Up or down

The idea

Linear growth is represented by a straight line with positive slope, and linear decay by a straight line with negative slope.

A plant and a candle

A plant grows 2 cm every week. A candle burns down 2 cm every hour. You draw each as a straight line, with time along x and height along y.

Which way do you think each line slopes as you go to the right?

• Both go up
• Growth goes up, burning down goes down
• Both go down

Growth and decay

Change a. The line goes up or down as you go to the right. Read what the toy says.

Sign of the slope

Linear growth is a straight line with positive slope. Linear decay is a straight line with negative slope.

Growth: as x grows, y goes up, and the line climbs to the right. Decay: as x grows, y goes down, and the line falls to the right.

Notes

Growth: a line with positive slope. Decay: a line with negative slope.

Check yourself

Which line shows linear decay?

Water in a tank rises by 5 litres every minute. The line that shows this has…

A line goes down as you move to the right. Its slope is…

The line y = ax passes through the point (1, −3). What is a?

Answer: -3

−3 = a × 1, so a = −3. The slope is negative: this is linear decay.

• y = 3x + 4. Its slope 3 is positive: that is growth.
• y = 3x − 4. Its slope 3 is positive: that is growth.
• y = −3x + 4 — correct. Yes. Its slope is −3, which is negative.
• a positive slope — correct. Yes. The amount grows, so the slope is positive.
• a negative slope. A negative slope shows the amount falling.
• no slope. The amount changes, so the line has a slope.
• positive. A positive slope climbs to the right.
• negative — correct. Yes. A line that falls to the right has a negative slope.
• the same as y = x. The line y = x climbs to the right.