PC 3 Unit Graphing Polynomials Worksheet: Identify end behavior of polynomial functions.

It looks like you've uploaded a worksheet titled "PC 3 Unit Graphing Polynomials Worksheet", which asks students to:

1. Graph the given polynomial functions.
2. Identify the end behavior of each function (i.e., what happens as $ x \to \infty $ and $ x \to -\infty $).
3. Fill in the blanks for:
- As $ x \to \infty $, $ f(x) \to $ ___
- As $ x \to -\infty $, $ f(x) \to $ ___

Let’s go through each problem one by one, analyze the polynomials, determine their end behaviors, and explain how to graph them.

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1. $ p(x) = x^4 - 4x^2 $

#### Step 1: Degree and Leading Coefficient
- Degree: 4 (even)
- Leading term: $ x^4 $
- Leading coefficient: +1 (positive)

#### End Behavior:
- For even degree with positive leading coefficient:
- As $ x \to \infty $, $ f(x) \to \infty $
- As $ x \to -\infty $, $ f(x) \to \infty $

✔ So:
> As $ x \to \infty $, $ f(x) \to \infty $
> As $ x \to -\infty $, $ f(x) \to \infty $

#### Graph Features:
- This is a quartic (degree 4) polynomial.
- Factor: $ p(x) = x^2(x^2 - 4) = x^2(x - 2)(x + 2) $
- Roots: $ x = 0 $ (double root), $ x = 2 $, $ x = -2 $
- The graph touches the x-axis at $ x = 0 $ (due to even multiplicity), crosses at $ x = \pm 2 $

You can sketch it as a "W"-like shape near the origin, but since leading term dominates, it rises on both ends.

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2. $ p(x) = (x + 1)^3 $

#### Step 1: Degree and Leading Coefficient
- Degree: 3 (odd)
- Leading term: $ x^3 $
- Leading coefficient: +1 (positive)

#### End Behavior:
- Odd degree, positive leading coefficient:
- As $ x \to \infty $, $ f(x) \to \infty $
- As $ x \to -\infty $, $ f(x) \to -\infty $

✔ So:
> As $ x \to \infty $, $ f(x) \to \infty $
> As $ x \to -\infty $, $ f(x) \to -\infty $

#### Graph Features:
- Cubic function shifted left by 1 unit.
- Root at $ x = -1 $ with multiplicity 3 → inflection point (crosses x-axis and flattens).
- Shape similar to $ y = x^3 $, but centered at $ x = -1 $

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3. $ p(x) = -x^3 - 2(x + 1) $

Let’s simplify:
$$
p(x) = -x^3 - 2x - 2
$$

#### Step 1: Degree and Leading Coefficient
- Degree: 3 (odd)
- Leading term: $ -x^3 $
- Leading coefficient: -1 (negative)

#### End Behavior:
- Odd degree, negative leading coefficient:
- As $ x \to \infty $, $ f(x) \to -\infty $
- As $ x \to -\infty $, $ f(x) \to \infty $

✔ So:
> As $ x \to \infty $, $ f(x) \to -\infty $
> As $ x \to -\infty $, $ f(x) \to \infty $

#### Graph Features:
- Cubic with negative leading coefficient → starts high on left, ends low on right.
- No obvious rational roots (check via Rational Root Theorem if needed), but you can plot a few points.

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4. $ p(x) = (x + 6)^2(x - 2) $

#### Step 1: Degree and Leading Coefficient
- Degree: 3 (since $ (x+6)^2 $ is degree 2, times $ (x-2) $ → total degree 3)
- Leading term: $ x^2 \cdot x = x^3 $
- Leading coefficient: +1 (positive)

#### End Behavior:
- Odd degree, positive leading coefficient:
- As $ x \to \infty $, $ f(x) \to \infty $
- As $ x \to -\infty $, $ f(x) \to -\infty $

✔ So:
> As $ x \to \infty $, $ f(x) \to \infty $
> As $ x \to -\infty $, $ f(x) \to -\infty $

#### Graph Features:
- Roots: $ x = -6 $ (multiplicity 2 → touches x-axis), $ x = 2 $ (crosses)
- Opens upward on the right, downward on the left.

