New NCERT Ganita Manjari · Class 9 Maths · Part I · Chapter 2: Introduction to Linear Polynomials
This chapter starts with algebraic expressions and polynomials in one variable, and then focuses on linear polynomials of the form ax + b. It shows how linear patterns, linear growth and linear decay all give straight-line graphs. We also learn what the slope and y-intercept of y = ax + b mean.
From the Class 9 Maths Formula Sheet, made as per the new NCERT book Ganita Manjari · Pages 7–11 of the book.
Key points
- In 4x + 5y + 3, the terms are 4x, 5y and 3; x and y are variables; 4 and 5 are coefficients; 3 is the constant.
- An expression in one variable and its powers is a one-variable (univariate) polynomial.
- The degree of a polynomial is the highest power of the variable.
- Degree 0, 1, 2 and 3 polynomials are called constant, linear, quadratic and cubic polynomials.
- A linear polynomial has degree 1 and the form ax + b, where a ≠ 0.
- A polynomial works like an input–output machine (a function): put in x and get p(x).
- A linear pattern is a sequence where the difference between two consecutive terms is constant; its nth term is a linear expression in n.
- In linear growth a quantity increases by a constant amount; in linear decay it decreases by a constant amount.
- The graph of y = ax + b is a straight line; plot any two points, join them and extend both ways.
- A point lies on the line if and only if its coordinates satisfy the equation.
- Lines y = ax pass through the origin; a > 0 means the line rises and a < 0 means it falls.
- Lines with the same slope a but different b are parallel.
Formulas
| Linear polynomial | ax + b, a ≠ 0 degree 1 |
| Degree | Degree = highest power of the variable e.g. x² + 5x + 1 has degree 2 |
| Linear relationship | y = ax + b a = slope, b = y-intercept |
| y-intercept point | The line y = ax + b cuts the y-axis at (0, b) b = 0 ⇒ the line passes through the origin |
| Line through the origin | y = ax a > 0 rises (growth), a < 0 falls (decay) |
| Tile pattern nth term | Stage n: (n − 1) + n = 2n − 1 constant difference 2 |
| Linear growth example | C(d) = 100 + 60d +₹60 for every km |
| Linear decay example | h(t) = 3 − 0.5t −0.5 m every month |
Diagrams to learn
- Table of polynomial types by degree
- Polynomial as an input–output function machine
- Tile pattern stages with table (2n − 1)
- Graphs of linear growth (C = 100 + 60d) and linear decay (h = 3 − 0.5t)
- Graph of y = 2x + 1
- Lines through the origin y = ax for positive and negative a
- Parallel lines and y-intercepts of y = ax + b
- y² means 1y², so its coefficient is 1, not 0. A term keeps its sign: in 9x³ + 5x² − 8x − 10 the constant term is −10.
- A rule like 15n − 5 holds only where the pattern holds (here for n ≥ 2). For 1 km it would give ₹10, but the actual fare is ₹25.
- First make y alone: 2y = 4x + 7 ⇒ y = 2x + 3.5, so the slope is 2 and the y-intercept is 3.5 (not 4 and 7).
A real page from the book
Get the full chapter in the book
This page is a summary. The Class 9 Maths Formula Sheet has the complete chapter in colour, with every diagram and worked example.
Frequently asked questions
What is a linear polynomial in Class 9 Maths?
A linear polynomial is a polynomial of degree 1. It has the form ax + b, where a ≠ 0, for example 2x + 3.
What are slope and y-intercept in y = ax + b?
a is the slope of the line and b is the y-intercept. The line cuts the y-axis at the point (0, b).
What is the difference between linear growth and linear decay?
In linear growth a quantity increases by a constant amount over equal intervals, so the line has a positive slope. In linear decay it decreases by a constant amount, so the line has a negative slope.
← Part I · Chapter 1: Orienting Yourself: The Use of CoordinatesPart I · Chapter 3: The World of Numbers →
