For 𝑓(𝑥) = (𝑥 + 2)2(𝑥 − 1)(𝑥 − 2), answer the following.
a) Show the leading coefficient and state the end behavior of the graph. You may use
arrows.
Leading Coefficient ___________ End Behavior ______________
b) Find the x-intercepts and their multiplicity and state whether the graph crosses or
turns at that intercept.
Intercept: _____________ Multiplicity: ___________ Cross or Touch: ___________
Intercept: _____________ Multiplicity: ___________ Cross or Touch: ___________
Intercept: _____________ Multiplicity: ___________ Cross or Touch: ___________
c) Find the y-intercept.

Let's break this problem down step by step.

### Given Function:
\[ f(x) = (x + 2)^2(x - 1)(x - 2) \]

### Part (a): Leading Coefficient and End Behavior

1. **Leading Coefficient:**
   To determine the leading coefficient, we need to examine the degree and leading term of the polynomial.

   - The factor \((x + 2)^2\) contributes a degree of 2.
   - The factor \((x - 1)\) contributes a degree of 1.
   - The factor \((x - 2)\) contributes a degree of 1.

   Therefore, the degree of the polynomial is \( 2 + 1 + 1 = 4 \). This is a degree 4 polynomial, and the highest degree term is obtained by multiplying the highest powers of \(x\) in each factor.

   - From \((x + 2)^2\), the highest power term is \(x^2\).
   - From \((x - 1)\), the highest power term is \(x\).
   - From \((x - 2)\), the highest power term is \(x\).

   Multiplying these together gives \( x^2 \cdot x \cdot x = x^4 \).

   The leading term is \(x^4\), and the leading coefficient is **1**.

2. **End Behavior:**
   For a polynomial of degree 4 (even degree) with a positive leading coefficient, the end behavior is:
   \[
   \text{As } x \to +\infty, \ f(x) \to +\infty \quad (\text{upward})
   \]
   \[
   \text{As } x \to -\infty, \ f(x) \to +\infty \quad (\text{upward})
   \]

   Thus, the end behavior is:
   \[
   \boxed{ \uparrow \quad \text{and} \quad \uparrow }
   \]

### Part (b): X-Intercepts, Multiplicity, and Crossing/Touching Behavior

1. **X-intercepts:**
   To find the x-intercepts, set \(f(x) = 0\).

   \[
   (x + 2)^2(x - 1)(x - 2) = 0
   \]

   This equation is 0 when any of the factors are 0.

   - \(x + 2 = 0 \Rightarrow x = -2\)
   - \(x - 1 = 0 \Rightarrow x = 1\)
   - \(x - 2 = 0 \Rightarrow x = 2\)

   Therefore, the x-intercepts are \(x = -2\), \(x = 1\), and \(x = 2\).

2. **Multiplicity and Crossing/Touching Behavior:**
   - For \(x = -2\), the factor \((x + 2)\) has multiplicity 2 (because it’s squared). When the multiplicity is even, the graph **touches** the x-axis and turns around at this point.
   
   - For \(x = 1\), the factor \((x - 1)\) has multiplicity 1 (odd). When the multiplicity is odd, the graph **crosses** the x-axis at this point.

   - For \(x = 2\), the factor \((x - 2)\) also has multiplicity 1 (odd). Therefore, the graph **crosses** the x-axis at this point as well.

Thus, the details are:
- Intercept: \(x = -2\), Multiplicity: 2, Cross or Touch: **Touch**
- Intercept: \(x = 1\), Multiplicity: 1, Cross or Touch: **Cross**
- Intercept: \(x = 2\), Multiplicity: 1, Cross or Touch: **Cross**

### Part (c): Y-Intercept

The y-intercept occurs when \(x = 0\).

Substitute \(x = 0\) into the function:
\[
f(0) = (0 + 2)^2(0 - 1)(0 - 2) = 2^2(-1)(-2) = 4 \cdot (-1) \cdot (-2) = 8
\]

Thus, the y-intercept is \( (0, 8) \).

---

### Summary:

(a) 
- **Leading Coefficient:** 1
- **End Behavior:** \(\uparrow \quad \text{and} \quad \uparrow\)

(b) 
- Intercept: \(x = -2\), Multiplicity: 2, Touch
- Intercept: \(x = 1\), Multiplicity: 1, Cross
- Intercept: \(x = 2\), Multiplicity: 1, Cross

(c) 
- **Y-Intercept:** \( (0, 8) \)

---

Would you like to dive into any specific detail, or have more questions? Here are 5 related questions:

1. What happens to the end behavior if the leading coefficient is negative?
2. Can you explain how to find the degree of a polynomial in more detail?
3. How does the multiplicity of an intercept affect the graph's behavior?
4. What is the difference between even and odd degree polynomials in terms of end behavior?
5. How can we determine the sign of the y-intercept without fully solving it?

**Tip:** The degree and leading coefficient of a polynomial dictate the graph's overall shape and direction.

For f(x) = (x + 2)^2(x - 1)(x - 2), answer the following.
a) Show the leading coefficient and state the end behavior of the graph.
   Leading Coefficient ___________ End Behavior ______________
b) Find the x-intercepts and their multiplicity and state whether the graph crosses or turns at that intercept.
   Intercept: _____________ Multiplicity: ___________ Cross or Touch: ___________
   Intercept: _____________ Multiplicity: ___________ Cross or Touch: ___________
   Intercept: _____________ Multiplicity: ___________ Cross or Touch: ___________
c) Find the y-intercept.

Explore how to analyze a polynomial function f(x) = (x + 2)^2(x - 1)(x - 2), including identifying the leading coefficient, determining end behavior, finding x-intercepts, and calculating the y-intercept. Ideal for students in Grades 10-12.

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