https://file.aiquickdraw.com/mathbot/file/1726175303861vg2v12lr.jpg

The problem in the image asks you to plot the graph of a piecewise function \( f(x) \). The function is defined as follows:

\[
f(x) = 
\begin{cases} 
-2x, & \text{if } x < 0, \\
2\sqrt{x}, & \text{if } 0 \leq x \leq 4, \\
x^2 - 4x + 4, & \text{if } x > 4.
\end{cases}
\]

### Explanation of each piece:
1. **For \( x < 0 \):**
   The function is linear, \( f(x) = -2x \). It is a straight line with slope \(-2\), passing through the origin.

2. **For \( 0 \leq x \leq 4 \):**
   The function is \( f(x) = 2\sqrt{x} \). This is a square root function scaled by 2. It is defined for non-negative values of \( x \) and will pass through points like \( (0, 0) \), \( (1, 2) \), \( (4, 4) \).

3. **For \( x > 4 \):**
   The function is quadratic, \( f(x) = x^2 - 4x + 4 \). This quadratic function can be rewritten as \( (x - 2)^2 \), which represents a parabola with a vertex at \( (2, 0) \). However, it is only valid for \( x > 4 \).

### Steps to Graph:
1. For \( x < 0 \), plot the line \( f(x) = -2x \).
2. For \( 0 \leq x \leq 4 \), plot the curve \( f(x) = 2\sqrt{x} \), connecting the points smoothly between \( x = 0 \) and \( x = 4 \).
3. For \( x > 4 \), plot the parabola \( f(x) = (x - 2)^2 \).

Would you like me to provide a detailed graph of this function?

Here are some questions you may consider after understanding the function:

1. What is the domain of each part of the piecewise function?
2. How does the function behave as \( x \to -\infty \)?
3. Can you find the points of intersection between the different parts of the function?
4. What is the derivative of each piece of the function?
5. How would you describe the continuity of the function at \( x = 0 \) and \( x = 4 \)?

**Tip:** When dealing with piecewise functions, always ensure the transition points (like \( x = 0 \) and \( x = 4 \)) are evaluated carefully for continuity and smoothness.

Побудуйте графік функції f(x) = { -2x, якщо x < 0; 2√x, якщо 0 ≤ x ≤ 4; x^2 - 4x + 4, якщо x > 4. }

Learn how to graph a piecewise function that includes a linear function (-2x for x < 0), a square root function (2√x for 0 ≤ x ≤ 4), and a quadratic function (x^2 - 4x + 4 for x > 4). This problem is suitable for students in Grades 10-12.

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