Five things, each one you can check yourself on paper.
OpenStax, Precalculus, §2.3 Modeling with Linear Functions §2.3, pp. 233-246 — the pages these objectives are drawn from
Warm-up
A model is a formula plus a claim about what it describes, and the formula carries none of the second part.
Discussion prompt
A tank holds 200 litres and drains at 3 litres per minute, modelled by 200 minus 3 times the minutes. What does that formula happily tell you that is nonsense?
Hint: Try putting in 100 minutes. Then try negative 10.
Answer:
At 100 minutes it returns negative 100 litres, which is not a volume. The tank emptied at about 66.7 minutes and stayed empty; the formula kept subtracting.
At negative 10 minutes it returns 230 litres, more than the tank holds. The formula does not know the tank has a capacity, or that time before the start has not happened.
So the formula is right on a limited stretch and meaningless outside it. Saying which stretch is part of the model, and the formula gives no hint that the question needs asking — which is exactly why it is the step that gets skipped.
Concept
Building a linear model means finding the rate and the initial value, writing the rule, and then saying what the numbers mean and where the rule applies. The last two are not optional extras.
\[ f(x) = \underbrace{m}_{\text{rate}}\,x + \underbrace{b}_{\text{initial value}}, \qquad x \in \text{[what the situation allows]} \]
The algebra here is entirely §2.1's. What is new is that every symbol now has a name, a unit, and a range of validity, and questions ask about those as often as they ask for a number. A bare formula with no units and no domain is an incomplete answer to a modelling question.
Figure (svg): A five-stage flow from a written situation to an interpreted answer, showing the modelling cycle: identify the variables, find the rate, find the initial value, write the rule, and check the domain
OpenStax, Precalculus, §2.3 Modeling with Linear Functions §2.3, pp. 233-236
Section
Section 1
Concept
Either of two related quantities can be taken as the input. The choice decides what the slope means and which questions the model answers directly.
It is worth noticing that both choices are legitimate models of the same situation. Neither is wrong; one is simply more convenient for the question at hand. If you build the inconvenient one, you can still answer, but you will be solving an equation where the other model would only have needed a substitution.
Figure (svg): A contrast showing the same pair of quantities modelled two ways, with the input and output swapped, giving two different slopes that are reciprocals of each other
OpenStax, Precalculus, §2.3 Modeling with Linear Functions §2.3, pp. 233-237
Picture it
One pair of quantities, two choices of input, two reciprocal slopes.
Figure (svg): A contrast showing the same pair of quantities modelled two ways, with the input and output swapped, giving two different slopes that are reciprocals of each other
Eighteen dollars per hour and one eighteenth of an hour per dollar describe the same job. The units tell you which model you have built, and they are the fastest check that you built the one you intended.
Worked example
Read the question first, then decide which quantity is the input.
\[ \text{A car uses fuel at } 0.08 \text{ litres per km from a } 45 \text{ litre tank. How far can it go?} \]
Read what is being asked for
Why: A distance is wanted.
Decide the input
Why: Distance is wanted, so it should be the input we solve for later, or the output.
Identify the rate and its sign
Why: Fuel decreases as distance grows.
\[ m = -0.08\text{ litres per } \text{km} \]
Identify the starting value
Why: The tank before any driving.
\[ b = 45\text{ litres} \]
Figure (svg): The solution to Worked example identify the variables shown as a ladder of expressions, one row per legal move
\[ F(d)=45-0.08d \]
Verify: check the units of the slope
Why: The slope is litres per kilometre, which is output units over input units — consistent with distance as the input and fuel as the output. Had the model been built the other way, the slope would have been kilometres per litre, about 12.5, which is the fuel economy rather than the consumption rate.
OpenStax, Precalculus, §2.3 Modeling with Linear Functions §2.3, pp. 234-236
Sorting
The verb in the sentence carries the sign, not the number.
Sort into buckets
Sort each described rate.
