Exercise 2.6 Practice

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Overview

This page provides comprehensive Ch 2: Polynomials - Exercise 2.6 Practice. Practice graphing sets of linear equations, analyzing the role of slope $a$ and y-intercept $b$ in the equation $y = ax + b$. Free step-by-step NCERT solutions and explanations.

Graphing Linear Equations and Parameter Role Analysis

Q1: Graphing & Slope/Intercept Roles

Draw the graphs of the following sets of lines. In each case, reflect on the role of '$a$' and '$b$'.
(i) $y = 4x$, $y = 2x$, $y = x$
(ii) $y = -6x$, $y = -3x$, $y = -x$
(iii) $y = 5x$, $y = -5x$
(iv) $y = 3x - 1$, $y = 3x$, $y = 3x + 1$
(v) $y = -2x - 3$, $y = -2x$, $y = 2x + 3$

Core Concept: The Slope-Intercept Form
In a linear equation $y = ax + b$:

Slope ($a$): Represents the steepness and direction of the line. If $a > 0$, the line rises from left to right. If $a < 0$, it falls. The larger the absolute value $|a|$, the steeper the line.
y-intercept ($b$): Represents the point $(0, b)$ where the line intersects the y-axis. It shifts the line vertically up or down. If $b = 0$, the line passes through the origin $(0, 0)$.

(i) $y = 4x$, $y = 2x$, $y = x$:

Here, $b = 0$ for all lines, so they all pass through the origin $(0, 0)$.
The slopes are $a = 4$, $a = 2$, and $a = 1$. Since all slopes are positive, the lines rise from left to right.
Role of $a$: As the value of $a$ decreases from 4 to 1, the lines become less steep and tilt closer to the x-axis.

Coordinates Table for graphing: x │ y = 4x │ y = 2x │ y = x ──────┼──────────┼──────────┼──────── -1 │ -4 │ -2 │ -1 0 │ 0 │ 0 │ 0 1 │ 4 │ 2 │ 1

(ii) $y = -6x$, $y = -3x$, $y = -x$:

Here, $b = 0$ for all lines, so they all pass through the origin $(0, 0)$.
The slopes are $a = -6$, $a = -3$, and $a = -1$. Since all slopes are negative, the lines fall from left to right (lie in Quadrants II and IV).
Role of $a$: As $|a|$ decreases from 6 to 1, the lines become less steep.

Coordinates Table for graphing: x │ y = -6x │ y = -3x │ y = -x ──────┼──────────┼──────────┼──────── -1 │ 6 │ 3 │ 1 0 │ 0 │ 0 │ 0 1 │ -6 │ -3 │ -1

(iii) $y = 5x$, $y = -5x$:

Both lines have $b = 0$ and pass through the origin $(0, 0)$.
The slopes are $a = 5$ and $a = -5$. They have the same steepness but opposite directions.
Reflection: These lines are mirror reflections of each other across the y-axis (and x-axis).

Coordinates Table for graphing: x │ y = 5x │ y = -5x ──────┼──────────┼───────── -1 │ -5 │ 5 0 │ 0 │ 0 1 │ 5 │ -5

(iv) $y = 3x - 1$, $y = 3x$, $y = 3x + 1$:

All three lines have the exact same slope ($a = 3$), which means they have the same steepness and direction. Therefore, these lines are parallel.
The y-intercepts are $b = -1$, $b = 0$, and $b = 1$.
Role of $b$: The parameter $b$ shifts the lines vertically. $y = 3x + 1$ is shifted 1 unit up relative to $y = 3x$, and $y = 3x - 1$ is shifted 1 unit down.

Coordinates Table for graphing: x │ y = 3x-1 │ y = 3x │ y = 3x+1 ──────┼──────────┼──────────┼───────── -1 │ -4 │ -3 │ -2 0 │ -1 │ 0 │ 1 1 │ 2 │ 3 │ 4

(v) $y = -2x - 3$, $y = -2x$, $y = 2x + 3$:

For $y = -2x - 3$ and $y = -2x$: The slope is $a = -2$, so these two lines are parallel. The line $y = -2x - 3$ is shifted 3 units vertically downwards relative to $y = -2x$.
For $y = 2x + 3$: The slope is positive ($a = 2$), so this line rises from left to right and intersects the other two lines. Its y-intercept is $b = 3$.
Reflection: Parallel lines share the same slope coefficient $a$. Different values of $b$ translate the lines vertically. Changing the sign of $a$ changes the direction of tilt.

Coordinates Table for graphing: x │ y = -2x-3│ y = -2x │ y = 2x+3 ──────┼──────────┼──────────┼───────── -1 │ -1 │ 2 │ 1 0 │ -3 │ 0 │ 3 1 │ -5 │ -2 │ 5

Exercise 2.5 Next End-of-Chapter Exercises