Students match polynomial functions to their graphs by analyzing degree, leading coefficient, end behavior, and turning points.

Let's solve each problem step by step, analyzing the behavior of the graph of each polynomial function based on its degree, leading coefficient, end behavior, and number of turning points. Then we'll match it to the correct graph from the options provided.

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🔹 Problem 1: $ f(x) = x^3 - 4x^2 + 5 $

#### Step 1: Degree
- The highest power of $ x $ is 3 → Degree = 3

#### Step 2: Sign of Leading Coefficient
- The leading term is $ x^3 $, so the coefficient is +1 → Positive

#### Step 3: End Behavior
- For odd-degree polynomials:
- If leading coefficient is positive:
- As $ x \to -\infty $, $ f(x) \to -\infty $ → Down to the left
- As $ x \to \infty $, $ f(x) \to \infty $ → Up to the right

> ✔ So: Down to the left and Up to the right

#### Step 4: Turning Points
- A polynomial of degree $ n $ has at most $ n-1 $ turning points.
- Degree = 3 → at most $ 3 - 1 = 2 $ turning points

#### Summary for Problem 1:
| Feature | Answer |
|--------|--------|
| Degree | 3 |
| Sign of Leading Coefficient | Positive |
| End Behavior | Down to the left, Up to the right |
| Turning Points | At most 2 |

🔍 Now look at the graphs on the right. Which one matches?
- Odd degree (3) → ends go in opposite directions
- Positive leading coefficient → down-left, up-right
- At most 2 turning points → likely 1 or 2 peaks/troughs

✔ Match: Third graph from the top (the one with a "W" shape but only two turns, going down-left, up-right)

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🔹 Problem 2: $ f(x) = -x^5 + 4x^3 - 4x $

#### Step 1: Degree
- Highest power: $ x^5 $ → Degree = 5

#### Step 2: Sign of Leading Coefficient
- Leading term: $ -x^5 $ → coefficient is -1 → Negative

#### Step 3: End Behavior
- Odd degree, negative leading coefficient:
- As $ x \to -\infty $, $ f(x) \to \infty $ → Up to the left
- As $ x \to \infty $, $ f(x) \to -\infty $ → Down to the right

> ✔ So: Up to the left and Down to the right

#### Step 4: Turning Points
- Degree = 5 → at most $ 5 - 1 = 4 $ turning points

#### Summary for Problem 2:
| Feature | Answer |
|--------|--------|
| Degree | 5 |
| Sign of Leading Coefficient | Negative |
| End Behavior | Up to the left, Down to the right |
| Turning Points | At most 4 |

🔍 Look at the graphs:
- Must be odd degree, ends go opposite directions
- Negative leading coefficient → up-left, down-right
- Can have up to 4 turns

✔ Match: Fifth graph from the top (bottom one) — goes up-left, down-right, has multiple turns (likely 3–4)

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🔹 Problem 3: $ f(x) = -x^2(x - 2)(x + 2) $

First, simplify the expression:

$$
f(x) = -x^2(x - 2)(x + 2)
= -x^2(x^2 - 4)
= -x^4 + 4x^2
$$

So, $ f(x) = -x^4 + 4x^2 $

#### Step 1: Degree
- Highest power: $ x^4 $ → Degree = 4

#### Step 2: Sign of Leading Coefficient
- Leading term: $ -x^4 $ → coefficient is -1 → Negative

#### Step 3: End Behavior
- Even degree, negative leading coefficient:
- As $ x \to -\infty $, $ f(x) \to -\infty $ → Down to the left
- As $ x \to \infty $, $ f(x) \to -\infty $ → Down to the right

> ✔ So: Down to the left and Down to the right

#### Step 4: Turning Points
- Degree = 4 → at most $ 4 - 1 = 3 $ turning points

#### Summary for Problem 3:
| Feature | Answer |
|--------|--------|
| Degree | 4 |
| Sign of Leading Coefficient | Negative |
| End Behavior | Down to the left, Down to the right |
| Turning Points | At most 3 |

🔍 Graph must:
- Be even degree → same direction on both ends
- Negative leading coefficient → both ends go down
- Have up to 3 turning points

✔ Match: Second graph from the top — it starts high, dips, goes up, then down again, both ends go down

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✔ Final Matching:

| Function | Graph Match |
|--------|-------------|
| 1. $ f(x) = x^3 - 4x^2 + 5 $ | Third graph (from top) |
| 2. $ f(x) = -x^5 + 4x^3 - 4x $ | Fifth graph (bottom) |
| 3. $ f(x) = -x^2(x - 2)(x + 2) $ | Second graph (from top) |

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📝 Completed Worksheet Answers:

#### 1. $ f(x) = x^3 - 4x^2 + 5 $
- Degree: 3
- Sign of Leading Coefficient: Positive
- End Behavior: Down to the left and Up to the right
- Turning Points: at most 2

#### 2. $ f(x) = -x^5 + 4x^3 - 4x $
- Degree: 5
- Sign of Leading Coefficient: Negative
- End Behavior: Up to the left and Down to the right
- Turning Points: at most 4

#### 3. $ f(x) = -x^2(x - 2)(x + 2) $
- Degree: 4
- Sign of Leading Coefficient: Negative
- End Behavior: Down to the left and Down to the right
- Turning Points: at most 3

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💡 Key Concepts Recap:

- End Behavior depends on:
- Degree: even or odd
- Sign of leading coefficient: positive or negative
- Turning Points: at most $ n - 1 $ for degree $ n $
- Graph matching: use end behavior and number of turns

Let me know if you'd like the sketch drawn or need help plotting key points!