## Algebra 1

The function for this relationship would be C(t) = 4t - 1. In 1 day the machine runs for 23 cycles. The reasonable domain for this graph would be from $\frac{1}{4}$ to 6 hours because the machine can run for 6 hours max and completes one cycle in 15 minutes. The range would be from 0 cycles which is minimum to 23 cycles which is maximum. The graph is shown below in the screenshot.
The function for this relationship would be C(t) = 4t - 1 because in one hour the machine can make 4 cycles as it takes 15 minutes which is $\frac{1}{4}$ hours and this after 1 hour you can make 4 cycles. The reasonable domain for this graph would be from $\frac{1}{4}$ to 6 hours because the machine can run for 6 hours max and completes one cycle in 15 minutes. We cannot count 0 hours in the domain because we cannot have negative cycles and therefore at t=$\frac{1}{4}$, the number of cycles is 0. The range would be from 0 cycles which is minimum to 23 cycles which is maximum. The graph is a linear function because in one hour the machine makes 4 cycles which is the slope of 4. We subtract 1 from 4t because in the first hour we make only 3 cycles in the first hour as it takes 10 minutes to warm up and thus we lose $\frac{1}{6}$ hours which is 10 minutes. The relationship is linear.