Graphs of y = ax: slope

Two points are enough for a straight line. For y = 2x + 1: x = 0 gives (0, 1) = A, and x = 3 gives (3, 7) = B. Plot them, join them and extend the line both ways.

Is a point on the line? Only if its coordinates make the equation true. (7, 15) is on y = 2x + 1 because 2 × 7 + 1 = 15.

Find the rule from the points.

• (–1, –3), (0, 0), (1, 3), (3, 9), (4, 12): each y is three times its x, so the line is y = 3x.
• (–3, 6), (–2, 4), (0, 0), (1, –2): each y is –2 times x, so the line is y = –2x.

Every line y = ax passes through the origin (0, 0). Compare y = ½x, y = x and y = 2x. The line y = x is equally tilted towards both axes. When a > 1 the line is steeper than y = x; when a < 1 it is less steep. a is called the slope.

Negative slope. y = –⅓x, y = –x and y = –3x fall from left to right. So linear growth is a line with positive slope, and linear decay is a line with negative slope.

The slope is the step. For the tiles, y = 2x – 1 has slope 2, the same 2 tiles added at every stage. The slope is the constant difference between neighbouring terms.