Real-World Mathematics: Graphs, Functions, Optimisation and Systems

World → variables → relationship → representation → constraints → calculation → optimum or prediction → validation → world.

1. A variable is a quantity allowed to change

Real systems contain quantities that vary. Time passes. Distance accumulates. Demand rises and falls. Temperature changes. Price affects quantity purchased. A variable gives Mathematics a way to represent one of these changing quantities without replacing it with a new symbol every time its value changes.

Variables are useful because they allow us to describe a family of situations at once. The formula C = 4n + 12 can represent a fixed charge of 12 plus 4 per unit for any allowed value of n. It is not one arithmetic problem; it is a rule generating many arithmetic problems.

2. A function is a controlled relationship between variables

A function tells us how an output is determined from an input. In a simple model, distance may depend on time, cost may depend on quantity, or height may depend on time after an object is launched.

The function is the relationship. The equation, table and graph are representations of that relationship.

• Equation: compact symbolic rule.
• Table: selected input-output pairs.
• Graph: visual structure across a range.
• Words: contextual interpretation.

A mathematically mature learner can move between these forms without treating them as separate topics.

3. Linear functions describe constant rates of change

A linear model has constant gradient. In y = mx + c, m gives the change in y for each unit change in x, while c gives the value of y when x = 0 if that interpretation makes sense in context.

Worked example: delivery cost

A delivery service charges a fixed $8 plus $1.50 per kilometre. Let d be distance in kilometres and C be cost. Then C = 1.5d + 8.

• Gradient 1.5 means cost rises by $1.50 for each additional kilometre.
• Intercept 8 represents the fixed charge.
• At 10 km, C = 1.5(10) + 8 = $23.

The equation makes the pricing structure visible. The graph would show every modelled distance-cost pair at once.

4. Gradient is a rate, not just a triangle calculation

In school, gradient is often calculated as rise/run. In applications it is better understood as a rate of change carrying units.

• Distance-time graph gradient can represent speed.
• Cost-quantity graph gradient can represent marginal cost in a simple linear model.
• Temperature-time graph gradient can represent rate of heating or cooling over an interval.
• Height-distance graph gradient can represent slope.

The axes determine the unit. If vertical axis is dollars and horizontal axis is kilograms, gradient has units dollars per kilogram.

5. The intercept may have a real meaning—or no sensible physical meaning

In a taxi-style fare model, the vertical intercept may represent a starting charge. In a temperature model fitted over a short range, extrapolating back to x = 0 may produce an intercept with little practical meaning. The mathematics allows an intercept; the context decides whether it deserves interpretation.

6. Piecewise functions model systems with rules that change

Many real pricing and operating systems are not described by one rule over the entire domain. A parking fee may use one rate for the first hour and another thereafter. Shipping may be free below one threshold and charged above it. A tax or tariff can change at a boundary.

A piecewise function uses different formulas in different regions. The mathematics matches the system by preserving the rule change instead of forcing one line through everything.

Worked example: simple parking model

Suppose parking costs $3 for the first hour or part thereof, then $2 for each additional full hour in a simplified model. A 1-hour stay costs $3; a 2-hour stay costs $5; a 4-hour stay costs $9. This is not a continuous straight-line relationship because the charging rule is discrete and stepped.

The lesson is broader: choose a representation that respects how the real rule works.

7. Systems of equations locate where conditions meet

When two relationships must hold at the same time, solving simultaneous equations finds their common solution. Graphically, this is the intersection point.

Worked example: break-even comparison

Plan A costs C = 20 + 2x. Plan B costs C = 8 + 3x. At the break-even quantity, costs are equal:

20 + 2x = 8 + 3x, so x = 12. At x = 12, both cost $44.

For quantities below 12, compare one test point; for quantities above 12, compare another. The intersection is not merely a coordinate. It is a decision boundary.

8. Quadratic functions model change that is not constant

Quadratic relationships appear when the rate of change itself changes in a regular way. School examples include projectile height and area problems. In optimisation, the turning point of a quadratic can represent a maximum or minimum.

Worked example: maximum area with fixed perimeter

A rectangle has perimeter 40 m. Let one side be x. Then the other side is 20 − x. Area A = x(20 − x) = 20x − x².

This quadratic opens downward. Its maximum occurs at x = 10, giving A = 100 m². The optimum rectangle is a square.

The real-world interpretation requires constraints: 0 < x < 20. Values outside that interval do not describe a physical rectangle.

9. Exponential functions model multiplicative change

Linear growth adds a constant amount. Exponential growth multiplies by a constant factor. Repeated percentage growth therefore naturally creates exponential structure.

A model P = P₀(1.03)^t represents 3% growth per period under the assumption that the same rate continues and compounds. Such models can describe idealised population growth, repeated financial change or decay processes over ranges where the assumptions are reasonable.

Exponential models should not be extrapolated carelessly. Real resources, policies, capacity and behaviour can change. A strong model always carries its assumptions with it.

