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For the function shown, determine the following.

the Period
the Maximum
the Minimum
the Midline. Express as a line:
the Amplitude
Knowing that this is a periodic function, what would the output value be when
?

Let's analyze the graph to determine the required characteristics of the function:

1. **Period**: The period is the distance along the x-axis after which the function starts repeating itself. Observing the graph, the function repeats its pattern approximately every 10 units along the x-axis. So, the period is **10**.

2. **Maximum**: The maximum value is the highest y-value the function reaches. From the graph, the maximum y-value is **5**.

3. **Minimum**: The minimum value is the lowest y-value the function reaches. The minimum y-value is **-5**.

4. **Midline**: The midline is the horizontal line that represents the average of the maximum and minimum values. It is calculated as:
   \[
   \text{Midline} = \frac{\text{Maximum} + \text{Minimum}}{2} = \frac{5 + (-5)}{2} = \frac{0}{2} = 0
   \]
   So, the midline is the line **\(y = 0\)**.

5. **Amplitude**: The amplitude is the distance from the midline to either the maximum or minimum value. Since the maximum is 5 and the midline is 0, the amplitude is:
   \[
   \text{Amplitude} = \text{Maximum} - \text{Midline} = 5 - 0 = 5
   \]

For the output value when \( x = ? \), please provide the specific x-value you are interested in.

Would you like more details on any of these calculations? 

Here are some related questions:

1. How do we calculate the period of a non-uniform periodic function?
2. What are the implications of a shifted midline for amplitude calculations?
3. How does the amplitude change if the midline is not at \(y = 0\)?
4. Can a function have multiple periods, and how would that affect the graph?
5. How would you determine the equation of this function based on its graph?

**Tip:** When identifying the period from a graph, always ensure that you check multiple cycles to confirm consistency.

Explore how to analyze periodic functions to determine the period, maximum, minimum, and midline. Understand concepts like amplitude and midline calculations with detailed explanations and examples. Suitable for high school students and beyond.

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