1. **State the problem:** We analyze the monthly charges of three telecom companies (Any Talk, Easy Talk, Talk More) based on their usage time (minutes) from the graph. 2. **Initial values:** The initial value is the cost at 0 minutes (y-intercept). - Any Talk starts at (0, 0), so initial value is $0$. - Easy Talk starts at (0, 20), so initial value is $20$. - Talk More starts at (0, 15), so initial value is $15$. The initial value represents the subscription charge or fixed monthly fee before any usage. 3. **Subscription charge:** Any Talk has initial value $0$, so it has no subscription charge. 4. **Constant rate:** The rate is the slope of each line (change in cost per minute). - Any Talk: from (0,0) to (400,40), slope $= \frac{40-0}{400-0} = \frac{40}{400} = 0.1$ per minute. - Easy Talk: from (0,20) to (400,28), slope $= \frac{28-20}{400-0} = \frac{8}{400} = 0.02$ per minute. - Talk More: from (0,15) to (400,23), slope $= \frac{23-15}{400-0} = \frac{8}{400} = 0.02$ per minute. 5. **Parallel lines:** Easy Talk and Talk More have the same slope $0.02$, so their lines are parallel. This means they increase cost at the same rate per minute. 6. **Gregory uses 300 minutes:** Calculate cost for each: - Any Talk: $0 + 0.1 \times 300 = 30$ - Easy Talk: $20 + 0.02 \times 300 = 20 + 6 = 26$ - Talk More: $15 + 0.02 \times 300 = 15 + 6 = 21$ Gregory should use Talk More because it costs the least at 300 minutes. 7. **Sabrina uses less than 200 minutes:** Calculate cost at 200 minutes: - Any Talk: $0 + 0.1 \times 200 = 20$ - Easy Talk: $20 + 0.02 \times 200 = 20 + 4 = 24$ - Talk More: $15 + 0.02 \times 200 = 15 + 4 = 19$ For less than 200 minutes, Sabrina should use Any Talk if usage is very low (close to 0) because it has no subscription fee, but Talk More is cheaper at 200 minutes. **Summary:** - Initial values: Any Talk $0$, Easy Talk $20$, Talk More $15$. - No subscription charge: Any Talk. - Rates: Any Talk $0.1$, Easy Talk $0.02$, Talk More $0.02$. - Parallel lines: Easy Talk and Talk More. - Gregory (300 min): Talk More best. - Sabrina (<200 min): Any Talk best for very low usage, Talk More better near 200 min.