Worked example
The question determines whether you evaluate or solve.
\[ \text{Using } F(d)=45-0.08d, \text{ how far can the car travel before the tank is empty?} \]
Translate the question
Why: Empty means the output is zero.
\[ s e t F(d) = 0 \]
Write the equation
Why: The rule equals zero.
\[ 45 - 0.08 d = 0 \]
Solve for the input
Why: Divide 45 by 0.08.
\[ d = 562.5 \]
Attach the units
Why: The input was in kilometres.
\[ 562.5 \text{km} \]
Figure (svg): The solution to Worked example answer with the model shown as a ladder of expressions, one row per legal move
\[ d = 562.5 \text{ km} \]
Verify: sanity-check the size
Why: At 0.08 litres per kilometre, 45 litres should last roughly 45 divided by 0.08, which is a bit over 500 kilometres — the right order of magnitude for a full tank. Note this was a SOLVE rather than an evaluation, because the question gave the output; building the model the other way round would have made it a substitution instead.
OpenStax, Precalculus, §2.3 Modeling with Linear Functions §2.3, pp. 236-237
Trap
\[ \text{drains at } 3 \text{ litres per minute} \;\Longrightarrow\; V(t)=200+3t \]
Read the rate as 3 and put it in the model
Why: The number 3 appears in the problem without a minus sign, so it is used as written.
The model predicts the tank filling rather than emptying.
Draining means the output decreases, so the rate is negative 3 litres per minute regardless of how the number was written in the sentence.
The words carry the sign: drains, falls, loses and decreases are all negative; fills, rises, gains and grows are positive. The digit in the sentence never carries it.
Check the model's direction against the story. After one minute this model gives 203 litres, more than the tank started with, which is visibly wrong. One substitution catches every sign error of this kind.
Prediction
A model has slope 20 dollars per hour, and the input and output are swapped.
Predict first
What is the new slope?
Correct: One twentieth of an hour per dollar.
Why: Swapping the roles inverts the fraction that the rate is, so the number becomes its reciprocal and the units turn upside down. This is the inverse function from §1.7, arriving in an applied setting: the two models undo each other, and the reciprocal slope is exactly what the reflection in the line y equals x produces.
Matching
A modelling sentence usually contains both constants, in different grammatical shapes.
Match the pairs
Why: Phrases containing the word 'per' are almost always rates and give the slope. Phrases naming a one-off amount or a starting condition give the initial value. The last pair shows the sign coming from the verb rather than the number, which is the distinction the trap slide is about.
Step zero
You are given a paragraph describing a situation and asked to model it.
Discussion prompt
Before writing any algebra, what two questions do you answer, and why in that order?
Hint: What decides which quantity goes on which axis?
Answer:
First: what is being asked for? That quantity is normally the output, because then answering is an evaluation rather than solving an equation.
Second: what quantity is it being asked for in terms of? That is the input. Time is the input whenever it appears, since nothing predicts time from something else.
Answering these before any algebra prevents the most expensive error in the section, which is building a correct model of the wrong relationship. Once the two variables are named, with their units, the rate and the initial value can be read out of the words almost mechanically.
Section
Section 2
Concept
A linear model is determined by either a rate together with one value, or by two values. Both routes lead to the same rule through §2.1's forms.
The last bullet does more work than it looks. In a problem about years, deciding whether input zero means the year 2000 or the year 2020 changes the intercept completely, and two students with the same correct model can appear to disagree. Stating the convention removes the ambiguity.
Figure (svg): A five-stage flow from a written situation to an interpreted answer, showing the modelling cycle: identify the variables, find the rate, find the initial value, write the rule, and check the domain
OpenStax, Precalculus, §2.3 Modeling with Linear Functions §2.3, pp. 237-242
Picture it
Four of them are algebra you already have; the fifth is what makes it a model.
Figure (svg): A five-stage flow from a written situation to an interpreted answer, showing the modelling cycle: identify the variables, find the rate, find the initial value, write the rule, and check the domain
The order matters only in that the variables must be named first. Everything after that is §2.1 with units attached.
Worked example
Slope first, then point-slope, then interpret.
\[ \text{A town had } 20\,000 \text{ people in } 2010 \text{ and } 26\,000 \text{ in } 2016. \text{ Model it.} \]
Define the input
Why: Years since 2010, so the first data point is at zero.
\[ t =\text{ years since } 2010 \]
Compute the slope
Why: Change in people over change in years.
\[ \frac{6000}{6} = 1000\text{ per year} \]
Read the initial value
Why: The population at t equal to zero.
\[ b = 20000 \]
Write the rule
Why: Slope-intercept, since we have the value at zero.
\[ P(t) = 20000 + 1000 t \]
Figure (svg): The solution to Worked example from two data points shown as a ladder of expressions, one row per legal move
\[ P(t)=20\,000+1000t \]
Verify: check the second data point
Why: At t equal to 6 the model gives 20000 plus 6000, which is 26000 — matching the second data point exactly. Choosing 2010 as the origin made the intercept one of the given values, which is why defining the input that way was worth doing before computing anything.
