Connect Table Rule and Graph

Learn points say which quantities go together

Write a point as (x, y): input first, output second. On these graphs, time is horizontal and total water is vertical. The origin is (0, 0). A proportional graph is a straight line through the origin over its allowed inputs. A straight line with a nonzero starting value is not proportional.

For 0 to 4 minutes, Empty start has y = 2 × x. Three liters at start has y = 2 × x + 3. Both gain 2 liters each minute, but their total-water relationships differ.

Two matched water-model graphs for time 0 to 4 minutes. Empty start passes through (0,0), (1,2), (2,4) and (4,8). Three liters at start passes through (0,3), (1,5), (2,7) and (4,11). Both are straight; only empty start passes through the origin.
Time (minutes)Empty start (liters)Three liters at start (liters)
003
125
247
4811

In the empty-start model, (1, 2) means 2 liters after 1 minute: k is 2 liters per minute. In the other model, (1, 5) includes the starting 3 liters. The output at one input unit is not its rate of increase.

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Connect Table Rule and Graph

Guided practice a fractional constant

An empty tank gains 1.5 liters each minute, for 0 to 4 minutes. Let x be time in minutes and y be total water in liters.

x (minutes)y (liters)
0
1
2
4

Complete the table. State k as a decimal or fraction with units and write a rule. On the supplied grid plot the four pairs; join them only within the stated continuous-time domain. Explain (0, 0) and the point at one minute.

Blank coordinate grid. Horizontal time axis runs from 0 to 4 minutes in steps of 1; vertical total-water axis runs from 0 to 12 liters in steps of 1. Use the same labeled pairs in an oral or written explanation if drawing is not your response method.

Rule, units and point meaning

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Connect Table Rule and Graph

Practice check a straightline shortcut

For 0 to 4 minutes, a different tank begins with 2 liters and gains 2 liters each minute.

Time (minutes)Total water (liters)
02
14
26
410

Fictional learner Ari says, “The graph is straight and passes through (1, 4), so total water is proportional to time with k = 4 liters per minute.” Plot or describe the given pairs. Decide whether the claim and proposed rule y = 4 × x fit. Give a rule matching the stated model and explain the starting point.

Blank coordinate grid: time 0 to 4 minutes horizontally, total water 0 to 12 liters vertically. Plot or describe the table’s four ordered pairs; the grid does not supply a line or classification.

Evidence, rule and point explanation

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