Rules, tables and dependent variables
Connect changing quantities through an equation, a table and a graph.
Before you begin
- Substitution into expressions
- Plotting ordered pairs
Choose input and output quantities
The independent variable is the input you choose or measure as the starting quantity in a model. The dependent variable is determined from it by a rule. If each ticket costs $6, the number of tickets n is the input and total cost C = 6n is the output. This naming describes the model; it does not by itself establish a scientific cause.
Connect representations and restrictions
A table lists selected input-output pairs, an equation states the general rule and a graph displays pairs as positions. Put the independent variable on the horizontal axis. Include zero if it is meaningful, and check whether intermediate values are allowed. Ticket counts are whole numbers, so a line through their plotted points may suggest fractional purchases that the actual situation does not permit.
Worked example
Water flows at a constant 4 L/min into an empty tank. Write a rule and table for 0, 1, 2 and 3 minutes.
Show the worked solution
- Let t be elapsed time in minutes and V be water volume in litres. The rule is V = 4t.
- Substitute t = 0, 1, 2, 3 to obtain V = 0, 4, 8, 12.
- Plot (0,0), (1,4), (2,8), (3,12). Time is independent and volume is dependent, until the tank fills.
V = 4t; volumes 0, 4, 8 and 12 L.
Try it yourself
For y = 3x + 2, what is the output when x = 4?
Common mistakes
- State the units and which variable is the input.
- A graph may need discrete points when intermediate inputs are not meaningful.
What can you explain now?
A notebook costs $2.50 with no fixed charge. Write a cost rule and list pairs for 0, 2 and 4 notebooks. Can n = 1.5 be used for an actual purchase?
Compare with the explanation
C = 2.5n; (0,0), (2,5), (4,10). No, whole notebooks require a whole-number n.
Multiply each allowed count by the unit price. The equation can accept decimal inputs algebraically, but the purchase context restricts them.
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Bring the ideas together
Equations and relationships
Try the unit reviewOriginal StudyForA content. AI-assisted checks completed; subject-teacher review is still pending. These lessons are not endorsed by an exam board.
Common Core mathematics standards ↗