Build a Rule and Graph

A linear rule can be written y = mx + b. x is input; y is output; m is change per input unit. b is the rule’s value at x = 0. If zero is an allowed input, b is the modeled output there. Two distinct input points determine one line when the relationship is explicitly assumed linear; two observations alone do not establish a physical linear rule.

Model from two nonzero points

For 0 ≤ t ≤ 6, a supplied linear model of total water V has points (2, 10) and (4, 13). Minutes come first and liters second.

m = (13 − 10) ÷ (4 − 2) = 3 ÷ 2 = 1.5 liters per minute. At t = 2, the changing part is 1.5 × 2 = 3 liters. Subtract it from the observed total: b = 10 − 3 = 7 liters. So V = 1.5t + 7. Check both points: 1.5 × 2 + 7 = 10 and 1.5 × 4 + 7 = 13.

The 10 liters is the total after two minutes; it is not the start. m has liters-per-minute units. b has liters units. At zero time, the modeled total is 7 liters, so this total is linear but not proportional to time.

1 Guided rule

A supplied linear model on 0 ≤ t ≤ 6 has points (1, 7) and (4, 13), in minutes and liters. Find m, recover b, write a rule and verify both points.

2 A new linear model

For 0 ≤ t ≤ 8, a supplied linear total-water model has points (2, 8) and (6, 18). Find its change rate and value at zero. Complete the table. Write a rule, then plot or describe its graph over the allowed domain.

3 Recover a negative-rate start

For 0 ≤ t ≤ 6, a supplied linear model of water remaining has points (2, 13) and (4, 8), with minutes first. Find m and b, write a rule, and predict the amount at five minutes. Check both original points.

4 A zero input outside the model

A supplied linear model is valid only for 2 ≤ t ≤ 8 minutes. Its points are (2, 11) and (5, 17) liters. Find a matching linear expression. Does its algebraic value at t = 0 establish an actual amount at zero minutes under these stated conditions? Explain.

Model a temperature below zero

A constructed continuous linear Celsius-temperature model is valid for 0 ≤ t ≤ 6 minutes. Its points are (2, −3) and (4, 1). Rate = [1 − (−3)] ÷ (4 − 2) = 4 ÷ 2 = 2 degrees Celsius per minute. Recover b = −3 − 2 × 2 = −7 degrees Celsius. C = 2t − 7 gives −3 at minute 2 and 1 at minute 4. The start is negative while the rate is positive; their signs describe different quantities.

5 Your signed-reference practice

Another supplied continuous linear Celsius-temperature model on 0 ≤ t ≤ 6 gives (1, −2) and (4, 7). Find its rate, value at zero and rule. Verify both observations. Explain what the two signs mean.

6 Your zero-start practice

A supplied continuous linear total-water model on 0 ≤ t ≤ 6 has points (2, 6) and (5, 15). Find its rate and value at zero. Write and verify a rule. Is this total proportional to elapsed time? Compare it with the constant 9-liter model.