To determine whether the function represents a direct variation, we need to check two key properties:
1. The function should pass through the origin (0,0).
2. The function should exhibit a constant rate of change, or slope, meaning the ratio of the change in cost to the change in time should be consistent.
### Step-by-Step Analysis:
Let's analyze the given data from the table:
| Time (hours) | Cost (\[tex]$) |
|--------------|-----------|
| 0 | 0 |
| 2 | 10 |
| 4 | 20 |
| 6 | 30 |
| 8 | 40 |
#### 1. Check if the function passes through the origin
Looking at the time and cost values, we see:
- When the time is 0 hours, the cost is \$[/tex]0.
So, it passes through the origin (0,0).
#### 2. Calculate the rate of change (slope)
We need to determine if the change in cost per hour is constant. Calculate the slope between successive points.
- From (0, 0) to (2, 10):
[tex]\[ \text{slope} = \frac{10 - 0}{2 - 0} = \frac{10}{2} = 5 \][/tex]
- From (2, 10) to (4, 20):
[tex]\[ \text{slope} = \frac{20 - 10}{4 - 2} = \frac{10}{2} = 5 \][/tex]
- From (4, 20) to (6, 30):
[tex]\[ \text{slope} = \frac{30 - 20}{6 - 4} = \frac{10}{2} = 5 \][/tex]
- From (6, 30) to (8, 40):
[tex]\[ \text{slope} = \frac{40 - 30}{8 - 6} = \frac{10}{2} = 5 \][/tex]
All calculated slopes are \[tex]$5 per hour, indicating the rate of change is constant.
### Conclusion
The function passes through the origin and has a constant rate of change of \$[/tex]5 per hour. Thus, the function represents a direct variation.
So, the correct explanation is:
"The function represents a direct variation because it passes through the origin and has a constant rate of change of \$5 per hour."
