Graphical Representation of a Linear Polynomial
In Class 9 NCERT Maths, a linear polynomial is a polynomial of degree 1 — for example, 2x + 3 or 5 − 4y. When such a polynomial is used to connect two quantities, we get a linear relationship that can always be written in the standard form:
y = ax + b
Here x and y are the two variables, a is a constant called the slope, and b is a constant called the y-intercept. The big idea behind the graphical representation of linear polynomials is simple but powerful: every relationship of the form y = ax + b, when plotted on a coordinate plane, produces a perfectly straight line. This is why these are called linear relationships in the first place.
1.0How to Plot a Linear Equation y = ax + b
Step 1: Write the Equation: Consider the linear equation:
y = 2x + 1
Step 2: Find Coordinates: Choose values of x and calculate corresponding values of y.
Both points satisfy the equation.
Step 3: Plot the Points: Mark the points (0,1) and (3,7) on graph paper.
Step 4: Join the Points: Using a ruler, join the two points and extend the line in both directions.
This produces the graph of: y = 2x + 1
The NCERT textbook further notes that other points such as (1,3) and (7,15) also lie on this same line because they satisfy the equation. A point belongs to a line only if its coordinates satisfy the equation of that line.
Understanding Slope (a) and y-Intercept (b)
The two constants in y = ax + b each control the line in a specific way:
- The slope a decides how steep the line is. For lines of the form y = ax (which always pass through the origin (0, 0)): when a > 1, the line is steeper than y = x; when a < 1, the line is less steep. The slope also equals the constant difference between consecutive terms of a linear pattern. For example, in the tile pattern giving y = 2x − 1, the slope 2 is exactly the amount the count increases at each stage.
- The y-intercept b is the value where the line crosses the y-axis. Any line written as y = ax + b cuts the y-axis at the point (0, b). So y = x + 3 cuts the y-axis at (0, 3), and y = 3x − 2 cuts it at (0, −2) — two units below the origin.
2.0Linear Growth vs Linear Decay on a Graph
The sign of the slope tells the whole story of how a quantity behaves over time:
- Linear growth — a quantity increasing by a fixed amount over equal intervals — is represented by a straight line with a positive slope (the line rises from left to right). Example: a journey cost C(d) = 100 + 60d increases by ₹60 for every extra kilometre.
- Linear decay — a quantity decreasing by a fixed amount over equal intervals — is represented by a straight line with a negative slope (the line falls from left to right). Example: water height h(t) = 3 − 0.5t drops by 0.5 m each month.
This single visual clue — does the line go up or down? — instantly classifies a real-world situation as growth or decay.
Parallel Lines: The Role of Changing a and b
By comparing several graphs, NCERT draws three neat conclusions:
- In y = ax + b, a is the slope and b is the y-intercept.
- If you change a and keep b fixed, the steepness changes but the line still cuts the y-axis at the same point.
- If you change b and keep a fixed, the line simply shifts up or down while staying parallel to the original.
So lines such as y = 2x − 1, y = 2x + 1 and y = 2x + 5 are all parallel — equal slopes, different y-intercepts. This is the foundation for recognising parallel lines purely from their equations.
3.0Solved Examples
Example 1 — Plot the linear relationship given by the equation y = 2x + 1 on a coordinate plane. Identify its slope and y-intercept.
Solution: Find two points on the line:
- Let x = 0. Substituting this into the equation gives y = 2(0) + 1 = 1. This gives us our first point, A(0, 1).
- Let x = 3. Substituting this into the equation gives y = 2(3) + 1 = 7. This gives us our second point, B(3, 7).
Plot and join: Mark points A(0, 1) and B(3, 7) on a graph paper, then draw a straight line passing through both points.
Identify parameters: Comparing y = 2x + 1 with the general form y = ax + b, we find:
- Slope (a): 2
- Y-intercept (b): 1, meaning it crosses the y-axis at (0, 1).
Example 2: Consider the three equations y = 2x - 1, y = 2x + 1, and y = 2x + 5. Describe the relationship between their graphical representations without plotting them.
Solution:
- Observe the slope (a) for all three lines: In every equation, the coefficient of x is 2 (a = 2). Since their slopes are equal, all three lines have the exact same steepness and inclination.
- Observe the y-intercept (b): The values of b are -1, 1, and 5, respectively. This means they will cross the y-axis at three different points: (0, -1), (0, 1), and (0, 5).
- Conclusion: Because they share an identical slope but have different y-intercepts, these three lines are parallel to each other and represent shifted versions of the same line.
4.0Common Mistakes Students Should Avoid
While drawing graphs, students often make these mistakes:
- Plotting coordinates incorrectly.
- Interchanging x and y values.
- Using unequal scales on axes.
- Forgetting to verify whether points satisfy the equation.
- Drawing a curved line instead of a straight line.
- Careful plotting and checking can prevent these errors.