Behavior of the Graph of a Polynomial Function worksheet | Live ... - Free Printable

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
- 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 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:
| 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 graphs on the right:

- Graph #3 (third from top): starts down on the left, goes up on the right, has two turns → matches this description.

✔ Match: Graph #3

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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 (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:
| 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 |

Now check the graphs:

- Graph #5 (bottom one): starts up on the left, ends down on the right, has multiple turns (likely 4 or more) → matches this.

✔ Match: Graph #5

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

First, simplify the expression to find degree and leading coefficient.

#### Step 1: Expand or identify degree
- $ 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 $

→ Degree = 4 (highest power)

#### 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 and Down to the right

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

#### ✔ Summary:
| 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 graphs:

- Graph #1 (top one): both ends go down, has three turns → matches this.

✔ Match: Graph #1

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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 and Up to the right
- Turning Points: at most 2
- Sketch: Match with Graph #3

#### Problem 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
- Sketch: Match with Graph #5

#### Problem 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
- Sketch: Match with Graph #1

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✔ Matching Summary:

| Function | Graph Match |
|--------|------------|
| 1. $ f(x) = x^3 - 4x^2 + 5 $ | Graph #3 |
| 2. $ f(x) = -x^5 + 4x^3 - 4x $ | Graph #5 |
| 3. $ f(x) = -x^2(x - 2)(x + 2) $ | Graph #1 |

> ⚠️ Note: The graphs are labeled vertically on the right side. From top to bottom:
> - Graph 1: Top one (down-left, down-right)
> - Graph 2: Second (up-left, up-right)
> - Graph 3: Third (down-left, up-right)
> - Graph 4: Fourth (up-left, down-right)
> - Graph 5: Bottom one (up-left, down-right, many turns)

So your sketch should be matched accordingly.

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✔ How to Fill in the Worksheet:

#### 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
- Sketch: Draw a cubic shape going down-left, up-right, with two turns.

#### 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
- Sketch: Quintic shape, starts high on left, ends low on right, multiple turns.

#### 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
- Sketch: Quartic shape, opens downward, symmetric-like but not symmetric, three turns.

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🎯 Key Concepts Recap:

- End Behavior depends on:
- Degree: even vs odd
- Sign of leading coefficient: positive or negative
- Turning Points: at most $ n - 1 $ for degree $ n $
- Use these to eliminate incorrect graphs quickly.

Let me know if you want me to help draw the sketches!