https://file.aiquickdraw.com/mathbot/file/1726078828943il6dxdz1.jpg

The graph above shows the piecewise function \( f(x) \) defined as:

\[
f(x) = \begin{cases} 
\sqrt{x}, & 0 \leq x < 4 \\
(x - 6)^2 - 2, & 4 \leq x \leq 10 
\end{cases}
\]

### Slope Analysis:

1. **For \( 0 \leq x < 4 \)** (\( f(x) = \sqrt{x} \)):
   - The slope, \( f'(x) = \frac{1}{2\sqrt{x}} \), is positive and decreasing as \( x \) increases. The slope starts off steep near \( x = 0 \) and gradually flattens as it approaches \( x = 4 \). This means the curve is concave down, slowing its rise.

2. **At \( x = 4 \)**:
   - The function has a sharp transition at \( x = 4 \), marked by a discontinuity in the derivative (slope) because \( \sqrt{x} \) and \( (x - 6)^2 - 2 \) meet with different slopes at this point.

3. **For \( 4 \leq x \leq 10 \)** (\( f(x) = (x - 6)^2 - 2 \)):
   - The slope, \( f'(x) = 2(x - 6) \), starts negative (since \( x - 6 < 0 \) for \( x < 6 \)), indicating the graph is decreasing from \( x = 4 \) to \( x = 6 \). 
   - After \( x = 6 \), the slope becomes positive, and the graph begins to increase. This indicates the parabola is concave up in this interval, with the minimum at \( x = 6 \).

Do you have any questions or need further clarification?

Here are five related questions for you to explore:

1. How would the graph change if the interval for \( \sqrt{x} \) extended beyond \( x = 4 \)?
2. Can you find the exact points where the slope of the function is zero?
3. What happens to the function if we shift the parabola horizontally by adding or subtracting a constant to \( x \)?
4. How does the second derivative help in understanding the concavity of each piece of the function?
5. How can we extend this piecewise function to make the transition at \( x = 4 \) smoother?

**Tip**: When graphing piecewise functions, pay close attention to how the slopes (first derivatives) behave at the boundaries to identify any discontinuities in the slope.

Carefully graph f(x) = sqrt(x) for 0 <= x < 4 and (x - 6)^2 - 2 for 4 <= x <= 10. Then, write a detailed slope statement that describes the changes in slope over the interval.

Learn how to graph the piecewise function f(x) with a square root and parabola over two intervals. This problem involves slope analysis and derivatives, suitable for high school students in Grades 10-12.

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