Functions and Graphs for Business: Cost, Revenue, Profit, Demand and Supply (Basics)

Table of Contents

• Meaning of function and notation
• Domain, range and real-life interpretation
• Linear functions and graphs (slope-intercept)
• Demand function and demand curve (linear case)
• Supply function and supply curve (linear case)
• Cost functions: fixed, variable, total and average
• Revenue functions: TR, AR, MR (basic)
• Profit function and break-even (graph + algebra)
• How to draw graphs quickly in exams (steps + tables)
• Typical exam questions and short notes
• Key terms explained
• Quick recap (1-minute revision)

Meaning of function and notation

A function is a rule that assigns exactly one output to each input.

• If output depends on input, we write: y = f(x).
• In business, output might be cost, revenue, demand, supply, profit, etc.

Business idea:

• Quantity demanded depends on price → Qd = f(P)
• Total cost depends on output → C = f(Q)
• Revenue depends on price and quantity → R = f(Q) when price is known or depends on Q.

Notation examples:

• C(Q) means total cost when output is Q.
• R(Q) means total revenue when output is Q.
• Π(Q) means profit when output is Q.

Domain, range and real-life interpretation

• Domain: set of allowed inputs.
• Range: set of possible outputs.

In business context:

• Output Q cannot be negative → Q ≥ 0.
• Price P cannot be negative → P ≥ 0.
• Sometimes Q must be an integer (units), but in maths we often treat it as continuous.

Exam tip: Always mention practical restrictions like Q ≥ 0 and P ≥ 0.

Linear functions and graphs (slope-intercept)

A linear function has the form:

• y = mx + c

Where:

• m = slope (rate of change)
• c = intercept (value when x = 0)

Interpretation:

• If m > 0, y increases with x.
• If m < 0, y decreases with x.

Example: y = 50x + 1000

• c = 1000 means fixed part
• m = 50 means for each 1 unit increase in x, y increases by 50

Graph basics:

• Two points are enough to draw a straight line.

Demand function and demand curve (linear case)

A demand function shows relationship between quantity demanded (Qd) and price (P).

A common linear demand form:

• Qd = a − bP (b > 0)

Meaning:

• As P increases, Qd decreases → downward sloping demand curve.

Small example

If Qd = 100 − 2P:

• When P = 10, Qd = 80
• When P = 30, Qd = 40

So demand curve can be plotted using (P, Qd) points.

Note: In economics, we often plot Q on x-axis and P on y-axis, but for functions either way is acceptable if labelled clearly.

Supply function and supply curve (linear case)

A supply function shows relationship between quantity supplied (Qs) and price.

A common linear supply form:

• Qs = c + dP (d > 0)

Meaning:

• As P increases, Qs increases → upward sloping supply curve.

Small example

If Qs = 20 + 3P:

• P = 10 ⇒ Qs = 50
• P = 20 ⇒ Qs = 80

Cost functions: fixed, variable, total and average

Fixed cost (FC)

Cost that does not change with output (within relevant range): rent, salaries.

Variable cost (VC)

Cost that changes with output: raw materials, direct labour.

Total cost (TC)

TC = FC + VC

A common linear cost function:

• TC = F + vQ Where:
• F = fixed cost (intercept)
• v = variable cost per unit (slope)

Average cost (AC)

AC = TC/Q (for Q>0)

Marginal cost (MC) (basic idea)

MC is the additional cost of producing one more unit. If TC is linear: TC = F + vQ, then MC = v (constant).

Exam note: Real cost curves are often U-shaped, but at basic level linear model is acceptable.

Revenue functions: TR, AR, MR (basic)

Total Revenue (TR)

TR = P × Q

If P is constant, TR is a straight line through origin.

If price depends on quantity (demand function), e.g., P = a − bQ, then: TR = (a − bQ)Q = aQ − bQ^2 (a quadratic).

Average Revenue (AR)

AR = TR/Q = P

Marginal Revenue (MR) (basic)

MR is additional revenue from selling one more unit.

• If P is constant, MR = P.
• If P falls with more quantity (downward demand), MR is less than price.

Profit function and break-even (graph + algebra)

Profit: Π = TR − TC

If TC = F + vQ and TR = pQ (constant price p), then: Π(Q) = pQ − (F + vQ) = (p − v)Q − F

This is a straight line:

• slope = (p − v) = contribution per unit
• intercept = −F

Break-even point

Break-even occurs when Π = 0. 0 = (p − v)Q − F Q = F/(p − v)

Small numeric example

Let:

• p = 500
• v = 300
• F = 40,000 Contribution per unit = 200 Break-even Q = 40,000/200 = 200 units

Diagram idea

How to draw graphs quickly in exams (steps + tables)

Step method for a straight line y = mx + c:

1. Find intercept: point when x = 0 → (0, c)
2. Choose a convenient x (like 10), compute y.
3. Plot the two points and join.

Table method (good for demand/supply): Pick 2–3 prices and compute quantities.

Example Qd = 100 − 2P:

Plot points and join with a straight line.

Common exam instruction: Always label axes and write the equation on the graph.

Typical exam questions and short notes

• Define function with one business example.
• Given cost function TC = 10,000 + 200Q, find TC for Q=150 and compute AC.
• Given demand Qd = 100 − 2P and supply Qs = 20 + 3P, find equilibrium price and quantity.
• Given p, v and F, compute break-even and interpret.
• Explain difference between TR, AR and MR.

Key terms explained

• Function — rule mapping input to output.
• Domain — allowed inputs.
• Range — possible outputs.
• Demand function — Qd as a function of price.
• Supply function — Qs as a function of price.
• Fixed cost — cost independent of output.
• Variable cost — cost that changes with output.
• Total revenue — P×Q.
• Profit — TR−TC.
• Break-even — profit = 0.

Quick recap (1-minute revision)

• Linear function: y = mx + c.
• Demand (linear): Qd = a − bP.
• Supply (linear): Qs = c + dP.
• TC = F + vQ.
• TR = P×Q.
• Profit Π = TR − TC.
• Break-even: Q = F/(p − v).