https://file.aiquickdraw.com/mathbot/file/1729717995555yxbxnm70.jpg

The graph you uploaded shows a polynomial curve, and there is a set of multiple-choice statements about its properties.

Let’s analyze the graph and the given statements:

1. **Degree of the polynomial (odd or even):** 
   - The ends of the polynomial graph are both pointing upwards. This suggests that the degree of the polynomial is even (because even-degree polynomials have their ends going in the same direction, either both up or both down).
   - **Correct statement:** "It has an even degree."

2. **Leading coefficient (positive or negative):**
   - Since the polynomial starts and ends by rising upward, the leading coefficient is positive. If it were negative, the ends would point downward.
   - **Incorrect statement:** "The leading coefficient is negative."
   - **Correct statement:** "The leading coefficient is positive."

3. **Turning points:**
   - The graph has 3 turning points (local maximum and minimum). A turning point occurs where the direction of the graph changes. Since the graph turns three times, it does **not** have 4 turning points.
   - **Incorrect statement:** "It has 4 turning points."

4. **Relative extrema (minima or maxima):**
   - The graph clearly has relative maxima and minima (peaks and valleys). So, the statement that there are no relative minimums or maximums is false.
   - **Incorrect statement:** "There are no relative minimums or maximums."

### Conclusion: 
The correct statements based on the graph are:
- "It has an even degree."
- "The leading coefficient is positive."

Would you like further details or have any additional questions?

Here are five related questions to consider:
1. What is the general form of an even-degree polynomial with a positive leading coefficient?
2. How does the number of turning points relate to the degree of a polynomial?
3. What does the behavior of a polynomial's ends tell us about its leading term?
4. How can we determine if a graph has relative extrema from the function's first and second derivatives?
5. What impact does a negative leading coefficient have on an even-degree polynomial?

**Tip:** The degree of a polynomial gives a clue about its behavior. For example, the number of turning points is always less than or equal to the degree of the polynomial.

Which of the following statements are true about the polynomial shown?

Learn how to determine the degree, leading coefficient, and turning points of a polynomial graph. This problem explores polynomial behavior and extrema, ideal for students in Grades 10-12.

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