OpenStax, Precalculus, §2.3 Modeling with Linear Functions §2.3, pp. 238-240
Faded example
A gym charges 60 to join and 35 a month. Model the total cost after m months.
Fill in the blanks
C(m) = 35m + 60
Why: The monthly charge is the rate and so multiplies the input; the joining fee is paid once and is the value at zero months. Checking: at m equal to 0 the cost is 60, the joining fee alone, and each further month adds 35. Swapping the two constants is the standard error and gives a model that is wrong from the first month onward.
Worked example
Point-slope handles it without finding the intercept first.
\[ \text{A plant is } 12 \text{ cm tall after } 3 \text{ weeks and } 26 \text{ cm after } 10 \text{ weeks.} \]
Compute the slope
Why: Change in height over change in weeks.
\[ \frac{14}{7} = 2 \text{cm}\text{ per week} \]
Use point-slope with either point
Why: The first is as good as the second.
\[ h - 12 = 2(w - 3) \]
Expand
Why: Distribute and collect.
\[ h = 2 w + 6 \]
Interpret the intercept
Why: The height at week zero.
\[ 6 \text{cm}\text{ when planted} \]
Figure (svg): The solution to Worked example when neither point is at zero shown as a ladder of expressions, one row per legal move
\[ h(w)=2w+6 \]
Verify: check both data points and the interpretation
Why: At 3 weeks: 6 plus 6 is 12. At 10 weeks: 20 plus 6 is 26. Both match. The intercept 6 is a genuine prediction rather than a given value — the model claims the plant was 6 cm at planting, which is a testable statement and might be wrong if growth was not linear from the start.
OpenStax, Precalculus, §2.3 Modeling with Linear Functions §2.3, pp. 240-242
Error analysis
A student models a population and reports the rule.
Annotate
On: \( P(t)=20\,000+1000t \)
Define every variable with its unit and its origin. It costs one sentence and it is the difference between a formula and a model.
Prediction
A model gives a plant's height in centimetres after w weeks, and its intercept is 6.
Predict first
What does the 6 claim?
Correct: The plant was 6 cm tall at week zero.
Why: The intercept is the output when the input is zero, so it is the height at the start. Growth per week is the slope, which is a different constant. Note that this is a prediction rather than an observation if week zero was not one of the measured times, and it may be a poor one if the plant did not grow linearly from planting.
Sorting
The information you are given decides the form.
Sort into buckets
Sort each set of given information.
Explain it to yourself
Modelling years, one student uses the calendar year and another uses years since 2010.
Discussion prompt
Do they get the same model? Explain what differs and what does not.
Hint: What happens to the slope, and what happens to the intercept?
Answer:
The slope is identical: the population grows at the same rate however you label the years, since the rate is a ratio of changes and shifting the origin changes neither change.
The intercepts differ enormously. Using years since 2010 gives an intercept of 20000, the population then. Using the calendar year gives the population the model would predict for the year zero, which is a large negative number and means nothing.
So the two models are the same line described on differently labelled axes, and both are correct provided each states its convention. The years-since version is preferred because its intercept is a meaningful quantity rather than a nonsensical extrapolation two thousand years back.
Section
Section 3
Concept
A linear formula accepts every real number. The situation it models usually does not, and the model's domain is the set of inputs for which its predictions mean something.
The discrete case is worth separate attention. A model of the cost of buying n items is only meaningful at whole numbers, so its graph is a set of dots rather than a line — even though drawing the line through them is standard practice and harmless as long as everyone knows the points between are not claims.
Figure (svg): A line drawn across a wide window with only part of it solid, the rest dashed, showing that a model's mathematical graph extends beyond the interval the situation permits
OpenStax, Precalculus, §2.3 Modeling with Linear Functions §2.3, pp. 242-245
Picture it
The dashed continuation is the formula; the solid stretch is the model.
