Represent relationships using tables

A table represents a linear relationship by pairing input values with corresponding output values, identifying the independent and dependent quantities and supporting the determination of missing values. For equally spaced inputs, a linear pattern has a constant change in output; the table can reveal the rate of change and initial value and distinguish proportional relationships, which have a constant ratio and include the pair $(0,0)$ , from nonproportional linear relationships. Nonlinear, piecewise, and more abstract function tables are not included.

Detailed Explanation: Represent relationships using tables

A table shows how an input and an output are connected.

The independent quantity is the input. You choose its value.
The dependent quantity is the output. Its value depends on the input.

Example

A bike rental costs a $2 starting fee plus $3 for each hour. Complete the table.

Hours rented, $x$	Cost, $y$
$0$	$2$
$1$	$5$
$2$	$8$
$3$	$11$
$4$	?

Step 1: Identify the quantities

The number of hours, $x$ , is the independent quantity because we choose how long to rent the bike.

The cost, $y$ , is the dependent quantity because it depends on the number of hours.

Step 2: Look for the change in the output

As the input increases by $1$ hour, the cost changes by:

5-2=3,\qquad 8-5=3,\qquad 11-8=3

The output has a constant change of $3$ . This means the relationship is linear.

Step 3: Find the missing value

The cost increases by $3$ for each additional hour:

11+3=14

So the completed table is:

Hours rented, $x$	Cost, $y$
$0$	$2$
$1$	$5$
$2$	$8$
$3$	$11$
$4$	$\boxed{14}$

The table represents the rule

y=3x+2

The $3$ is the rate of change, and the $2$ is the starting value.

This relationship is not proportional because the output is not $0$ when the input is $0$ . A proportional relationship must include the pair $(0,0)$ .

Learn by doing: Represent relationships using tables

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