Q1. Page 36: Draw the graphs of the following sets of lines. In each case, reflect on the role of 'a' and 'b'. y = 4x, y = 2x, y = x y = -6x, y = -3x, y = -x y = 5x, y = -5x y = 3x - 1, y = 3x, y = 3x + 1 y = -2x - 3, y = -2x, y = 2x + 3

Answer:

(i) y = 4x, y = 2x, y = x: 

Table of values

[Insert graph: y = 4x, y = 2x, y = x plotted on the same axes, all passing through origin]

Observations:All three lines pass through the origin (0,0) since b = 0 in each case. As slope a increases (1 → 2 → 4), the line becomes steeper.

ii) y = –6x, y = –3x, y = –x:

[Insert graph: y = –6x, y = –3x, y = –x plotted on the same axes, all passing through origin with negative slopes]

Observations: All pass through the origin with negative slopes; as |a| increases, the line becomes steeper (but sloping downward).

(iii) y = 5x, y = –5x

[Insert graph: y = 5x and y = –5x plotted on the same axes, mirror images through origin]

Observations: : Both pass through the origin with equal steepness (|a| = 5) but slope in opposite directions. — mirror images of each other about the axes.

LineTwo Pointsy = 5x(0,0), (1,5)y = –5x(0,0), (1,–5)

(iv) y = 3x – 1, y = 3x, y = 3x + 1: 

[Insert graph: y = 3x – 1, y = 3x, y = 3x + 1 plotted on the same axes, three parallel lines]

Observations: All have the same slope (a = 3), so the lines are parallel; only b (y-intercept) differs, shifting the lines up or down.

Liney-intercepty = 3x – 1(0,–1)y = 3x(0,0)y = 3x + 1(0,1)

(v) y = –2x – 3, y = –2x, y = 2x + 3:

 [Insert graph: y = –2x – 3, y = –2x, y = 2x + 3 plotted on the same axes — two parallel lines and one non-parallel line]

Observations: 

The first two lines have the same slope (a = –2) and are parallel to each other, with y-intercepts –3 and 0.

The third line has a different (positive) slope, so it is not parallel to the other two.

LineSlopey-intercepty = –2x – 3–2–3y = –2x–20y = 2x + 323

Note:

To draw the graphs, plot two points for each line and connect them.

Reflections on the role of 'a' and 'b':

• Role of 'a' (slope): The value of 'a' determines the steepness and direction of the line. A larger positive 'a' makes the line steeper (increasing from left to right). A negative 'a' makes the line slope downwards (decreasing from left to right). The sign of 'a' indicates whether the line represents linear growth (positive) or linear decay (negative).
• Role of 'b' (y-intercept): The value of 'b' determines where the line intersects the y-axis. Changing 'b' shifts the line vertically up or down, but keeps it parallel to the original line (if 'a' is constant).