Figure (svg): A line drawn across a wide window with only part of it solid, the rest dashed, showing that a model's mathematical graph extends beyond the interval the situation permits
The two ends are bounded for different reasons: the left because time before the start has not happened, the right because the tank is empty and the physical process stops.
Worked example
One end from the start of the situation, the other from where it breaks down.
\[ \text{For } V(t)=200-3t \text{ litres after } t \text{ minutes, state the domain.} \]
Consider the lower end
Why: Time before the process started has not happened.
\[ t \ge 0 \]
Consider the upper end
Why: Volume cannot be negative.
\[ 200 - 3 t \ge 0 \]
Solve that condition
Why: Divide 200 by 3.
\[ t \le 66.67 \]
Combine
Why: Both conditions at once.
\[ 0 \le t \le 66.67 \]
Figure (svg): A line drawn across a wide window with only part of it solid, the rest dashed, showing that a model's mathematical graph extends beyond the interval the situation permits
\[ 0 \le t \le \tfrac{200}{3} \approx 66.7 \text{ minutes} \]
Verify: check what happens at the endpoints
Why: At t equal to zero the model gives 200 litres, the full tank. At about 66.7 minutes it gives zero, the empty tank. Beyond that the formula gives negative volumes, which is where the model stops describing the tank — the water does not go negative, it simply stops.
OpenStax, Precalculus, §2.3 Modeling with Linear Functions §2.3, pp. 243-244
Discrimination
Each restriction comes from the situation, not the algebra.
Sort into buckets
Sort each restriction by which end it bounds.
Worked example
Some situations only permit whole-number inputs.
\[ \text{Tickets cost } 12 \text{ each with a } 4 \text{ booking fee. State the model and its domain.} \]
Write the rule
Why: Rate times tickets, plus the one-off fee.
\[ C(n) = 12 n + 4 \]
Ask what inputs are possible
Why: You cannot buy part of a ticket.
Consider the lower end
Why: Buying no tickets is possible, or not, by convention.
\[ n \ge 1\text{ typically} \]
State the domain
Why: Positive whole numbers.
\[ n = 1, 2, 3,... \]
Figure (svg): The solution to Worked example a discrete domain shown as a ladder of expressions, one row per legal move
\[ C(n)=12n+4, \quad n \in \{1,2,3,\dots\} \]
Verify: check what the line between the dots would claim
Why: At n equal to 2.5 the formula gives 34, which would be the cost of two and a half tickets — not a thing. The graph is genuinely a row of dots, and drawing a line through them is a convenience for seeing the trend rather than a claim about the points in between.
OpenStax, Precalculus, §2.3 Modeling with Linear Functions §2.3, pp. 244-245
Trap
\[ V(t)=200-3t \;\Longrightarrow\; \text{domain: all real numbers} \]
Apply the domain rules from Section 1.2
Why: There is no denominator and no even root, so the formula accepts every real number.
The domain is reported as the whole real line.
That is the formula's domain, not the model's. The model describes a tank, and the tank imposes limits the formula knows nothing about.
Negative t is time before the draining started, and t beyond about 66.7 gives negative volumes. Neither is a statement about a tank.
For a model, ask the situation rather than the formula. §1.2's rules find where the algebra breaks; here nothing algebraic breaks at all, and the limits come entirely from the story.
Prediction
A model gives a balance of 800 minus 45 times the number of weeks.
Predict first
Where does the model stop describing the account?
Correct: At about 17.8 weeks, when the balance reaches zero.
Why: Setting the output to zero and solving gives 800 divided by 45, which is about 17.8. Beyond that the formula predicts a negative balance. Whether that is meaningful depends on the account: an overdraft would make it real, which is a good illustration that the domain comes from the situation and not from the mathematics.
Faded example
A candle is 25 cm tall and burns 4 cm per hour. Find the upper end of the domain.
Fill in the blanks
25 - 4t \ge 0 \;\Longrightarrow\; t \le 6.25 \text___, \text___ 0 \le t \le ___
Why: Setting the height at or above zero and solving gives 25 over 4, which is 6.25 hours. After that the candle is gone; the formula would report negative heights, which describe nothing. The lower end is zero because the candle was not burning before it was lit.
