Chapter 2: Linear Polynomials
Exploring variables, terms, linear growth, linear decay, and straight-line graphs.
📖 The Anatomy of an Expression
- Variables: The letter-numbers that can change value (x and y).
- Coefficients: The numbers multiplying the variables (4 and 5).
- Constant: The fixed number without a variable (3).
- Terms: The individual parts separated by + or − (4x, 5y, and 3).
📏 Classification by Degree
The degree is the highest power of the variable in a univariate polynomial.
| Polynomial | Degree | Name |
|---|---|---|
| 8 | 0 | Constant |
| 3z + 7 | 1 | Linear |
| x2 + 5x + 1 | 2 | Quadratic |
| 5y3 + y2 − 1 | 3 | Cubic |
🔍 The Anatomy of a Linear Relationship (y = ax + b)
A linear relationship is represented by a straight line. The equation has two critical components:
'a' = The Slope: Represents the rate of change or the steepness of the line.
'b' = The y-intercept: The exact point (0, b) where the line cuts the vertical y-axis.
📈 Linear Growth (Positive Slope)
Quantity increases by a fixed amount over equal intervals. The straight line goes upwards (a > 0).
Base fare ₹25, plus ₹15 per km.
Equation:
y = 15x + 25
📉 Linear Decay (Negative Slope)
Quantity decreases by a fixed amount over equal intervals. The straight line goes downwards (a < 0).
Starts at 3m, drops 0.5m per month.
Equation:
h = -0.5t + 3
⚙️ Algorithm: Finding 'a' and 'b' from Real Data
Problem: 10 GB data costs ₹350. 20 GB data costs ₹550. Find the linear relationship y = ax + b.
Extract Data Points
Identify x (GB) and y (Cost).
Point 1: (10, 350)
Point 2: (20, 550)
Set Up Equations
Substitute into y = ax + b:
Eq 1: 350 = 10a + b
Eq 2: 550 = 20a + b
Solve for 'a'
Subtract Eq 1 from Eq 2:
200 = 10a
a = 20 (Cost per GB)
Solve for 'b'
Put 'a' back into Eq 1:
350 = 10(20) + b
b = 150 (Fixed Fee)
Final Equation: y = 20x + 150
⭐ Key Visual Properties of Linear Graphs
- Passing through Origin: If b = 0, the equation becomes y = ax. This straight line will always pass directly through the origin (0,0).
- Changing the Slope (a): If we keep the y-intercept fixed but change 'a', the steepness of the line changes. It pivots around the y-intercept.
- Parallel Lines: If two equations have the exact same slope 'a' but different y-intercepts 'b' (e.g., y = 2x + 1 and y = 2x + 5), the lines shift up or down but remain perfectly parallel to each other.
⚠️ Common Pitfalls & Exam Traps
Pitfall 1 (Confusing Exponents): A polynomial like 10x − x2 is NOT linear. Because the highest power of x is 2, it is a quadratic polynomial. Linear polynomials must have a maximum degree of 1.
Pitfall 2 (Mixing up Intercepts): The y-intercept 'b' is where x = 0 (the point is (0, b)). Students often incorrectly plot 'b' on the x-axis instead of the vertical y-axis.
Pitfall 3 (Constant Difference Rule): For a pattern to be considered linear, the difference between consecutive terms must remain exactly the same. If the difference changes (e.g., +2, then +4, then +8), the relationship is not linear.