Linear Function Graphs

Activities to try

1. 1

   Build the line from a table of values

   Why it works: A straight line is a consequence of a constant rate of change, not an assumption — 'straight' is something you can test from the numbers.

   1. Take the equation y = 2x + 1 and complete a table of values for x = -2, -1, 0, 1, 2 (giving y = -3, -1, 1, 3, 5).
   2. Plot each (x, y) pair on axes and join them.
   3. Observe that equal steps in x give equal steps in y, and that the points fall exactly on a straight line.
   4. State in your own words why a constant rate of change produces a straight line.
2. 2

   Vary m and c independently

   Why it works: m controls steepness AND direction (sign), while c slides the whole line to its y-intercept — each parameter is bound to a distinct visual effect.

   1. Keep c fixed and change m: try m = 1, then m = 3, then m = -2, replotting or predicting each line.
   2. Describe what happens as positive m gets larger (the line steepens) and as m goes negative (the line flips from climbing to falling, pivoting about the same intercept); steepness depends on the size of m ignoring its sign, so m = -2 is steeper than m = 1.
   3. Now keep m fixed and change c: try c = 0, then c = 2, then c = -3.
   4. Describe what happens (the whole line slides up or down and crosses the y-axis exactly at c), and predict the crossing point before plotting.
3. 3

   Distance-time race: steepness is speed, not height

   Why it works: Speed is the gradient (steepness), not the height of the line; a runner can start ahead (large c) yet be slower (small m). Emma is faster because her line is steeper, covering more distance each second, even though Maggie's line sits higher at 4 seconds — until the lines meet at 5 seconds, after which Emma pulls ahead.

   1. Plot two runners' distance over time on the same axes, e.g. Emma d = 6t and Maggie d = 4t + 10.
   2. Before computing, predict who is faster and write your reason.
   3. At 4 seconds, note that Maggie's line is higher (26 m versus Emma's 24 m), but Emma's line is steeper.
   4. Take two points on each line and compute distance-per-second (Emma gains 6 per second; Maggie gains 4 per second).
   5. Identify where/whether the lines cross — here they meet at 5 seconds (both at 30 m), after which Emma pulls ahead — and say what that crossing means in the race.