Math 1 · Unit 7 · Lesson 1 · part 2 of 6
Lines add. This family multiplies.
In Unit 3 you met the first family of graphs: the lines, y = mx + b.
Lines add. This family multiplies.
In Unit 3 you met the first family of graphs: the lines, y = mx + b. Every line adds the same amount every step, and that amount is the slope. Today a second family arrives, and it does something different: it multiplies by the same number every step.
Here is the story. A rumor starts with one person. Every day, each person who knows it tells two new people. Day 0: 1 person. Day 1: 2. Day 2: 4. Day 3: 8. Day 4: 16. Each day the number doubles: it is multiplied by 2, not increased by 2. Fold a sheet of paper and count the layers and you get the same list: 1, 2, 4, 8, 16.
| x | y | subtract to compare |
|---|---|---|
| 0 | 1 | |
| 1 | 3 | 3 − 1 = 2 |
| 2 | 5 | 5 − 3 = 2 |
| 3 | 7 | 7 − 5 = 2 |
| 4 | 9 | 9 − 7 = 2 |
| x | y | divide to compare |
|---|---|---|
| 0 | 1 | |
| 1 | 2 | 2 ÷ 1 = 2 |
| 2 | 4 | 4 ÷ 2 = 2 |
| 3 | 8 | 8 ÷ 4 = 2 |
| 4 | 16 | 16 ÷ 8 = 2 |
Down the line's table, the outputs go up by a constant difference: subtract any output from the next and you always get 2. Down the doubling table, the outputs are related by a constant ratio: divide any output by the one before and you always get 2. Subtract to find a difference; divide to find a ratio. A function that multiplies by the same number every step is an exponential function.
Build both tables one step at a time and watch the two columns pull apart.