Chapter 51: Linear Functions
1. What is a Linear Function? (The Simplest & Most Honest Definition)
A linear function is a relationship between two quantities where:
when one quantity changes by a fixed amount, the other quantity changes by a constant multiple of that amount.
In other words:
The change is always proportional — no curves, no sudden jumps, no slowing down or speeding up.
The graph of a linear function is always a perfectly straight line.
That’s why we call it linear — from the Latin word “linea” meaning “line”.
The most common way to write it:
y = mx + c (or sometimes f(x) = mx + b)
- y = output / dependent variable (what you want to find)
- x = input / independent variable (what you control or measure)
- m = slope / gradient / rate of change (how steep the line is)
- c (or b) = y-intercept (where the line crosses the y-axis when x = 0)
2. Everyday Hyderabad Examples of Linear Functions
You already see and use linear functions every day — even if you never studied them.
Example 1 – Ola / Uber fare (almost linear)
Fare = ₹50 (base fare) + ₹12 per kilometer
Written as a function:
fare = 12 × distance + 50
- Here m = 12 (₹12 increase per extra km)
- c = 50 (you always pay at least ₹50 even for 0 km)
- Graph: straight line starting at ₹50 when distance = 0, rising ₹12 every km
Real ride:
- 3 km → ₹50 + 12×3 = ₹86
- 5 km → ₹50 + 12×5 = ₹110
- 10 km → ₹50 + 12×10 = ₹170
Perfectly linear (until surge pricing or waiting time kicks in — then it becomes piecewise linear).
Example 2 – Monthly mobile recharge
You have a plan: ₹399 for 28 days unlimited calls + 2 GB/day data.
If you use exactly 2 GB every day, your cost per day is fixed:
daily_cost = 399 ÷ 28 ≈ ₹14.25 per day
This is linear if you think of cost over time (ignoring validity tricks).
Example 3 – Tiffin center pricing (very common)
A aunty near your home sells idli-vada tiffin for ₹60 per plate.
If you order for your family:
total_cost = 60 × number_of_plates
This is the purest linear function:
- Slope m = 60 (₹60 extra per additional plate)
- Intercept c = 0 (no plate = ₹0)
3. The Four Key Features You Must Understand
Every linear function has these four things:
- Slope (m) — tells how fast y changes when x increases by 1
- m > 0 → line goes up (positive relationship)
- m < 0 → line goes down (negative relationship)
- m = 0 → horizontal line (no change — constant function)
- Y-intercept (c) — value of y when x = 0
- Where the line crosses the y-axis
- Straight line — no curves, no bends
- Constant rate of change — the increase/decrease is always the same size
Quick table to remember:
| Slope value | What the graph looks like | Real-life meaning (Hyderabad example) |
|---|---|---|
| m = 3 | Steep upward line | Petrol price rises ₹3 per litre per month |
| m = 0.5 | Gentle upward slope | Savings account gives 50 paise interest per ₹100 per month |
| m = –2 | Steep downward line | Phone battery percentage drops 2% every 10 minutes of gaming |
| m = 0 | Flat horizontal line | Fixed monthly rent — doesn’t change with days used |
4. How to Spot a Linear Function in Real Life (Quick Checklist)
Ask these four questions:
- Does one thing increase/decrease by a fixed amount when the other changes by 1 unit?
- Is the graph perfectly straight (when plotted)?
- Is there a constant rate (same change every time)?
- Can you write it in the form y = mx + c?
If yes to most → it’s linear.
Step 5: Quick Exercise You Can Do Right Now
Think of your daily life in Hyderabad. Find three linear relationships you encounter.
Example answers most students give:
- Ola fare ≈ 15 × km + 40
- Monthly electricity bill = 8 × units consumed + fixed charge
- Cost of printing notes = 2 × number of pages + binding charge
Now try it yourself — write three of your own.
Final Teacher Summary (Repeat This to Anyone!)
A linear function is the simplest, most predictable relationship in mathematics:
y = mx + c
- m tells how much y changes per unit of x (slope / rate)
- c tells where it starts when x = 0 (starting value / fixed cost)
- The graph is always a straight line
- The change is constant — no surprises, no curves
In Hyderabad you meet linear functions every day:
- Auto fare
- Petrol pump price
- Tiffin bill
- Savings interest
- Phone recharge validity cost per day
It is the foundation of almost all mathematics, physics, economics, and machine learning — because so many real-world relationships are approximately linear (at least over a small range).
Understood the heart of linear functions now? 🌟
Want to go deeper?
- How to draw a linear function by hand (step-by-step)?
- What happens when a relationship is not linear (quadratic, exponential examples)?
- Real machine learning example where linear functions are used (linear regression)?
- How to find m and c from two points in daily life?
Just tell me — next class is ready! 🚀
