Math 1 · Unit 2 · Lesson 6 · part 3 of 5
Three ways to find it: a table, a graph, a rule
A plant's height in inches, week by week.
Three ways to find it: a table, a graph, a rule
From a table
A plant's height in inches, week by week.
| week | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| inches | 2 | 4 | 5 | 7 | 8 | 12 |
Interval from week 1 to week 5. Pick the two rows: (1, 4) and (5, 12).
Top: outputs, 12 − 4 = 8 inches.
Bottom: inputs, 5 − 1 = 4 weeks.
Rate: 84 = 2. About 2 inches per week between week 1 and week 5.
From a graph
Read two points off the picture, then do the same subtraction. On the graph below, the interval from x = 2 to x = 6 runs from (2, 5) to (6, 13).
Top: 13 − 5 = 8. Bottom: 6 − 2 = 4. Rate: 84 = 2 inches per week. The dashed steps show it: over 4 to the right, up 8.
Now the interval from x = 0 to x = 2: (0, 2) to (2, 5). Top: 3. Bottom: 2. Rate: 32 = 1.5 inches per week. A different interval, a different rate. That is why you always name the interval.
From a rule
f(x) = x × x + 1. Interval from x = 1 to x = 3.
Evaluate at the two inputs first: f(1) = 1 + 1 = 2, and f(3) = 9 + 1 = 10. Now you have two points, (1, 2) and (3, 10).
Top: 10 − 2 = 8. Bottom: 3 − 1 = 2. Rate: 82 = 4. The output grows 4 for each step of x, on average, between 1 and 3.
Same three moves. The only extra step is finding the outputs first.