7 min readConcept sheetShort calculation; 5-mark cost-line question

Linear functions and straight-line graphs

The graph of y = mx + c is a straight line. m is the slope (rate of change); c is the y-intercept (value when x is 0).

Slopem = (y₂ − y₁) ÷ (x₂ − x₁)

Slope-intercept formy = mx + c

Point-slope formy − y₁ = m(x − x₁)

Parallel linesSame slope

Straight-line formulas

Slope through (x₁, y₁) and (x₂, y₂)m = (y₂ − y₁) ÷ (x₂ − x₁)

Slope-intercept formy = mx + c

Point-slope formy − y₁ = m(x − x₁)

General formax + by = k (slope = −a/b)

y-interceptPut x = 0

x-interceptPut y = 0

Parallel / perpendicularm₁ = m₂ / m₁ × m₂ = −1

Reading the slope in business

Slope	Line	Example
m > 0	Rises left to right	Total cost rises with output
m < 0	Falls left to right	Demand: higher price, fewer sales
m = 0	Horizontal	Fixed cost: same at any output
Undefined	Vertical	A fixed capacity limit, x = 500

The line through two points

Problem

Find the equation of the line through (2, 5) and (6, 17).

Working

m = (17 − 5) ÷ (6 − 2) = 12 ÷ 4 = 3.
Point-slope form with (2, 5): y − 5 = 3(x − 2).
y − 5 = 3x − 6, so y = 3x − 1.
Check with (6, 17): 3(6) − 1 = 17. ✓

Answer

y = 3x − 1

A cost line from two data points

Problem

A Negombo printing shop's total cost is Rs. 16,000 for 100 wedding cards and Rs. 22,000 for 250 cards. Assuming a linear cost function, find it, interpret it, and find the cost of 400 cards.

Working

Points (100, 16,000) and (250, 22,000).
m = (22,000 − 16,000) ÷ (250 − 100) = 6,000 ÷ 150 = 40.
C − 16,000 = 40(x − 100), so C = 40x − 4,000 + 16,000 = 40x + 12,000.
Interpretation: variable cost Rs. 40 per card (slope); fixed cost Rs. 12,000 (intercept).
C(400) = 40 × 400 + 12,000 = 16,000 + 12,000 = 28,000.

Answer

C(x) = 40x + 12,000; 400 cards cost Rs. 28,000.

Intercepts of a budget line

Problem

A firm spends exactly Rs. 100,000 on advertising: radio spots cost Rs. 2,000 each (x) and newspaper adverts Rs. 5,000 each (y). Find the intercepts and slope of the budget line.

Working

Budget line: 2,000x + 5,000y = 100,000.
x-intercept (y = 0): x = 100,000 ÷ 2,000 = 50 radio spots.
y-intercept (x = 0): y = 100,000 ÷ 5,000 = 20 adverts.
Slope = −2,000 ÷ 5,000 = −0.4, so y = 20 − 0.4x.
Meaning: each extra radio spot means giving up 0.4 of a newspaper advert.

Answer

Intercepts (50, 0) and (0, 20); slope −0.4.

In Sri Lankan business

A Sri Lankan printing shop or catering business can split its costs into a fixed part (rent, salaries) and a variable part per unit. That split is just the intercept and the slope of a straight line.

Exam angle

Short answer: “Find the slope of the line through (2, 3) and (6, 11).” (11 − 3) ÷ (6 − 2) = 8 ÷ 4 = 2.

Structured question: “Cost is Rs. 16,000 for 100 cards and Rs. 22,000 for 250. Find the linear cost function and interpret it.” m = 40, C = 40x + 12,000: Rs. 40 variable per card, Rs. 12,000 fixed.

Short answer: “Find the x- and y-intercepts of 3x + 4y = 24.” x = 8 (when y = 0) and y = 6 (when x = 0).

Remember it

Slope = rise ÷ run. In a cost line, slope = variable cost per unit and intercept = fixed cost.