https://file.aiquickdraw.com/mathbot/file/1728488698531bdk8q1xo.jpg

Let's go over the questions in the image and the corresponding graphs.

### Problem 3:
The question asks which polynomial function touches the x-axis at (-4, 0). A polynomial function touches the x-axis at a point if the factor corresponding to that point has an even multiplicity (like a squared factor).

- Option D: \( f(x) = (x - 4)^2 \) has a squared factor at \( x = 4 \), so the graph touches the x-axis at \( (4,0) \).
- However, none of the options specifically mention \( x = -4 \). Thus, this question might have an issue with its options.

### Questions 4-10:
You are asked to match the graphs (A, B, C, D) with the following descriptions:

#### 4. **Graph of a function with an even degree and a negative leading coefficient**:
   - A polynomial with an even degree and a negative leading coefficient will have both ends of the graph pointing down.
   - **Answer: Graph A** fits this description as both ends of the graph are going down.

#### 5. **Graph of a function with an even degree and a positive leading coefficient**:
   - A polynomial with an even degree and a positive leading coefficient will have both ends pointing upwards.
   - **Answer: Graph B** fits this description.

#### 6. **Graph of a function with an odd degree and a positive leading coefficient**:
   - A polynomial with an odd degree and a positive leading coefficient will have the left end going down and the right end going up.
   - **Answer: Graph D** fits this description.

#### 7. **Graph of a function with an odd degree and a negative leading coefficient**:
   - A polynomial with an odd degree and a negative leading coefficient will have the left end going up and the right end going down.
   - **Answer: Graph C** fits this description.

Do you want more details on any part of this solution?

Here are 5 related questions to expand your understanding:
1. How do the leading coefficient and degree affect the shape of polynomial graphs?
2. What is the difference between an odd-degree and even-degree polynomial in terms of end behavior?
3. How does the multiplicity of a root affect the graph at that root?
4. How can you determine the number of turning points in a polynomial graph?
5. What does it mean for a polynomial function to cross or touch the x-axis?

**Tip**: Always check the degree and sign of the leading coefficient to quickly predict the end behavior of polynomial graphs.

What is the graph of the function with an even degree and a negative leading coefficient? What is the graph of the function with an even degree and a positive leading coefficient? What is the graph of the function with an odd degree and a positive leading coefficient? What is the graph of the function with an odd degree and a negative leading coefficient?

Understand how to identify polynomial function graphs by analyzing their degree and leading coefficients. Learn to match the graph to the characteristics of even/odd degrees and positive/negative leading coefficients. Suitable for high school students studying polynomial behavior.

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