Trace or copy the graph of the given function *f*.

The *x* *y*-coordinate plane is given. The curve enters the window in the second quadrant just above the negative *x*-axis, goes up and right, changes direction in the second quadrant, goes down and right, passes through the origin, changes direction in the fourth quadrant, goes up and right, and exits the window in the fourth quadrant just below the positive *x*-axis.

Sketch the graph of *f* ′ on the same coordinate axes.

To trace or copy the graph of the function \( f(x) \), let's break down the key features of its description, and then determine the sketch for its derivative, \( f'(x) \), based on these characteristics.

### Key Observations About \( f(x) \):
- The curve enters the window in the second quadrant **just above the negative x-axis**.
- It goes **up and to the right**, indicating that the slope is positive here.
- It changes direction (reaches a maximum) in the second quadrant, which implies the slope becomes zero at this point.
- After the turning point, it **goes down and to the right**, meaning the slope becomes negative.
- The curve **passes through the origin**. At this point, the function has a value of zero but does not necessarily imply any specific behavior for the derivative at this moment.
- In the fourth quadrant, it changes direction again (reaches a minimum), implying that the slope becomes zero again at that point.
- Finally, the curve **goes up and to the right** and exits the window in the fourth quadrant **just below the positive x-axis**, meaning the slope is positive as it leaves the view.

### Sketch of \( f'(x) \):
To sketch the graph of the derivative \( f'(x) \), we focus on how the slope of the curve behaves at different parts:

1. **In the second quadrant (before the first turning point)**:
   - The function starts with a positive slope, so \( f'(x) \) starts positive.
   - As the curve approaches its maximum (where the slope becomes zero), the graph of \( f'(x) \) approaches zero from above.

2. **At the first turning point (in the second quadrant)**:
   - The slope of \( f(x) \) is zero here, so \( f'(x) = 0 \) at this point.
   
3. **Between the first turning point and the origin**:
   - The curve is going down and to the right, meaning the slope is negative, so \( f'(x) \) will be below the x-axis.
   - As it approaches the origin, the slope remains negative, though we don’t have enough information to say if \( f'(x) \) is exactly zero at the origin.

4. **In the fourth quadrant (after the origin, before the second turning point)**:
   - The function hits a minimum, so the slope decreases until it becomes zero again.
   - Therefore, \( f'(x) \) will increase and approach zero as the curve reaches its minimum.

5. **After the second turning point**:
   - The curve rises again, meaning the slope becomes positive, and thus, \( f'(x) \) will be above the x-axis.

### Summary of the Sketch for \( f'(x) \):
- The derivative starts **positive**, decreases to **zero** at the first turning point, then goes **negative** as the curve descends.
- It reaches another **zero** at the second turning point, then becomes **positive** again as the function ascends out of the window.

If you need help with further details on the sketch, feel free to ask!

Here are five related questions for deeper understanding:
1. How would the graph of \( f'(x) \) change if there were more turning points in \( f(x) \)?
2. What does the second derivative, \( f''(x) \), tell us about the concavity of \( f(x) \)?
3. How does the behavior of \( f'(x) \) at the origin affect the nature of the graph of \( f(x) \)?
4. What types of functions commonly produce a graph with two turning points like the one described?
5. How can critical points help in sketching the graph of a function’s derivative?

**Tip**: When sketching \( f'(x) \), always pay attention to where \( f(x) \) reaches a maximum or minimum—those points are key locations where the derivative will be zero!

Trace or copy the graph of the given function f. The x y-coordinate plane is given. The curve enters the window in the second quadrant just above the negative x-axis, goes up and right, changes direction in the second quadrant, goes down and right, passes through the origin, changes direction in the fourth quadrant, goes up and right, and exits the window in the fourth quadrant just below the positive x-axis. Sketch the graph of f' on the same coordinate axes.

Learn how to trace the graph of a function and sketch its derivative. This problem explains the behavior of f(x) and its derivative f'(x), focusing on turning points, slopes, and curve analysis. Suitable for high school students in Grades 10-12.

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