A linear function adds the same amount over equal input intervals. With two known charges and a target time, compare the hour gaps first. If the target is one equal gap beyond a known charge, repeat the same dollar increase; Desmos can handle the arithmetic. Don’t treat a known total as a price per hour, because the total may include a fixed charge.
Hints
- Hint 1
Compare the hour gap between the two given times with the gap from the later time to the requested time. In a linear function, equal time gaps produce equal changes in the charge. Do these gaps match?
- Hint 2
Find the increase in the total charge between the two given times. If the next time gap is the same length, add that increase once more. Don’t divide a total charge by its number of hours and assume that’s the hourly rate.
Step-by-step
Repeat the increase over an equal time gap
Step 1Compare the hour gaps
Type and in Desmos. Both show , so the two time gaps are equal: each covers hours.
- Step 2
Connect equal time gaps to equal charges
A linear function changes by the same dollar amount over equal time gaps. Since both gaps are hours, their charge increases must match:
- Step 3
Find the first increase
Type in Desmos; it shows . That’s the increase for the first hours, so the next hours add another $105. The $105 is not an hourly rate.
- Step 4
Find the 8-hour charge
Start with the 5-hour charge of $195. Type in Desmos; it shows . So the tutor charges $300 for hours. Grid in 300.