The function p(x) = -0.3x^2 + 20x^1/2 + 35 gives the price of a stock, in dollars, over
the first 25 days of the month.
a. Find the five-day moving average value of the stock after the first 5 days (that is,
from x = 0 to x = 5 days). (Round your answer to the nearest cent.)

To find the five-day moving average of the stock price given by the function \( p(x) = -0.3x^2 + 20x^{\frac{1}{2}} + 35 \) from \( x = 0 \) to \( x = 5 \), we need to calculate the average price over this interval.

### Steps:

1. **Calculate the values of \( p(x) \) from \( x = 0 \) to \( x = 5 \)**:
   We need to evaluate the function at integer points: \( x = 0, 1, 2, 3, 4, \) and \( 5 \).

2. **Find the average of these values**:
   The five-day moving average is calculated by averaging the function values over these points.

Let's compute each value of \( p(x) \) and then find the average: 

\[
\text{Five-day average} = \frac{p(0) + p(1) + p(2) + p(3) + p(4) + p(5)}{6}
\]

Let's calculate this step-by-step.The values of \( p(x) \) from \( x = 0 \) to \( x = 5 \) are approximately:

- \( p(0) = 35.00 \)
- \( p(1) = 54.70 \)
- \( p(2) = 62.08 \)
- \( p(3) = 66.94 \)
- \( p(4) = 70.20 \)
- \( p(5) = 72.22 \)

The five-day moving average value of the stock after the first 5 days is:

\[
\text{Moving Average} \approx 60.19
\]

### Questions for Further Exploration:

1. What is the price of the stock on the 10th day?
2. How would the moving average change if we considered a different interval, e.g., days 5 to 10?
3. How does the quadratic term \(-0.3x^2\) affect the price of the stock over time?
4. What is the maximum price of the stock over the first 25 days?
5. How does the five-day moving average compare to a ten-day moving average?

**Tip:** To gain insights into the stock’s trend, try calculating moving averages over different intervals and comparing them.

The function p(x) = -0.3x^2 + 20x^1/2 + 35 gives the price of a stock, in dollars, over the first 25 days of the month. a. Find the five-day moving average value of the stock after the first 5 days (that is, from x = 0 to x = 5 days). (Round your answer to the nearest cent.)

Calculate the five-day moving average of a stock price using the quadratic function p(x) = -0.3x^2 + 20x^(1/2) + 35. Follow a detailed step-by-step approach to evaluate the function from x = 0 to x = 5. Suitable for Grades 10-12.

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