Edge cases
A model's domain is bounded above at the point where the output reaches zero.
Discussion prompt
Is the endpoint itself included, and does it matter?
Hint: Does the situation happen at that exact instant?
Answer:
Usually yes, it is included: the tank really is empty at that instant, and the model's prediction of zero litres is correct there. So the interval is closed at that end.
It matters when the situation changes character at the endpoint rather than merely stopping. A candle that has burnt out has height zero at that instant, which is a real state; a projectile hitting the ground is a different case, since the model of free flight stops applying at impact rather than continuing to predict correctly.
The habit worth building is to say why each endpoint is where it is, in the situation's words. That usually settles the bracket without a rule, and it produces the sentence a modelling question is actually asking for.
Section
Section 4
Concept
Predicting inside the range of the data is interpolation and is comparatively safe. Predicting outside it is extrapolation, and the further out you go the more you are assuming.
model breakdown — The point beyond which a model's predictions stop being reliable, usually because the assumption of a constant rate of change no longer holds in the situation being described.
The distinction is not about the arithmetic, which is identical in both cases. It is about how much of the answer rests on evidence and how much on an assumption. That is why questions ask whether a prediction is reasonable as well as what it is, and the two answers are genuinely separate.
Figure (svg): A line fitted through a cluster of data points, extended far beyond them, with the far region shaded in warning colours and labelled as extrapolation
OpenStax, Precalculus, §2.3 Modeling with Linear Functions §2.3, pp. 245-246
Picture it
The same line, the same arithmetic, two very different levels of confidence.
Figure (svg): A line fitted through a cluster of data points, extended far beyond them, with the far region shaded in warning colours and labelled as extrapolation
Nothing about the line changes at the edge of the data. What changes is how much evidence supports the prediction, and the graph gives no warning at all.
Worked example
Same model, same arithmetic, different standing.
\[ \text{Data covers } 2010\text{--}2016 \text{ with } P(t)=20\,000+1000t. \text{ Predict } 2014 \text{ and } 2080. \]
Compute the first
Why: Four years after 2010.
\[ P(4) = 24000 \]
Classify it
Why: 2014 lies inside the data range.
Compute the second
Why: Seventy years after 2010.
\[ P(70) = 90000 \]
Classify it
Why: Far beyond the data.
Figure (svg): A line fitted through a cluster of data points, extended far beyond them, with the far region shaded in warning colours and labelled as extrapolation
\[ P(4)=24\,000 \text{ (interpolation)}; \; P(70)=90\,000 \text{ (extrapolation)} \]
Verify: say what the second one assumes
Why: It assumes the town grows by exactly 1000 people every year for seventy years, through every change in economy, housing policy and infrastructure. That is a very strong claim, supported by six years of data. The arithmetic is right and the prediction is not trustworthy, and those are separate judgements.
OpenStax, Precalculus, §2.3 Modeling with Linear Functions §2.3, pp. 245-246
Sorting
The data covers the inputs from 2 to 10.
Sort into buckets
Sort each prediction.
Worked example
Sometimes the breakdown point can be identified from the situation alone.
\[ \text{A child grows } 6 \text{ cm a year, modelled from ages } 4 \text{ to } 8. \text{ When must the model fail?} \]
State the model's assumption
Why: The same 6 cm every year, forever.
Test it far out
Why: At age 40, that is 216 cm added to the age-4 height.
Identify the real cause
Why: Human growth stops in the late teens.
State the breakdown
Why: The model fails from around the late teens onward.
Figure (svg): The solution to Worked example find where a model must break shown as a ladder of expressions, one row per legal move
\[ \text{Breakdown once growth ceases, in the late teens.} \]
Verify: note that the data could not have revealed this
Why: Nothing in the ages 4 to 8 data hints that growth will ever stop — those years really are close to linear. The breakdown was identified from knowledge of the situation, not from the numbers. That is generally true, and it is why a model needs someone who understands what it describes.
OpenStax, Precalculus, §2.3 Modeling with Linear Functions §2.3, pp. 246-246
Error analysis
A student uses a model built from six years of data.
Annotate
On: \( P(t)=20\,000+1000t \;\Longrightarrow\; P(500)=520\,000 \text{ people in } 2510 \)
A correct calculation and a reasonable prediction are two different things. Modelling questions usually want both judgements, and giving only the number answers half of what was asked.
