To derive a function rule for the given table, we first need to identify the pattern between the number of days and the cost to rent the truck.
The table shows the following values:
| Days | Cost to Rent a Truck | |------|-----------------------| | 1 | 34 | | 2 | 57 | | 3 | 80 | | 4 | 103 |
Next, let's calculate the difference in the cost as the number of days increases:
- From 1 day to 2 days: \(57 - 34 = 23\)
- From 2 days to 3 days: \(80 - 57 = 23\)
- From 3 days to 4 days: \(103 - 80 = 23\)
The cost increases by a constant amount of 23 for each additional day. Therefore, we can deduce the cost function is linear.
Now, let's find the function rule. A linear function can be represented in the form:
\[ \text{Cost} = m \times \text{Days} + b \]
Where:
- \(m\) is the slope (the rate of increase),
- \(b\) is the y-intercept (the cost when days = 0).
From our calculations above, we know:
- The slope \(m = 23\) (the cost increases by 23 for each additional day).
To find \(b\), we can use one of the points. Let's use the point where \(Days = 1\) and \(Cost = 34\):
\[ 34 = 23 \times 1 + b \] \[ 34 = 23 + b \] \[ b = 34 - 23 = 11 \]
Now we can write the function rule:
\[ \text{Cost} = 23 \times \text{Days} + 11 \]
So the final function rule for the cost to rent a truck based on the number of days is:
\[ C(d) = 23d + 11 \]
Where \(C(d)\) is the cost and \(d\) is the number of days you rent the truck.