**Using graphs to find rates of change :**

A rate of change is a ratio of the amount of change in the dependent variable to the amount of change in the independent variable.

In this section, we are going to see, how to use a graph to find rates of change.

**Example 1 :**

The graph shows the distance Nathan bicycled over time. What is Nathan’s rate of change ?

**Solution :**

**Step 1 :**

Identify the independent and dependent variables.

Independent : Time

Dependent : Distance

**Step 2 :**

Find the rates of change.

**0 hour to 1 hour :**

Change in distance / Change in time = (15 - 0) / (1 - 0)

Change in distance / Change in time = 15 / 1

Change in distance / Change in time = 15

**1 hours to 4 hours :**

Change in distance / Change in time = (60 - 15) / (4 - 1)

Change in distance / Change in time = 45 / 3

Change in distance / Change in time = 15

**2 hours to 4 hours :**

Change in distance / Change in time = (60 - 30) / (4 - 2)

Change in distance / Change in time = 30 / 2

Change in distance / Change in time = 15

Nathan’s rate of change is 15 miles per hour.

**Reflect : **

1. Recall that the graph of a proportional relationship is a line through the origin. Explain whether the relationship between Nathan’s time and distance is a proportional relationship.

Yes ; the graph is a line through the origin.

2. Does it matter what interval you use when you find the rate of change of a proportional relationship ? Explain.

No ; in a proportional relationship, the rate of change is constant.

**Example 2 :**

The graph shows the rate at which water is leaking from a tank. Find the rate at which the water is leaking from the tank per minute.

**Solution :**

**Step 1 :**

Identify the independent and dependent variables.

Independent : Time

Dependent : Leakage

**Step 2 :**

Find the rates of change.

**0 hours to 4 hours :**

Leakage / Change in time = (3 - 0) / (4 - 0)

Leakage / Change in time = 3 / 4

Leakage / Change in time = 0.75

**4 hours to 8 hours :**

Leakage / Change in time = (6 - 3) / (8 - 4)

Leakage / Change in time = 3 / 4

Leakage / Change in time = 0.75

**1 hour to 8 hours :**

Leakage / Change in time = (6 - 0) / (8 - 0)

Leakage / Change in time = 6 / 8

Leakage / Change in time = 3 / 4

Leakage / Change in time = 0.75

The water is leaking at the rate of 0.75 gallons per minute.

After having gone through the stuff given above, we hope that the students would have understood, how to use graphs to find rates of change.

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