Prediction
A linear model describes a population growing steadily.
Predict first
What is the model assuming about the distant future?
Correct: That the growth rate stays exactly constant forever.
Why: Linearity is precisely the claim that the rate of change never varies, so extending the model into the future extends that claim. Real populations saturate, decline, or change rate for a hundred reasons, which is why growth is more often modelled with the exponential and logistic functions of Chapter 4.
Two truths and a lie
Two of these are true of extrapolation and one is false.
Eliminate the wrong options
One of these claims is wrong.
Survives elimination: B
Why: B is the false claim. The arithmetic in a far extrapolation is perfectly correct — that is exactly what makes it dangerous. What fails is the assumption that the linear relationship still holds out there, which is a judgement about the situation rather than about the calculation.
Real world
Linear extrapolation is the default forecast in a great many places.
Discussion prompt
A company projects next year's sales by extending this year's growth linearly. What should they check, and what would a more honest forecast include?
Hint: How far beyond the data is one year, and what could change?
Answer:
They should check how far the projection reaches beyond the data and whether anything in the situation is likely to change the rate — market size, competitors, capacity, the economy.
A more honest forecast gives a range rather than a number, and states the assumption explicitly: this figure assumes growth continues at the current rate. That converts a hidden assumption into a stated one, which is the main thing the distinction in this section buys.
One year ahead is usually a modest extrapolation and often reasonable. Five years is not, and the same arithmetic produces both — which is why the judgement has to be made separately from the calculation, every time.
Section
Section 5
Concept
A modelling answer is not finished when the formula is written. The slope and intercept have interpretations, and comparing two models usually means comparing those.
The crossing point is where §2.2's intersection work pays off. Two options, one cheaper to start and one cheaper per unit, cross exactly once, and that input is the answer to 'when is it worth switching'. It is the most common applied question in the whole chapter.
Figure (svg): A contrast showing the same pair of quantities modelled two ways, with the input and output swapped, giving two different slopes that are reciprocals of each other
OpenStax, Precalculus, §2.3 Modeling with Linear Functions §2.3, pp. 236-242
Picture it
The same number means different things depending on which model it came from.
Figure (svg): A contrast showing the same pair of quantities modelled two ways, with the input and output swapped, giving two different slopes that are reciprocals of each other
Eighteen dollars per hour and one eighteenth of an hour per dollar are the same fact. Which one a question wants is decided by what it asked for.
Worked example
Two models, and the crossing point decides between them.
\[ \text{Plan A: } 30 + 2n. \text{ Plan B: } 10 + 4n. \text{ Which is cheaper?} \]
Compare the starting values
Why: B starts cheaper.
\[ B\text{ cheaper at } n = 0 \]
Compare the rates
Why: A grows more slowly.
Find where they cross
Why: Set the two equal.
\[ 30 + 2 n = 10 + 4 n \]
Solve
Why: Collect and divide.
\[ n = 10 \]
Figure (svg): The solution to Worked example compare two plans shown as a ladder of expressions, one row per legal move
\[ \text{Equal at } n=10; \; B \text{ cheaper below}, \; A \text{ above.} \]
Verify: test one input on each side
Why: At n equal to 5: A costs 40 and B costs 30, so B wins as claimed. At n equal to 20: A costs 70 and B costs 90, so A wins. The crossing point genuinely divides the two regions, and testing one point each side is the check that the inequality was read in the right direction.
OpenStax, Precalculus, §2.3 Modeling with Linear Functions §2.3, pp. 241-242
Comparison
Every linear model has both, and questions ask about them separately.
Comparison matrix
| the slope | the intercept | |
|---|---|---|
| what it is | the rate of change | the value at input zero |
| typical name | per-unit cost, speed, growth rate | fixed cost, starting amount |
| units | output units per input unit | output units |
| always meaningful? | yes | only if input zero makes sense |
The last row is worth remembering. An intercept found by extrapolating far back — a population in the year zero, say — is a number the formula produces and not a fact about anything.
