Let y=f(x)
 be a periodic function whose values are given below. Find the period, amplitude, and midline.
x
6	36	66	96	126	156	186	216	246
f(x)
10	6	-10	10	6	-10	10	6	-10
The period is 
.

The middle line is 
.  
(Enter your answer in the form of y=1).

The amplitude is 
.



Let's analyze the periodic function based on the values provided for \( x \) and \( f(x) \):

| \( x \)  | 6 | 36 | 66 | 96 | 126 | 156 | 186 | 216 | 246 |
|----------|----|----|-----|-----|-----|-----|-----|-----|-----|
| \( f(x) \)| 10 | 6 | -10 | 10  | 6   | -10  | 10  | 6  | -10 |

### 1. **Finding the Period:**

The period of a function is the length of the interval after which the function repeats itself. Looking at the sequence of \( f(x) \) values, the function repeats every three \( x \)-values: \( 6, 36, 66 \), etc.

- From \( x = 6 \) to \( x = 96 \), the function repeats (e.g., at \( x = 6 \) and \( x = 96 \), \( f(x) = 10 \)).

The difference between these values is:
\[
96 - 6 = 90
\]
Thus, the **period** is \( \boxed{90} \).

### 2. **Finding the Midline:**

The midline is the average value around which the function oscillates. The maximum value is 10 and the minimum value is -10, so the midline is the average of these two:

\[
\text{Midline} = \frac{10 + (-10)}{2} = 0
\]
Thus, the equation for the midline is \( \boxed{y = 0} \).

### 3. **Finding the Amplitude:**

The amplitude is the distance from the midline to the maximum (or minimum) value. The maximum value is 10, and the midline is 0, so the amplitude is:

\[
\text{Amplitude} = |10 - 0| = 10
\]
Thus, the **amplitude** is \( \boxed{10} \).

### Summary of Results:
- **Period:** \( 90 \)
- **Midline:** \( y = 0 \)
- **Amplitude:** \( 10 \)

Would you like more details or have any questions? Here are some related questions:

1. How would the graph of this periodic function look over several cycles?
2. What effect does changing the amplitude have on the function's graph?
3. How can we use the midline to shift the graph vertically?
4. What if the period of the function was doubled?
5. How can we find the equation of this periodic function?

**Tip:** The period of a function determines how frequently it repeats its values over a given interval.

Let y = f(x) be a periodic function whose values are given below. Find the period, amplitude, and midline. x: 6, 36, 66, 96, 126, 156, 186, 216, 246 f(x): 10, 6, -10, 10, 6, -10, 10, 6, -10. The period is __. The middle line is __. (Enter your answer in the form of y = __). The amplitude is __.

Learn how to find the period, amplitude, and midline of a periodic function with specific values of x and f(x). This guide walks through key concepts such as periodicity, midline calculation, and amplitude determination for high school students.

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