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5. $ p(x) = (x - 1)^2(x + 4)(x - 2) $

#### Step 1: Degree and Leading Coefficient
- Degree: 4 (sum of exponents: 2 + 1 + 1 = 4)
- Leading term: $ x^2 \cdot x \cdot x = x^4 $
- Leading coefficient: +1 (positive)

#### End Behavior:
- Even degree, positive leading coefficient:
- As $ x \to \infty $, $ f(x) \to \infty $
- As $ x \to -\infty $, $ f(x) \to \infty $

✔ So:
> As $ x \to \infty $, $ f(x) \to \infty $
> As $ x \to -\infty $, $ f(x) \to \infty $

#### Graph Features:
- Roots: $ x = 1 $ (double root → touches x-axis), $ x = -4 $, $ x = 2 $
- Since degree 4, even, and positive lead coefficient → U-shaped overall
- Multiple turns due to 4 roots

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6. $ p(x) = (x - 3)^2(x - 4)^2(x + 1) $

#### Step 1: Degree and Leading Coefficient
- Degree: 5 (2 + 2 + 1 = 5)
- Leading term: $ x^2 \cdot x^2 \cdot x = x^5 $
- Leading coefficient: +1 (positive)

#### End Behavior:
- Odd degree, positive leading coefficient:
- As $ x \to \infty $, $ f(x) \to \infty $
- As $ x \to -\infty $, $ f(x) \to -\infty $

✔ So:
> As $ x \to \infty $, $ f(x) \to \infty $
> As $ x \to -\infty $, $ f(x) \to -\infty $

#### Graph Features:
- Roots: $ x = 3 $ (multiplicity 2 → touches), $ x = 4 $ (multiplicity 2 → touches), $ x = -1 $ (crosses)
- Since multiplicities are even at 3 and 4, the graph touches and turns around at those points.
- Overall shape: odd degree, positive — comes from bottom-left, goes to top-right.

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✔ Summary Table:

| Problem | Function | Degree | Lead Coeff | End Behavior |
|--------|---------|--------|------------|---------------|
| 1 | $ x^4 - 4x^2 $ | 4 (even) | + | $ \infty $, $ \infty $ |
| 2 | $ (x+1)^3 $ | 3 (odd) | + | $ \infty $, $ -\infty $ |
| 3 | $ -x^3 - 2x - 2 $ | 3 (odd) | - | $ -\infty $, $ \infty $ |
| 4 | $ (x+6)^2(x-2) $ | 3 (odd) | + | $ \infty $, $ -\infty $ |
| 5 | $ (x-1)^2(x+4)(x-2) $ | 4 (even) | + | $ \infty $, $ \infty $ |
| 6 | $ (x-3)^2(x-4)^2(x+1) $ | 5 (odd) | + | $ \infty $, $ -\infty $ |

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🔍 How to Graph:

For each:
1. Find roots (set $ p(x) = 0 $)
2. Determine multiplicity of each root:
- Even multiplicity → touches x-axis
- Odd multiplicity → crosses x-axis
3. Use end behavior to know direction at extremes
4. Plot key points and sketch smooth curve

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Final Answers to Fill In:

#### 1. $ p(x) = x^4 - 4x^2 $
- As $ x \to \infty $, $ f(x) \to \boxed{\infty} $
- As $ x \to -\infty $, $ f(x) \to \boxed{\infty} $

#### 2. $ p(x) = (x + 1)^3 $
- As $ x \to \infty $, $ f(x) \to \boxed{\infty} $
- As $ x \to -\infty $, $ f(x) \to \boxed{-\infty} $

#### 3. $ p(x) = -x^3 - 2(x + 1) $
- As $ x \to \infty $, $ f(x) \to \boxed{-\infty} $
- As $ x \to -\infty $, $ f(x) \to \boxed{\infty} $

#### 4. $ p(x) = (x + 6)^2(x - 2) $
- As $ x \to \infty $, $ f(x) \to \boxed{\infty} $
- As $ x \to -\infty $, $ f(x) \to \boxed{-\infty} $

#### 5. $ p(x) = (x - 1)^2(x + 4)(x - 2) $
- As $ x \to \infty $, $ f(x) \to \boxed{\infty} $
- As $ x \to -\infty $, $ f(x) \to \boxed{\infty} $

#### 6. $ p(x) = (x - 3)^2(x - 4)^2(x + 1) $
- As $ x \to \infty $, $ f(x) \to \boxed{\infty} $
- As $ x \to -\infty $, $ f(x) \to \boxed{-\infty} $

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If you'd like, I can help you sketch these graphs or provide step-by-step plotting instructions for any specific one!