10. A graph reveals structure faster than a table—but can also mislead

Graphs reveal trend, turning points, intersections, outliers and rates of change. They are powerful precisely because human vision can detect structure quickly.

That same power makes graphs vulnerable to misleading choices. A truncated vertical axis can exaggerate differences. Unequal intervals can distort change. Connecting discrete categories with a line can imply continuity that does not exist. A smoothed curve can hide raw variation.

Always read the axes, units, scale and domain before interpreting the shape.

11. Domain and range are real constraints

The domain of a model is the set of allowed inputs. In real work, this is not merely notation. Negative quantities may be impossible; a machine may operate only between certain temperatures; an age model may be valid only for a specific population; a pricing rule may apply only to a stated quantity range.

An algebraic formula may produce an answer outside the valid domain. Mathematics then has two stages: solve the equation, and decide whether the solution belongs to the real system.

12. Inequalities represent feasible regions

Many real decisions are constrained rather than exact. A budget cannot exceed a limit. Capacity must not be violated. A minimum service level must be met. A component must stay within tolerance. These are inequality conditions.

Several inequalities together define a feasible region: the set of choices that satisfy every constraint.

Worked example: production constraints

Suppose product X uses 2 hours of machine time and product Y uses 1 hour, with at most 40 machine hours available. Then 2x + y ≤ 40. If at least 10 units in total must be produced, x + y ≥ 10. Also x ≥ 0 and y ≥ 0.

The valid production choices are not one point but a region satisfying all inequalities.

13. Optimisation means choosing the best feasible option for a stated objective

Optimisation requires three ingredients:

• Decision variables: what can be chosen?
• Constraints: what choices are allowed?
• Objective: what does “best” mean?

Without an objective, there is no optimum. Without constraints, the mathematical optimum may be impossible. Without correctly chosen variables, the model may optimise the wrong system.

Objectives can include minimising cost, time, distance or error; maximising capacity, profit, coverage or reliability; or balancing several competing goals.

14. “Best” is conditional on what was asked

The shortest route may not be the fastest. The cheapest option may not be the most reliable. The highest output may use unacceptable energy. A model can optimise only the objective it is given.

An optimisation result is never simply “the best.” It is the best under this objective, these constraints and these assumptions.

15. Networks turn places and connections into mathematical objects

Roads, rail lines, computer networks, supply chains and social connections can be represented as graphs in the graph-theory sense: nodes connected by edges.

An edge can carry a weight such as distance, time, cost or capacity. A shortest-path problem then searches for the route with minimum total weight. Change the weight, and the “best” route may change.

Worked mini network

Suppose A→B costs 4, B→D costs 5, A→C costs 3 and C→D costs 8. Route A-B-D has total 9; A-C-D has total 11. If weight means distance, A-B-D is shorter. If congestion changes B-D to 10, A-C-D becomes preferable at 11 versus 14. The structure is stable; the data can change the decision.

16. Queues are systems of competing rates

A queue forms when arrivals and service interact. If customers arrive faster than they can be served for long enough, the queue grows. If service capacity comfortably exceeds average demand, the system may remain stable, though random bursts can still create waiting.

Even without advanced queueing theory, school Mathematics can identify the central quantities: arrival rate, service rate, number of servers, waiting time and utilisation. Ratios, averages, probability and functions all meet here.

17. Discrete and continuous models are not interchangeable

People, buses and boxes are counted discretely. Time, distance and temperature are often modelled continuously. A formula may suggest 3.6 workers, but a staffing decision requires whole people. A continuous optimum may need to be checked against nearby integer choices.

Rounding is not always a neutral final step. If capacity must be met, rounding down may violate a constraint. The direction of rounding can be part of the decision.

18. Calculus studies local change and accumulation

At more advanced levels, differentiation gives a precise language for instantaneous rate of change and optimisation, while integration describes accumulation. These ideas extend the same real-world questions already visible in gradients and areas under graphs.

A derivative equal to zero may identify a stationary point, but the context still decides whether it is a maximum, minimum or irrelevant point. An integral can accumulate flow over time, but only if the rate function and units are correctly defined.

19. Sensitivity asks whether the decision survives small changes

A model can produce one exact-looking optimum even when small changes in assumptions would produce a different choice. Sensitivity analysis changes important inputs and observes how much the output moves.

If a route remains best across a wide range of travel-time assumptions, the decision is robust. If a 1% cost change flips the recommendation, the boundary matters and should be communicated.

20. Interpolation is safer than uncontrolled extrapolation

Interpolation estimates within the range of observed data. Extrapolation extends beyond it. Extrapolation is riskier because the relationship may change outside the observed range.

A linear trend fitted between x = 10 and x = 20 may be useful for x = 15. Predicting x = 100 assumes the same relationship continues much farther than the evidence shows.

21. A practical modelling loop for functions and systems

1. State the real question.
2. Choose variables and units.
3. Identify the relationship. Linear, quadratic, exponential, piecewise, networked or something else?
4. State the domain and constraints.
5. Choose a representation. Equation, table, graph or network.
6. Calculate or optimise.
7. Test sensitivity and limits.
8. Return the result to the real decision.

