MATH1202 Chap.6 Graphical and Business Applications
Graphical and Business Applications
This chapter translates commerce stories into functions. It defines variables and units before building revenue, cost and profit, uses intersections as break-even points, and interprets a vertex as both a decision input and an objective value. Exponential threshold models extend the same discipline by requiring a time unit and a justified rounding decision.
Equations, tables and graphs are cross-checked so a result must be algebraically correct and commercially meaningful. Feasible domains exclude impossible negative quantities or values beyond capacity. The distinction between a decision variable and the resulting revenue, cost or profit is kept explicit in every final statement.
What this chapter covers
- 01
Variable and unit definitions
- 02
Revenue as price times quantity
- 03
Fixed and variable cost
- 04
Profit and break-even points
- 05
Quadratic maximum decisions
- 06
Exponential thresholds and rounding
Worked example · free
Optimise a quadratic profit
- +1Multiply price by the scaled quantity to form revenue.
- +1Subtract the complete cost expression.
- +1Find the vertex input of the profit parabola.
- +1Evaluate profit and interpret both coordinates with units.
Key terms
- Revenue
- Income modelled as selling price multiplied by quantity sold.
- Fixed Cost
- A cost that remains when the modelled production quantity is zero.
- Profit
- Revenue minus the complete cost function.
- Break-Even Point
- A quantity where revenue equals cost and profit equals zero.
Graphical and Business Applications FAQ
Why must I define the variable first?
The variable definition carries scale and units into every formula. Without it, a mathematically correct vertex may be interpreted as individual units instead of hundreds or thousands, changing the business answer by a large factor.
How do graphs show a profitable interval?
Profit is positive where the revenue curve lies above the cost curve. Their intersections are break-even boundaries. State the interval in the feasible domain rather than reporting only the two crossing quantities.
How should a threshold time be rounded?
Solve continuously first and preserve the inequality. If a process updates only at whole periods, move to the first whole period that actually satisfies the threshold. Explain why the rounding direction follows the context.
How are revenue, cost and profit related?
Revenue is the amount received from sales, usually price multiplied by quantity. Cost represents the total expense of producing that quantity, including any fixed and variable parts. Profit is revenue minus cost, with brackets around the complete cost expression. A break-even quantity makes profit zero, while positive profit occurs where revenue exceeds cost.
Why can an algebraic solution be infeasible?
The equation may allow values that the business setting does not. Negative production, demand beyond a stated capacity, non-integer items or a price outside the model range may all be inadmissible. State the feasible domain before solving, reject candidates outside it, and explain the restriction using the variable's meaning rather than only an inequality.
What does the vertex mean in a profit model?
For a downward-opening quadratic profit function, the vertex gives the continuous quantity that maximises predicted profit and the corresponding maximum value. Read both coordinates with units. If only whole items can be produced, compare the feasible integers on either side and select the one giving the larger actual profit.
How should a break-even answer be reported?
Give each feasible break-even input with its units and identify the profitable interval between or beyond those values according to the graph. An intersection is not itself a profit amount; it is the input where revenue equals cost. If the model uses hundreds or thousands of units, restore that scale in the conclusion.
How can units reveal a modelling error?
Write units beside every variable and intermediate expression. Price times quantity should produce money, profit must have the same units as revenue and cost, and an exponential exponent must be dimensionless after its time scale is considered. If addition combines unlike units or the final coordinate is interpreted in the wrong scale, revisit the translation before trusting the numerical answer.
Exam move
Translate in a fixed order: define the variable and units, write price and cost, calculate revenue, subtract cost with brackets, state the feasible domain, and then solve or optimise. Build a unit column beside the algebra so price, quantity, revenue and profit cannot be confused. Check a maximum from both vertex form and the graph, then compare nearby permitted quantities if decisions are discrete.
Practise explaining intercepts, intersections and turning points in full business sentences rather than reporting bare coordinates. For thresholds, identify the initial value and period multiplier, keep exact logarithms until the final line and test adjacent whole periods. End every problem by asking whether the sign, scale and units make sense in the original commercial setting, not just in the formula.
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