Steeper or flatter?

The idea

In y = ax, a is the slope of the line; for positive a, the line is steeper than y = x (which is equally inclined to both axes) when a > 1 and less steep when a < 1.

Three lines

Look at three lines through the origin: y = ½x, y = x and y = 2x. The line y = x is the middle one: it is equally inclined to both axes.

Which line do you think is the steepest?

• y = ½x
• y = x
• y = 2x

Compare with y = x

The dashed line is y = x. Change a and compare your line with it.

Slope a

In y = ax, a is the slope of the line. When a > 1 the line is steeper than y = x. When a < 1 it is less steep. The line y = x is equally inclined to both axes.

Here are y = ½x (less steep), y = x and y = 2x (steeper).

Notes

In y = ax, a is the slope. a > 1: steeper than y = x. a < 1: less steep.

Check yourself

Which line is steeper than y = x?

Which line is less steep than y = x?

The line y = ax passes through (2, 6). What is a? Is it steeper than y = x?

Answer: 3

6 = a × 2, so a = 3. That is more than 1, so the line is steeper than y = x.

The line y = x is equally inclined to both axes. What is its slope a?

• y = 3x — correct. Yes. a = 3 is more than 1.
• y = ½x. a = ½ is less than 1, so it is less steep.
• y = x. y = x is the line itself: equally steep.
• y = 2x. a = 2 is more than 1, so it is steeper.
• y = ½x — correct. Yes. a = ½ is less than 1.
• y = 4x. a = 4 is more than 1, so it is steeper.
• 0. a = 0 gives y = 0, which is the x-axis itself.
• 1 — correct. Yes. y = x is y = 1x, so a = 1.
• 2. a = 2 gives y = 2x, which is steeper.