https://file.aiquickdraw.com/mathbot/file/172799156482282boapvo.jpg

Let's break down the function provided: 

\[ f(x) = -2x(x - 3)^2(x + 4) \]

### 3. Name the real zeros of \( f(x) \).
To find the real zeros, set \( f(x) = 0 \):

\[ -2x(x - 3)^2(x + 4) = 0 \]

Solve for \(x\) by setting each factor to zero:
- \( -2x = 0 \implies x = 0 \)
- \( (x - 3)^2 = 0 \implies x = 3 \)
- \( x + 4 = 0 \implies x = -4 \)

So, the real zeros are: **0, 3, and -4**.

### 4. State the end behavior of \( f(x) \).
The end behavior of a polynomial is determined by the leading term, which comes from multiplying the highest degree terms in each factor. Here:

\[
f(x) = -2x(x-3)^2(x+4)
\]

The highest degree term is found by multiplying:

\[
-2x \times x^2 \times x = -2x^4
\]

Since the degree is 4 (an even number) and the leading coefficient is negative, the end behavior is:
- As \( x \to +\infty \), \( f(x) \to -\infty \)
- As \( x \to -\infty \), \( f(x) \to -\infty \)

### 5. Use the coordinate plane to sketch \( f(x) \).
Let's sketch the function considering the following:
- The zeros are at \(x = -4\), \(x = 0\), and \(x = 3\).
- At \(x = 3\), the factor \((x - 3)^2\) indicates a double root, so the graph will "bounce" off the x-axis at this point.
- Since the end behavior as \(x \to \infty\) and \(x \to -\infty\) is \(-\infty\), the graph falls on both sides.

Would you like a visual sketch of the graph, or more detailed steps to plot it?

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Here are 5 related questions:
1. How does the multiplicity of a root affect the graph's shape near the zero?
2. How do you determine the end behavior of a polynomial?
3. What is the effect of a negative leading coefficient on the graph?
4. How would you calculate the y-intercept of \(f(x)\)?
5. How does changing the coefficient of the polynomial affect the graph?

**Tip:** Always analyze the degree and leading coefficient to determine the end behavior of polynomials.

Consider the function f(x) = -2x(x - 3)^2(x + 4), to answer the following: 
3. Name real zeros of f(x). 
4. State the end behavior of f(x). 
5. Use the coordinate plane to sketch f(x). Make sure to label the real zeros.

Learn how to solve the polynomial function f(x) = -2x(x - 3)^2(x + 4) by finding its real zeros, analyzing its end behavior, and sketching its graph. This exercise involves key concepts such as polynomial factorization and end behavior analysis. Suitable for high school students in Grades 9-12.

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