Worked example
Questions ask for the meaning as often as for the number.
\[ \text{For } C(n)=30+2n \text{ dollars for } n \text{ items, interpret } 30 \text{ and } 2. \]
Interpret the constant term
Why: It is the cost at zero items.
\[ 30\text{ is } a\text{ fixed } \cos t \]
Give it a name from the situation
Why: Paid regardless of how many items.
Interpret the coefficient
Why: The extra cost per additional item.
\[ 2\text{ dollars per item} \]
State its units
Why: Output units per input unit.
Figure (svg): The solution to Worked example interpret both constants shown as a ladder of expressions, one row per legal move
\[ \text{fixed cost } 30 \text{ dollars}; \text{ marginal cost } 2 \text{ dollars per item} \]
Verify: check each interpretation against the formula
Why: Setting n to zero gives 30, confirming the fixed cost. Going from 5 items to 6 raises the cost from 40 to 42, a rise of 2, confirming the per-item figure. Each interpretation is tested by the calculation it claims to describe, which is what distinguishes an interpretation from a guess.
OpenStax, Precalculus, §2.3 Modeling with Linear Functions §2.3, pp. 236-238
Trap
\[ A: 30+2n, \quad B: 10+4n \;\Longrightarrow\; B \text{ is cheaper} \]
Compare the two fixed costs
Why: Ten is less than thirty, so B is declared the cheaper plan.
Plan B is recommended without reference to how many items are involved.
Which is cheaper depends on n. B starts cheaper but rises twice as fast, so A overtakes it at 10 items and is cheaper for every larger order.
A complete answer names the crossing point and says which plan wins on each side of it. That is the question the situation is actually asking.
Compare the rates as well as the starting values, and find where they cross. A recommendation with no quantity attached is answering a question nobody asked.
Prediction
Plan A has a high fixed cost and a low rate; plan B the reverse.
Predict first
Which is cheaper for a very large order?
Correct: A, because its rate is lower.
Why: For large inputs the rate dominates: the fixed cost is a one-off and the per-unit cost is paid every time. So the plan with the smaller slope eventually wins, however large its fixed cost. The fixed cost decides only the small orders, and the crossing point is where the advantage changes hands.
Faded example
Plan A costs 50 plus 3 per unit; plan B costs 20 plus 6 per unit.
Fill in the blanks
50 + 3n = 20 + 6n \;\Longrightarrow\; 30 = 3n \;\Longrightarrow\; n = 10
Why: Subtracting 20 and 3n from both sides gives 30 equal to 3n, so n is 10. Below 10 units B is cheaper because of its lower fixed cost; above 10 A wins because of its lower rate. Testing one value each side confirms the direction.
Explain it
Two plans, one cheap to start and one cheap per unit, is an extremely common situation.
Discussion prompt
Explain to someone choosing between two phone plans how to decide, in terms of slope and intercept.
Hint: What do they need to know about themselves before the mathematics can help?
Answer:
They need to estimate their own usage, because the answer depends entirely on it. Without that number the mathematics cannot recommend anything, and any recommendation made without it is guessing.
Then: the plan with the lower intercept wins for light usage, and the plan with the lower slope wins for heavy usage. The crossing point is the usage at which they cost the same, and it is the number worth computing.
The useful advice is therefore not 'plan A' but 'plan A if you use more than X, plan B otherwise', with X computed. That is a complete answer to a comparison question, and it is what the crossing point is for.
Comparison
Fill the blanks from memory. This is the distinction the section exists to make.
Comparison matrix
| the formula's domain | the model's domain | |
|---|---|---|
| found by asking | what breaks the algebra | what the situation allows |
| for a linear rule | every real number | usually a bounded interval |
| lower end typically | none | zero, where the situation starts |
| upper end typically | none | where the output becomes impossible |
| can be discrete | no | yes, when only whole inputs make sense |
The first row is the whole difference. §1.2's rules find where the algebra breaks, and for a linear rule they find nothing at all — every limit here comes from the story.
Pattern
Six steps, and the last two are the ones a formula alone will not remind you of.
Steps 5 and 6 are where the marks are in a modelling question, and they are the two steps with no algebraic trigger to remind you they exist.
OpenStax Algebra and Trigonometry 2e, §4.2 Modeling with Linear Functions §4.2
Check
The verb carries the sign.
Check your understanding
A pool holds 3000 litres and loses 25 litres an hour to evaporation. Which model is right?
