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 using its degree, leading coefficient, end behavior, and number of turning points. Then we’ll match it to the correct sketch from the given options.

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🔹 Problem 1:

Function: $ 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
- Leading term is $ x^3 $ → coefficient is +1 → Positive

#### Step 3: End Behavior
- For odd degree (3) with positive leading coefficient:
- 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 #1:
| Feature | Answer |
|--------|--------|
| Degree | 3 |
| Sign of Leading Coefficient | Positive |
| End Behavior | Down to the left and Up to the right |
| Turning Points | at most 2 |

✔ Now look at the sketches on the right:
- The third sketch (from top) shows a cubic-like shape: down on the left, up on the right, with two turns → matches this function.

➡️ Match: Sketch #3

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🔹 Problem 2:

Function: $ 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 (5), 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 #2:
| Feature | Answer |
|--------|--------|
| Degree | 5 |
| Sign of Leading Coefficient | Negative |
| End Behavior | Up to the left and Down to the right |
| Turning Points | at most 4 |

✔ Look at the sketches:
- The fifth sketch (bottom one) shows a degree 5 behavior: starts high on the left, ends low on the right, with multiple turns (likely 4 or fewer).
- It goes up to the left, down to the right, and has several wiggles → matches this.

➡️ Match: Sketch #5

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🔹 Problem 3:

Function: $ f(x) = -x^2(x - 2)(x + 2) $

First, expand to identify the degree and leading coefficient.

But we can do it without expanding:

#### Step 1: Degree
- Factors: $ x^2 $, $ (x - 2) $, $ (x + 2) $
- Total degree = $ 2 + 1 + 1 = 4 $

#### Step 2: Sign of Leading Coefficient
- Multiply the leading terms:
- $ -x^2 \cdot x \cdot x = -x^4 $
- So leading term is $ -x^4 $ → coefficient is negative

#### Step 3: End Behavior
- Even degree (4), negative leading coefficient:
- As $ x \to -\infty $, $ f(x) \to -\infty $
- As $ x \to \infty $, $ f(x) \to -\infty $
- 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 #3:
| Feature | Answer |
|--------|--------|
| Degree | 4 |
| Sign of Leading Coefficient | Negative |
| End Behavior | Down to the left and Down to the right |
| Turning Points | at most 3 |

✔ Now check the sketches:
- The second sketch (from top): starts down on the left, ends down on the right, has two "humps" → likely a quartic with negative leading coefficient.
- It dips down on both ends and has two turning points (which is ≤ 3) → valid.

→ This matches the end behavior: down on both ends

➡️ Match: Sketch #2

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✔ Final 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
- Match: Sketch #3

#### 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
- Match: Sketch #5

#### 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
- Match: Sketch #2

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📌 Bonus: How to Remember End Behavior

| Degree | Leading Coeff | End Behavior |
|-------|----------------|--------------|
| Even | Positive | ↑ both ends |
| Even | Negative | ↓ both ends |
| Odd | Positive | ↓ left, ↑ right |
| Odd | Negative | ↑ left, ↓ right |

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✔ Final Matching:

- #1 → Sketch #3
- #2 → Sketch #5
- #3 → Sketch #2

You can now fill in the table accordingly and sketch the graphs based on the analysis above.