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The given piecewise function \( f(x) \) is defined as:

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

### Step 1: Graphing the function

#### For \( 0 \leq x < 4 \), the function is \( f(x) = \sqrt{x} \):

- This is the graph of a square root function.
- At \( x = 0 \), \( f(x) = \sqrt{0} = 0 \).
- At \( x = 4 \), \( f(x) = \sqrt{4} = 2 \), but we stop just before \( x = 4 \).

So, this part of the graph is the square root curve, starting from (0, 0) and ending near (4, 2) but not including the point (4, 2).

#### For \( 4 \leq x \leq 10 \), the function is \( f(x) = (x - 6)^2 - 2 \):

- This is a quadratic function (a parabola) that opens upwards, shifted to the right by 6 units and down by 2 units.
- At \( x = 4 \), \( f(4) = (4 - 6)^2 - 2 = (-2)^2 - 2 = 4 - 2 = 2 \), so the parabola starts from \( (4, 2) \).
- At \( x = 6 \), \( f(6) = (6 - 6)^2 - 2 = 0 - 2 = -2 \), so the vertex is at \( (6, -2) \).
- At \( x = 10 \), \( f(10) = (10 - 6)^2 - 2 = 16 - 2 = 14 \), so the parabola extends to \( (10, 14) \).

### Step 2: Describing the Slope Changes

- **For \( 0 \leq x < 4 \)**: 
  - The slope of \( f(x) = \sqrt{x} \) is positive but decreasing as \( x \) increases. The function is concave down, meaning that as you move to the right, the slope of the tangent line decreases (it becomes flatter).
  - The slope approaches zero as \( x \) approaches 4 from the left, making the graph less steep.

- **At \( x = 4 \)**: 
  - The function switches from \( f(x) = \sqrt{x} \) to the quadratic function \( (x - 6)^2 - 2 \).
  - The value of the function at \( x = 4 \) is continuous (both pieces meet at \( (4, 2) \)), but the slopes are not continuous. The slope of \( \sqrt{x} \) approaches zero from the left, but the slope of the quadratic function is positive at \( x = 4 \).

- **For \( 4 \leq x \leq 10 \)**: 
  - The slope is initially negative, as the parabola is decreasing until the vertex at \( x = 6 \).
  - At \( x = 6 \), the slope is zero (the minimum point or vertex).
  - For \( x > 6 \), the slope becomes positive again, increasing as \( x \) moves toward 10. The parabola is concave up, so the slope increases rapidly as you move away from the vertex.

Would you like a detailed sketch of this graph or a more specific breakdown of the slope behavior?

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Here are five related questions to explore further:

1. What happens to the slope of a square root function as \( x \to 0^+ \)?
2. How do you find the derivative of a piecewise function at the boundary point (in this case, at \( x = 4 \))?
3. How do you interpret concavity in relation to the sign of the second derivative for each piece of the function?
4. What effect does shifting a quadratic function horizontally and vertically have on its graph and slope?
5. Can a piecewise function be differentiable at the boundary point? What conditions must be met?

**Tip:** When analyzing piecewise functions, always check the continuity and differentiability at the points where the pieces join.

Carefully graph f(x) = {√x ; 0 ≤ x < 4, (x - 6)^2 - 2 ; 4 ≤ x ≤ 10} below. Then, write a detailed slope statement that describes the changes in slope over the interval.

Learn how to graph a piecewise function consisting of a square root and a quadratic segment. This problem involves detailed slope analysis and descriptions of changes in slope, concavity, and continuity over the interval. Suitable for Grades 10-12.

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