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:

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: $ 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, 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 | Value |
|--------|-------|
| 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:

- Graph 1 (top): Starts high on the left, goes down, up, down → ends low → Not matching (should go up on right)
- Graph 2: U-shaped (quadratic-like), but degree 3 → not possible
- Graph 3: Starts low on left, goes up, then down, then up → ends high → matches down-left, up-right and has 2 turns → ✔ This is the one!

So, Graph 3 matches Problem 1.

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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, Down to the right

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

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

Now check graphs:

- Graph 1: Ends up on both sides → no
- Graph 2: U-shape → degree 2 → no
- Graph 3: Already used → down-left, up-right → no
- Graph 4: Starts high on left, goes down, up, down → ends low → up-left, down-right → ✔ matches!
- Graph 5: Starts low, goes up → ends high → opposite end behavior → no

So, Graph 4 matches Problem 2.

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🔹 Problem 3:

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

First, simplify or expand to find degree and leading coefficient.

#### Step 1: Expand (optional):
$ f(x) = -x^2(x - 2)(x + 2) $

Note: $ (x - 2)(x + 2) = x^2 - 4 $

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

#### 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 (4), 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, Down to the right

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

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

Now check remaining graphs:

- Graph 1: Starts high on left, ends low on right → doesn't match (both ends should be down)
- Graph 2: U-shaped → opens up → wrong sign
- Graph 3: Used
- Graph 4: Used
- Graph 5: Starts low, goes up, then down → ends low → both ends down → ✔ matches!

Also, Graph 5 has 3 turning points (peaks and valleys): one local max, two minima? Let’s see: it dips, rises, dips again → yes, 3 turning points → matches.

So, Graph 5 matches Problem 3.

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✔ Final Answers:

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

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

#### Problem 3: $ f(x) = -x^2(x - 2)(x + 2) $
- Degree: 4
- Sign of Leading Coefficient: Negative
- End Behavior: Down to the left, Down to the right
- Turning Points: At most 3
- Matched Graph: Graph 5

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📌 Matching Summary:

| Function | Matches Graph |
|---------|---------------|
| 1. $ x^3 - 4x^2 + 5 $ | Graph 3 |
| 2. $ -x^5 + 4x^3 - 4x $ | Graph 4 |
| 3. $ -x^2(x-2)(x+2) $ | Graph 5 |

You can now fill in the tables accordingly and sketch the graphs as per the matched ones.

Let me know if you'd like a hand-drawn sketch explanation or how to plot key points!