To determine the instructor's initial charge from the graph, we can use the information given about the points on the graph and the hourly rate.
From the points provided on the graph, we can represent the total cost \( C(t) \) as:
\[ C(t) = I + 8t \]
where:
- \( I \) represents the initial fee
- \( t \) represents the time in hours
- \( 8t \) represents the total hourly charges at a rate of $8.00 per hour.
We have the following points from the graph:
- At \( t = 4 \), \( C(4) = 52 \)
- At \( t = 5 \), \( C(5) = 60 \)
- At \( t = 6 \), \( C(6) = 68 \)
- At \( t = 7 \), \( C(7) = 76 \)
We can use one of these points to form an equation. Using the point \( (4, 52) \):
\[ 52 = I + 8 \times 4 \] \[ 52 = I + 32 \]
Now, solving for \( I \):
\[ I = 52 - 32 \] \[ I = 20 \]
So the instructor's initial charge is \( \boxed{20} \) dollars.
We can verify this with additional points:
Using the point \( (5, 60) \): \[ 60 = I + 8 \times 5 \] \[ 60 = I + 40 \] \[ I = 60 - 40 \] \[ I = 20 \]
Using the point \( (6, 68) \): \[ 68 = I + 8 \times 6 \] \[ 68 = I + 48 \] \[ I = 68 - 48 \] \[ I = 20 \]
Using the point \( (7, 76) \): \[ 76 = I + 8 \times 7 \] \[ 76 = I + 56 \] \[ I = 76 - 56 \] \[ I = 20 \]
All points confirm that the initial charge is \( \boxed{20} \) dollars.