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Let's go through the steps to solve this problem involving polynomial functions.

### 1. **Understanding the Problem**
   - **a)** You are asked to graph a 4th degree polynomial function with specific characteristics:
     - 3 turning points
     - 2 x-intercepts
     - Negative y-intercept
     - The function is non-negative (\( f(x) \geq 0 \)) between the x-intercepts

   - **b)** After graphing, you need to write a possible equation for the function.
   - **c)** Finally, you need to state the coordinates of at least four points that lie on your graph.

### 2. **Step-by-Step Solution**

#### a) **Graph the Polynomial Function**

Given the conditions:
- A 4th degree polynomial typically has up to 3 turning points.
- The polynomial has 2 x-intercepts, so we know the polynomial has at least two real roots.
- The polynomial touches the x-axis at two points, indicating the roots have even multiplicity (likely multiplicity 2 to maintain the 4th degree).
- Since the polynomial is non-negative between the x-intercepts, it suggests the function has a parabolic shape between these intercepts and doesn't cross the x-axis within that interval.
- The y-intercept is negative, so the function dips below the x-axis before rising.

Given these clues, one possible graph might look like this:
- **x-intercepts**: Let's assume they are at \( x = -2 \) and \( x = 2 \).
- **Turning points**: Likely occur near the x-intercepts and at a point below the x-axis where the y-intercept is negative.

You can draw the graph based on this description, ensuring that the polynomial is touching the x-axis at \( x = -2 \) and \( x = 2 \), has a negative y-intercept, and rises at the ends of the graph.

#### b) **Possible Equation of the Function**

A possible equation that fits these criteria is:
\[ f(x) = -a(x+2)^2(x-2)^2 \]
Here:
- \( -a \) ensures a negative y-intercept (since \( f(0) < 0 \)).
- \( (x+2)^2(x-2)^2 \) ensures the x-intercepts are at \( x = -2 \) and \( x = 2 \) and the function is non-negative between them.
- \( a \) is some positive constant, which would typically be chosen based on how steep or flat you want the curve to be.

#### c) **State Coordinates of Points on the Graph**

Based on the graph:
- **x-intercepts**: \( (-2, 0) \) and \( (2, 0) \)
- **y-intercept**: \( (0, -a(2)^4) = (0, -16a) \)
- Additional turning points can be calculated based on further analysis of the derivative of the polynomial, but typically would be symmetric around the y-axis given the form.

You can pick any suitable values of \( a \) to determine specific coordinates.

### Final Answers
1. **Graph**: Draw the described polynomial function with the given characteristics on the provided grid.
2. **Equation**: \( f(x) = -a(x+2)^2(x-2)^2 \)
3. **Coordinates**:
   - \( (-2, 0) \)
   - \( (2, 0) \)
   - \( (0, -16a) \)
   - One additional point from the curve, which can be calculated or estimated from the graph.

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**Do you need any further details or have any questions?**

### Follow-up Questions:
1. How do you determine the multiplicity of a root in a polynomial function?
2. What is the effect of the leading coefficient on the end behavior of a polynomial function?
3. How do you find turning points using the first derivative of a function?
4. What is the significance of the y-intercept in a polynomial function?
5. How do symmetry properties help in graphing polynomial functions?
6. What does it mean for a polynomial function to be non-negative over an interval?
7. How can you determine if a polynomial function will have a relative maximum or minimum at a turning point?
8. Why is it important to verify all conditions (like intercepts and turning points) when writing the equation for a polynomial function?

**Tip**: Always check the degree and leading coefficient when predicting the end behavior of a polynomial function.

Learn how to graph a 4th degree polynomial function with specific characteristics, including multiple turning points, x-intercepts, and a negative y-intercept. Includes step-by-step instructions and a possible equation for the function. Suitable for advanced high school students.

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