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The given polynomial function is \( f(x) = (x + 2)^2(x - 1) \).

Let's break down the different parts of the question to help you answer each one.

### (a) End Behavior
The function \( f(x) = (x + 2)^2(x - 1) \) is a cubic polynomial because when expanded, the highest degree term will be \( x^3 \). To determine the end behavior:
- Since the leading coefficient is positive (after expanding, the \(x^3\) term is positive), the graph will go **upwards as \( x \to +\infty \)** and **downwards as \( x \to -\infty \)**.
- So, the end behavior of the function is:
  - As \( x \to \infty \), \( f(x) \to \infty \).
  - As \( x \to -\infty \), \( f(x) \to -\infty \).

### (b) Zeros
To find the real zeros of the function \( f(x) = (x + 2)^2(x - 1) \):
- **Zero at \( x = -2 \)**: The factor \( (x + 2)^2 \) gives a zero at \( x = -2 \), but since it's squared, the graph touches the x-axis at this point but does not cross it.
- **Zero at \( x = 1 \)**: The factor \( (x - 1) \) gives a zero at \( x = 1 \), and since this factor is linear (degree 1), the graph crosses the x-axis here.

Thus:
- Zero where the graph crosses the x-axis: \( x = 1 \).
- Zero where the graph touches but does not cross the x-axis: \( x = -2 \).

### (c) Y-intercept
To find the y-intercept, substitute \( x = 0 \) into the function:
\[
f(0) = (0 + 2)^2(0 - 1) = 2^2(-1) = 4(-1) = -4.
\]
So, the y-intercept is \( -4 \).

### (d) Graphing
You can now plot the function using the following key points:
- **Zeros**: \( x = -2 \) (touches) and \( x = 1 \) (crosses).
- **Y-intercept**: \( (0, -4) \).
- **End behavior**: Downwards to the left, upwards to the right.

Would you like more details on any part of this explanation? Here are some further exploration questions:

1. What would happen to the graph if \( (x + 2)^2 \) were not squared but linear?
2. How would the graph's shape change if the leading coefficient were negative?
3. Can you explain why the graph touches but doesn't cross the x-axis at \( x = -2 \)?
4. How does the multiplicity of the zeros affect the graph's behavior near those points?
5. How can you determine the overall shape of a polynomial function from its degree?

**Tip:** Always look at the degree and leading coefficient of a polynomial to determine the end behavior of its graph!

Given the polynomial function f(x) = (x + 2)^2(x - 1), answer the questions regarding the graph of f, including end behavior, zeros, y-intercept, and graphing.

Learn how to graph the cubic polynomial function f(x) = (x + 2)^2(x - 1) by analyzing its end behavior, zeros, and y-intercept. This detailed explanation covers key concepts such as the multiplicity of zeros and how they affect the graph's shape.

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