Answer: A
Why: The starting value is 3000 litres and the rate is negative because the pool loses water. Checking after one hour gives 2975 litres, slightly less than the start, as it should be.
Check
Ask the situation, not the formula.
Check your understanding
For the model V(t) = 3000 - 25t litres after t hours, what is the model's domain?
Answer: A
Why: Time starts at zero, and the volume reaches zero when 25t equals 3000, which is at 120 hours. Beyond that the formula reports negative volumes, which describe nothing.
Check
How far from the data?
Check your understanding
A model is built from data covering years 1 through 8. Using it to predict year 30 is called what?
Answer: A
Why: Year 30 lies well outside the range the data covers, so the prediction rests on the assumption that the linear pattern continues rather than on evidence. The arithmetic is fine; the confidence is what is reduced.
Real world
Almost every projection you see in the news is a linear model with its domain unstated.
Discussion prompt
A headline says a trend, continued, means a certain figure by 2050. What is the phrase 'continued' hiding, and what should you check?
Hint: How much data supports the claim, and how far beyond it does the projection reach?
Answer:
'Continued' is doing the work of the entire modelling assumption: it claims the rate of change stays constant for every year between now and 2050, which is exactly the linearity assumption.
What to check is the ratio of the extrapolation to the data. A projection reaching thirty years beyond five years of data is resting almost entirely on the assumption; one reaching two years beyond twenty is resting mostly on evidence.
The honest version of such a headline states the assumption and gives a range. This section's vocabulary is exactly what lets you say what is missing: it is an extrapolation, and the question is where model breakdown is likely to fall.
Commit first
State your confidence along with your answer.
Predict first
A linear model of a cost has intercept 40 and slope 6. What does the 40 mean?
Correct: The cost when the input is zero, such as a fixed fee.
Why: The intercept is the output at input zero, which in a cost model is what is paid before any units are bought — a setup charge, joining fee or base rate. The per-unit cost is the slope, 6, and confusing the two produces a model that is wrong for every input except one.
Explain it
The test of understanding is being able to say why, not just what.
Discussion prompt
Explain to a classmate why stating a model's domain is part of the answer rather than an extra, using an example where ignoring it gives nonsense.
Hint: What does the tank formula say at 100 minutes?
Answer:
Give the tank: 200 litres draining at 3 a minute. At 100 minutes the formula reports negative 100 litres, which is not a quantity of water. The formula has no way to know the tank stopped.
So the formula is a correct description on a limited stretch and a meaningless one outside it. Saying which stretch is the difference between a model and an equation, and nothing in the algebra prompts you to say it.
A good explanation stresses that §1.2's domain rules cannot help here: there is no denominator and no root, so the algebra reports no restriction at all. The limits come from the situation, and only someone who understands the situation can find them.
Exit ticket
One honest answer, so the next lesson can start in the right place.
Predict first
Which idea from this lesson would you most want to see again?
Correct: Any of these is a legitimate answer; the useful one is the honest one.
Why: There is no correct choice here. The third is the most commonly omitted step and the easiest marks in the section once the habit is built. The second causes wrong answers that look entirely plausible, which makes it worth a few minutes of deliberate practice.
Connect it up
One page, drawn from memory, is worth more than rereading the section.
Draw it
Take any situation with a starting amount and a steady rate — a tank, a savings account, a candle. Write its model, and beside it write four things: what the input is with its units, what the slope means in the situation's words, what the intercept means, and the domain with a reason for each endpoint. Then sketch the graph with the allowed stretch solid and the rest dashed.
If both of your endpoints have a reason stated in the situation's words rather than in algebra, you have the step that separates a model from a formula.
Recap
Five things, and two of them have no algebraic prompt to remind you.
| if you remember one thing | it should be this |
|---|---|
| about the variables | define them with units before writing any algebra |
| about the rate's sign | the verb carries it, never the number |
| about the domain | ask the situation, not the formula |
| about predictions | correct arithmetic and a trustworthy prediction are separate judgements |
Section 2.4 handles the case this one assumed away: data that is nearly but not exactly linear, and how to fit a line to it and judge how well it fits.
OpenStax, Precalculus, §2.3 Modeling with Linear Functions §2.3, pp. 233-246 — everything on these slides traces back here
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