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📐 Chapter 2: Introduction to Linear Polynomials • Exercise Set 2.6

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Q1

Exercise Set 2.6

1. Draw the graphs of the following sets of lines. In each case, reflect on the role of 'a' and 'b'.

(i) y = 3x - 1, y = 3x, y = 3x + 1

(ii) y = 4x, y = 2x, y = x

(iii) y = -6x, y = -3x, y = -x

(iv) y = 5x, y = -5x

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

Reflections on the parameters 'a' (slope) and 'b' (y-intercept):

  • (i) y = 3x - 1, y = 3x, y = 3x + 1: Here, the slope a is constant (3), so the three lines are parallel. The y-intercept b changes (-1, 0, 1), shifting the parallel lines vertically up and down the y-axis.
  • (ii) y = 4x, y = 2x, y = x: Here, the y-intercept b is fixed at 0 (all pass through the origin). The slope a changes (4, 2, 1). As a increases, the lines become progressively steeper (closer to the y-axis).
  • (iii) y = -6x, y = -3x, y = -x: All pass through the origin. Since slopes are negative, they represent linear decay. As the magnitude of a increases (from 1 to 6), the lines get steeper in the negative direction.
  • (iv) y = 5x, y = -5x: Both pass through the origin. They have the same steepness but opposite directions; y = 5x represents growth (rising left to right), while y = -5x represents decay (falling left to right). They are reflections of each other across the y-axis.
  • (v) y = -2x - 3, y = -2x, y = 2x + 3: The first two lines (y = -2x - 3 and y = -2x) are parallel to each other with a negative slope, but with different y-intercepts (-3 and 0). The third line (y = 2x + 3) has a positive slope (intersecting the others) and cuts the y-axis at +3.
Parallel Lines: y = 3x − 1, y = 3x, y = 3x + 1 x y y = 3x y = 3x + 1 y = 3x − 1
Role of 'b': Parallel lines with same slope (a = 3) shifted vertically by y-intercept b