Skill: Represent relationships using graphs

A graph shows how one quantity changes when another quantity changes.

• The independent quantity is the input or quantity you choose. Place it on the horizontal $x$-axis.
• The dependent quantity changes because of the independent quantity. Place it on the vertical $y$-axis.
• A point is written as an ordered pair $(x,y)$.

Example

A bike rental company charges a $10 starting fee plus $5 for each hour rented.

Hours rented, $x$	Cost, $y$
$0$	$10$
$1$	$15$
$2$	$20$
$3$	$25$

Step 1: Identify the quantities

The number of hours is the independent quantity because it determines the cost.

• Horizontal axis: hours rented
• Vertical axis: cost in dollars

Step 2: Choose scales and labels

The hours go from $0$ to $3$, so label the horizontal axis from $0$ to $3$.

The costs go from $10$ to $25$. A scale counting by $5$ is useful, so label the vertical axis $0,5,10,15,20,25$.

Step 3: Write the ordered pairs

Use the hours for $x$ and the cost for $y$:

$$(0,10),\ (1,15),\ (2,20),\ (3,25)$$

Step 4: Plot the points

On the coordinate plane, plot:

• $(0,10)$
• $(1,15)$
• $(2,20)$
• $(3,25)$

The points form a straight-line pattern because the cost increases by $5 for every additional hour.

Step 5: Describe the graph

The graph shows a constant rate of change: the cost increases by $5 per hour. It starts at $(0,10)$, meaning the initial cost is $10 even when the bike is rented for $0$ hours.

Because the graph does not pass through $(0,0)$, this is not a proportional relationship. A proportional relationship has no starting value and passes through the origin $(0,0)$.