On Tuesday nights, a skating rink offers a special student price.

It costs $2.00 for the first 20 minutes.
After the first 20 minutes, it costs $0.05 per minute.
Andrea skates every Tuesday night for a minimum of 30 minutes and a maximum of 1 hour. The time she skates places a restriction on the domain of the function that represents the cost for skating.

On the coordinate plane below, graph the portion of the function that best represents Andrea's cost for skating on Tuesday nights.



To graph the portion of the function that represents Andrea's cost for skating on Tuesday nights, we need to define a piecewise function that takes into account the two different rates: 

1. **For the first 20 minutes:** The cost is fixed at $2.00. This corresponds to the first segment of the graph, where time $ t $ (in minutes) is between 0 and 20.
   
   \[
   C(t) = 2 \quad \text{for} \quad 0 \leq t \leq 20
   \]
   In this range, the cost remains constant at $2.00.

2. **After the first 20 minutes:** For each additional minute after 20 minutes, it costs an extra $0.05 per minute. Thus, for $ t > 20 $, the cost increases linearly by $0.05 for each minute beyond 20 minutes. 

   \[
   C(t) = 2 + 0.05(t - 20) \quad \text{for} \quad 20 < t \leq 60
   \]
   Here, $ t - 20 $ represents the number of minutes skated beyond the initial 20, and multiplying by 0.05 gives the additional cost.

### Step-by-step for graphing:

1. **First segment (0 to 20 minutes):** This is a horizontal line from $ t = 0 $ to $ t = 20 $ at a height of $2.00. The cost is constant at $2.00 for the first 20 minutes.
   
2. **Second segment (20 to 60 minutes):** Starting from $ t = 20 $, the cost starts to increase. At $ t = 20 $, the cost is $2.00, and for each additional minute, the cost increases by $0.05. Therefore, at $ t = 60 $ (1 hour), the cost is:

   \[
   C(60) = 2 + 0.05(60 - 20) = 2 + 0.05(40) = 2 + 2 = 4
   \]

   So, at $ t = 60 $, the cost is $4.00.

### Graph Characteristics:
- The first part is a horizontal line from $ t = 0 $ to $ t = 20 $, at $ C(t) = 2 $.
- The second part is a straight line with a positive slope of 0.05 from $ t = 20 $ to $ t = 60 $, starting at $2.00 and ending at $4.00.

To graph:
1. Draw a horizontal line from $ t = 0 $ to $ t = 20 $ at the cost $ C = 2 $.
2. From $ t = 20 $, draw a line with a slope of 0.05 that increases up to $ t = 60 $, where the cost is $4.00.

Would you like help plotting this graph or more details on any specific part of this process?

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**Related questions:**

1. How would the cost function change if the rate per minute after 20 minutes increased to $0.10?
2. What would be the total cost for skating exactly 45 minutes on Tuesday night?
3. How can we represent the cost function algebraically if Andrea skated for more than 1 hour?
4. What is the domain and range of the cost function in this situation?
5. How can you interpret the slope of the second portion of the graph?

**Tip:** When interpreting piecewise functions, be sure to carefully observe the transition points where the rate or behavior changes.

On Tuesday nights, a skating rink offers a special student price. It costs $2.00 for the first 20 minutes. After the first 20 minutes, it costs $0.05 per minute. Andrea skates every Tuesday night for a minimum of 30 minutes and a maximum of 1 hour. The time she skates places a restriction on the domain of the function that represents the cost for skating. On the coordinate plane below, graph the portion of the function that best represents Andrea's cost for skating on Tuesday nights.

Learn how to graph a piecewise function representing Andrea's cost for skating at a rink. The function includes a flat rate for the first 20 minutes and a linear increase after that. Suitable for students in Grades 8-10, this problem involves core concepts like linear functions and graphing.

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