Theory
The auto-rickshaw meter mystery
You take the same auto three days in a row.
2 km costs ₹43. 4 km costs ₹61. 6 km costs ₹79.
Without asking the driver, can you predict the fare for 10 km? By the end of this topic you will crack it in under a minute, because hidden in those numbers is a function, and you are about to learn how to rebuild its formula from data.
Theory
Three views of one story
A function is a story about how one quantity depends on another. A table tells it as a list of events, a graph as a picture, a formula as a one-line summary. Construction means hearing the story in one form and retelling it in another. Nothing new is invented; you translate.
Theory
The straight-line formula: y = mx + c
Most construction questions at this level are linear: the data follows
y = mx + c
- m is the slope, how much y changes each time x grows by 1
- c is the y-intercept, the value of y when x = 0, the "starting amount"
Find those two numbers and the function is fully rebuilt.
Theory
From a table: constant difference
Table: x = 0, 1, 2, 3 with y = 3, 5, 7, 9.
1. Check the y-differences: 5−3 = 2, 7−5 = 2, 9−7 = 2. Constant, so the data is linear and m = 2.
2. Read y at x = 0: y = 3, so c = 3.
3. Write it: y = 2x + 3.
No x = 0 row? Use any known point: put its x and y into y = mx + c and solve for c.
Quiz
A table shows x = 1, 2, 3, 4 with y = 10, 13, 16, 19. Which function fits?
- y = 3x + 7
- y = 3x + 10
- y = 10x + 3
- y = 4x + 6
Show the answer
y = 3x + 7
The y-difference is 3 each step, so m = 3. There is no x = 0 row, so use the point (1, 10): 10 = 3·1 + c gives c = 7. Option B grabbed the first y value as the intercept, the classic shortcut error: 10 is y at x = 1, not at x = 0.
Theory
From a graph: two points
Given a straight line through (0, 1) and (2, 5):
1. Slope: m = (y₂ − y₁)/(x₂ − x₁) = (5 − 1)/(2 − 0) = 2
2. Intercept: the point (0, 1) sits on the y-axis, so c = 1 directly
3. Equation: y = 2x + 1
Check with a third point if the graph shows one; every point on the line must satisfy the equation.
Think first
Back to the auto: 2 km costs ₹43, 4 km costs ₹61, 6 km costs ₹79.
Work out m (per-km rate) and c (base fare) in your head, then predict the 10 km fare.
Show the answer
Fare rises ₹18 per 2 km, so m = 9 rupees per km. Using (2, 43): 43 = 9·2 + c, so c = 25 (the base fare). Formula: fare = 9x + 25.
For 10 km: 9·10 + 25 = ₹115. That is construction: data in, formula out, prediction made.
Theory
From words to everything else
Word problems hand you m and c in disguise: "a fixed charge plus a constant rate" is always y = mx + c.
- fixed charge, base fare, starting salary → c
- per km, per hour, per unit → m
Once the formula exists, you can generate a table from it and plot the graph, completing the full circle: description → formula → table → graph.
Quiz
A cyber cafe charges ₹20 to sit down plus ₹15 per hour. Which function gives the total cost y for x hours?
- y = 20x + 15
- y = 15x + 20
- y = 35x
- y = 15x − 20
Show the answer
y = 15x + 20
The per-hour rate multiplies the hours (m = 15) and the one-time sitting charge stands alone (c = 20). Option A swaps the roles, the most common word-problem slip: always ask "which number repeats per unit?" That one is m.
Watch out
Check before you commit
Two habits prevent nearly all construction errors: verify the y-differences really are constant before assuming linearity (if they vary, the function is not linear and y = mx + c does not apply), and always substitute one unused point into your final equation. If it fails, the intercept is usually the culprit.
Theory
Where you will meet this again
Data science calls this line fitting, and machine learning's simplest model, linear regression, is exactly y = mx + c learned from a table of data. Every pricing API, EMI calculator and physics lab graph uses today's translation skill between table, graph and formula.
Summary
Key takeaways
- Table, graph and formula are three views of the same function; construction converts between them.
- Linear data obeys y = mx + c: m is the change per unit x, c the value at x = 0.
- From a table: constant y-difference gives m; y at x = 0 (or any point solved) gives c.
- From a graph: two points give m = (y₂ − y₁)/(x₂ − x₁), then one point pins c.
- Memory hook: the repeating number is m, the one-time number is c.