22. Worked real-world mini cases

Case A: phone-plan comparison

Plan A costs 18 + 0.08m, where m is usage units. Plan B costs 30 + 0.03m. Break-even occurs when 18 + 0.08m = 30 + 0.03m, so 0.05m = 12 and m = 240. Below 240 units, Plan A is cheaper; above 240, Plan B is cheaper under the simplified pricing assumptions.

Case B: container dimensions

A rectangular pen has 60 m of fencing and one side is x. If all four sides are fenced, the other side is 30 − x and area is A = x(30 − x). Maximum occurs at x = 15, so the square gives greatest area, 225 m².

Case C: route disruption

A shortest route through a network can change instantly when one edge becomes unavailable or its travel-time weight rises. Network Mathematics explains why resilient systems often maintain alternative paths rather than relying on one globally shortest route.

Case D: capacity threshold

A room allows at most 80 occupants. If groups contain 6 people each, 13 groups give 78 occupants and 14 groups give 84. The continuous division 80/6 ≈ 13.33 must be converted to an integer decision by rounding down because capacity is a hard upper constraint.

23. Common failure modes

• Axis blindness: graph shape is interpreted without reading scales and units.
• Intercept over-interpretation: y-intercept is given a physical meaning outside the valid domain.
• Wrong function family: a linear rule is used for multiplicative or curved behaviour.
• Constraint omission: an algebraically valid optimum is physically impossible.
• Continuous-to-discrete error: fractional people or vehicles are accepted without an integer decision.
• Objective confusion: “best” is claimed without stating what is being minimised or maximised.
• Extrapolation failure: a local trend is projected far outside the evidence.
• No sensitivity test: a fragile optimum is reported as though it were robust.

24. Practice set

1. For C = 5x + 12, identify gradient and intercept.
2. A service charges $7 fixed plus $2 per unit. Write a function for total cost C in terms of x.
3. Using that function, find cost for 15 units.
4. Plan A: C = 10 + 4x. Plan B: C = 28 + 2x. Find break-even x.
5. For y = 3x − 5, find y when x = 7.
6. A distance-time graph rises from 20 km at 1 h to 80 km at 3 h. Find average gradient over the interval.
7. A rectangle has perimeter 24 m. Write area as a quadratic in one side x.
8. Find the maximum area of that rectangle.
9. A population model is P = 500(1.02)^t. What does 1.02 mean?
10. State a reasonable domain for number of boxes x in a packing model.
11. If 3x + 2y ≤ 30, does x = 4, y = 8 satisfy the constraint?
12. A route has edge weights 5, 3 and 4. Find total route weight.
13. Two routes have total times 18 min and 21 min. If a delay adds 5 min to the first route, which becomes faster?
14. A system requires at least 100 seats. Vehicles hold 36 seats each. What minimum number of vehicles is needed?
15. Explain why 100/36 cannot simply be rounded to 2.
16. A quadratic objective has its maximum at x = 12, but the allowed domain is 0 ≤ x ≤ 10. Where must the constrained maximum occur?
17. A graph uses a vertical axis from 98 to 102. Why can a small difference look visually dramatic?
18. Explain the difference between interpolation and extrapolation.

25. Answers and reasoning

1. Gradient 5; intercept 12.
2. C = 2x + 7.
3. $37.
4. 10 + 4x = 28 + 2x gives 2x = 18, so x = 9.
5. 16.
6. (80 − 20)/(3 − 1) = 60/2 = 30 km/h.
7. If sides are x and 12 − x, A = x(12 − x) = 12x − x².
8. Maximum at x = 6, giving 36 m².
9. A 2% multiplicative increase per period under the model.
10. Non-negative integers, often with an upper capacity determined by the context.
11. 3(4) + 2(8) = 28 ≤ 30, so yes.
12. 12.
13. First becomes 23 min, so the 21-minute route becomes faster.
14. Ceiling of 100/36 = 3 vehicles.
15. Two vehicles provide only 72 seats, violating the minimum-capacity constraint.
16. At the boundary x = 10 if the quadratic is increasing over the allowed interval toward its unconstrained maximum at 12.
17. The truncated axis magnifies the visual proportion of small numerical changes.
18. Interpolation estimates within observed range; extrapolation predicts beyond it and relies more heavily on continuation assumptions.

26. What systems Mathematics teaches beyond graphs

Functions train us to make dependence explicit. Graphs train us to see structure. Simultaneous equations locate decision boundaries. Inequalities define what is feasible. Optimisation forces us to name an objective. Networks show that relationships can matter more than isolated objects. Sensitivity analysis reminds us that an exact-looking answer can still be fragile.

A model becomes useful when it preserves the relationships that matter, rejects impossible choices and returns a result that can survive the real system.

Continue the Real-World Mathematics series

• Measurement, Units, Scale and Estimation
• Ratios, Rates, Percentages and Proportions
• Probability, Statistics, Data and Decisions
• Return to the BTT Mathematics Hub