Exercise 1
Draw the graphs of the following sets of lines. In each case, reflect on the role of ' ' and ' '.
(i)
(ii)
(iii)
(iv)
(v)
Not cross-checked
This solution has not been cross-checked against the answer printed in NCERT.
Two points fix a line; then read off what \(\displaystyle a\) and \(\displaystyle b\) do.Every equation here has the form \(\displaystyle y = ax + b\), which is a polynomial of degree $\displaystyle 1$, so each graph is a straight line. That means you only need two points to draw it — plot a third as a check that the three are in a row.Two facts to keep in mind while plotting, and to confirm afterwards from the picture:
Plot each row against the \(\displaystyle x\) row and join. Here \(\displaystyle b = 0\) for all three, so all three lines pass through the origin. All three rise to the right because \(\displaystyle a > 0\), and the larger \(\displaystyle a\) is, the steeper the climb: \(\displaystyle y = 4x\) is the steepest and \(\displaystyle y = x\) the gentlest.(ii) \(\displaystyle y = -6x,\ y = -3x,\ y = -x\)
Again \(\displaystyle b = 0\), so all three pass through the origin. But now \(\displaystyle a < 0\), so all three fall to the right. Steepness still grows with the size of \(\displaystyle a\) ignoring its sign: \(\displaystyle y = -6x\) is the steepest of the three.(iii) \(\displaystyle y = 5x,\ y = -5x\)
The two slopes have the same size but opposite signs, so the lines are equally steep but tilted opposite ways, and they cross at the origin. Each is the mirror image of the other in the y-axis (equivalently, in the x-axis). Only the sign of \(\displaystyle a\) has changed, and only the direction of the tilt has changed with it.(iv) \(\displaystyle y = 3x - 1,\ y = 3x,\ y = 3x + 1\)
Now \(\displaystyle a = 3\) is the same for all three while \(\displaystyle b\) changes. The three lines are parallel — same slope, so same tilt, and they never meet. They cut the y-axis at \(\displaystyle (0, -1)\), \(\displaystyle (0, 0)\) and \(\displaystyle (0, 1)\) respectively. Changing \(\displaystyle b\) simply slides the whole line up or down without turning it.(v) \(\displaystyle y = -2x - 3,\ y = -2x,\ y = 2x + 3\)
The first two both have \(\displaystyle a = -2\): they are parallel falling lines, one slid down $\displaystyle 3$ units from the other, cutting the y-axis at \(\displaystyle (0, -3)\) and \(\displaystyle (0, 0)\).The third has \(\displaystyle a = +2\), so it is not parallel to them; it rises. Notice that \(\displaystyle 2x + 3 = -(-2x - 3)\), so \(\displaystyle y = 2x + 3\) is the mirror image of \(\displaystyle y = -2x - 3\) in the x-axis, and the two meet exactly on the x-axis: setting \(\displaystyle y = 0\) in either gives \(\displaystyle x = -\tfrac{3}{2}\), so they cross at \(\displaystyle \left(-\tfrac{3}{2},\, 0\right)\).(If your copy of the book prints the third line as \(\displaystyle y = -2x + 3\), then all three have \(\displaystyle a = -2\), all three are parallel, and their y-intercepts \(\displaystyle -3, 0, 3\) illustrate the same point as part (iv).)Answer: \(\displaystyle a\) controls the direction and steepness of the line — positive \(\displaystyle a\) rises, negative \(\displaystyle a\) falls, and a larger \(\displaystyle |a|\) is steeper; two lines with the same \(\displaystyle a\) are parallel. \(\displaystyle b\) controls the height — the line meets the y-axis at \(\displaystyle (0, b)\), so changing \(\displaystyle b\) slides the line up or down without changing its tilt. When \(\displaystyle b = 0\) the line passes through the origin.
\(\displaystyle b\) is the value of \(\displaystyle y\) when \(\displaystyle x = 0\), so the line crosses the y-axis at \(\displaystyle (0, b)\).
\(\displaystyle a\) is the slope: it tells you how much \(\displaystyle y\) changes when \(\displaystyle x\) increases by 1. Positive \(\displaystyle a\) means the line rises to the right, negative \(\displaystyle a\) means it falls, and a larger \(\displaystyle |a|\) means a steeper line.
(i) \(\displaystyle y = 4x,\ y = 2x,\ y = x\)| \(\displaystyle x\) | \(\displaystyle -1\) | $\displaystyle 0$ | $\displaystyle 1$ |
| \(\displaystyle y = 4x\) | \(\displaystyle -4\) | $\displaystyle 0$ | $\displaystyle 4$ |
| \(\displaystyle y = 2x\) | \(\displaystyle -2\) | $\displaystyle 0$ | $\displaystyle 2$ |
| \(\displaystyle y = x\) | \(\displaystyle -1\) | $\displaystyle 0$ | $\displaystyle 1$ |
| \(\displaystyle x\) | \(\displaystyle -1\) | $\displaystyle 0$ | $\displaystyle 1$ |
| \(\displaystyle y = -6x\) | $\displaystyle 6$ | $\displaystyle 0$ | \(\displaystyle -6\) |
| \(\displaystyle y = -3x\) | $\displaystyle 3$ | $\displaystyle 0$ | \(\displaystyle -3\) |
| \(\displaystyle y = -x\) | $\displaystyle 1$ | $\displaystyle 0$ | \(\displaystyle -1\) |
| \(\displaystyle x\) | \(\displaystyle -1\) | $\displaystyle 0$ | $\displaystyle 1$ |
| \(\displaystyle y = 5x\) | \(\displaystyle -5\) | $\displaystyle 0$ | $\displaystyle 5$ |
| \(\displaystyle y = -5x\) | $\displaystyle 5$ | $\displaystyle 0$ | \(\displaystyle -5\) |
| \(\displaystyle x\) | \(\displaystyle -1\) | $\displaystyle 0$ | $\displaystyle 1$ |
| \(\displaystyle y = 3x - 1\) | \(\displaystyle -4\) | \(\displaystyle -1\) | $\displaystyle 2$ |
| \(\displaystyle y = 3x\) | \(\displaystyle -3\) | $\displaystyle 0$ | $\displaystyle 3$ |
| \(\displaystyle y = 3x + 1\) | \(\displaystyle -2\) | $\displaystyle 1$ | $\displaystyle 4$ |
| \(\displaystyle x\) | \(\displaystyle -1\) | $\displaystyle 0$ | $\displaystyle 1$ |
| \(\displaystyle y = -2x - 3\) | \(\displaystyle -1\) | \(\displaystyle -3\) | \(\displaystyle -5\) |
| \(\displaystyle y = -2x\) | $\displaystyle 2$ | $\displaystyle 0$ | \(\displaystyle -2\) |
| \(\displaystyle y = 2x + 3\) | $\displaystyle 1$ | $\displaystyle 3$ | $\displaystyle 5$ |