Solved Characteristics P... Name: Date: Per Math 3 Unit 3 | Chegg.com - Free Printable

Let's solve the worksheet step by step. This is a Math 3 Unit 3 Worksheet focused on End Behavior of Polynomial Functions.

We'll go through each problem, identify the degree, leading coefficient, and end behavior for each polynomial function, then analyze the graphs to determine their degree (odd/even) and leading coefficient (positive/negative).

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Part 1: Identify Leading Coefficient, Degree, and End Behavior

#### 1. $ f(x) = 5x^4 + 7x - 3 $

- Degree: The highest power of $ x $ is $ 4 $ → Degree = 4
- Leading Coefficient: The coefficient of $ x^4 $ is $ 5 $ → Leading Coeff = 5
- End Behavior:
- Even degree (4), positive leading coefficient → both ends go up
- As $ x \to \infty $, $ f(x) \to \infty $
- As $ x \to -\infty $, $ f(x) \to \infty $
- So, End Behavior: Up on both ends

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#### 2. $ y = -2x^4 - 3x + 4 $

- Degree: Highest power is $ 4 $ → Degree = 4
- Leading Coefficient: $ -2 $
- End Behavior: Even degree, negative leading coefficient → both ends go down
- As $ x \to \infty $, $ y \to -\infty $
- As $ x \to -\infty $, $ y \to -\infty $
- End Behavior: Down on both ends

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#### 3. $ g(x) = x^3 - 9x^2 + 2x + 6 $

- Degree: $ 3 $ → Degree = 3
- Leading Coefficient: $ 1 $
- End Behavior: Odd degree, positive leading coefficient
- As $ x \to \infty $, $ g(x) \to \infty $
- As $ x \to -\infty $, $ g(x) \to -\infty $
- End Behavior: Down on left, up on right

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#### 4. $ y = -7x^3 + 3x^2 + 12x - 1 $

- Degree: $ 3 $ → Degree = 3
- Leading Coefficient: $ -7 $
- End Behavior: Odd degree, negative leading coefficient
- As $ x \to \infty $, $ y \to -\infty $
- As $ x \to -\infty $, $ y \to \infty $
- End Behavior: Up on left, down on right

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#### 5. $ h(x) = -2x^5 + 5x^4 - 3x $

- Degree: $ 5 $ → Degree = 5
- Leading Coefficient: $ -2 $
- End Behavior: Odd degree, negative leading coefficient
- As $ x \to \infty $, $ h(x) \to -\infty $
- As $ x \to -\infty $, $ h(x) \to \infty $
- End Behavior: Up on left, down on right

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#### 6. $ g(x) = 8x^4 + 4x^7 + 7x^4 - 9x $

First, combine like terms:
- $ 4x^7 + (8x^4 + 7x^4) - 9x = 4x^7 + 15x^4 - 9x $
- Degree: $ 7 $ → Degree = 7
- Leading Coefficient: $ 4 $
- End Behavior: Odd degree, positive leading coefficient
- As $ x \to \infty $, $ g(x) \to \infty $
- As $ x \to -\infty $, $ g(x) \to -\infty $
- End Behavior: Down on left, up on right

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#### 7. $ f(x) = 4x^5 - 3x^4 + 2x^3 $

- Degree: $ 5 $ → Degree = 5
- Leading Coefficient: $ 4 $
- End Behavior: Odd degree, positive leading coefficient
- As $ x \to \infty $, $ f(x) \to \infty $
- As $ x \to -\infty $, $ f(x) \to -\infty $
- End Behavior: Down on left, up on right

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#### 8. $ y = -x^6 + x^3 - x^2 + 1 - 1 $

Simplify: $ -x^6 + x^3 - x^2 $
- Degree: $ 6 $ → Degree = 6
- Leading Coefficient: $ -1 $
- End Behavior: Even degree, negative leading coefficient → both ends go down
- As $ x \to \infty $, $ y \to -\infty $
- As $ x \to -\infty $, $ y \to -\infty $
- End Behavior: Down on both ends

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#### 9. $ h(x) = 3x^5 - 7x^4 - 2x^3 $

- Degree: $ 5 $ → Degree = 5
- Leading Coefficient: $ 3 $
- End Behavior: Odd degree, positive leading coefficient
- As $ x \to \infty $, $ h(x) \to \infty $
- As $ x \to -\infty $, $ h(x) \to -\infty $
- End Behavior: Down on left, up on right

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Part 2: Analyze Graphs

We now analyze the graphs to determine:
- Degree: Odd or Even?
- Leading Coefficient: Positive or Negative?
- Justify based on end behavior

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#### 10.

Graph: Left side goes down, right side goes up
→ End behavior: Down on left, up on right
This matches an odd degree with positive leading coefficient.

- deg: odd
- coeff: positive
- justify: As $ x \to \infty $, $ f(x) \to \infty $; as $ x \to -\infty $, $ f(x) \to -\infty $. This is typical of odd-degree polynomials with positive leading coefficients.

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#### 11.

Graph: Both ends go down
→ Even degree, negative leading coefficient

- deg: even
- coeff: negative
- justify: As $ x \to \pm\infty $, $ f(x) \to -\infty $. Even degree with negative leading coefficient causes both ends to go down.

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#### 12.

Graph: Left side goes up, right side goes down
→ Down on right, up on left → odd degree, negative leading coefficient

- deg: odd
- coeff: negative
- justify: As $ x \to \infty $, $ f(x) \to -\infty $; as $ x \to -\infty $, $ f(x) \to \infty $. This is characteristic of odd-degree polynomials with negative leading coefficients.

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#### 13.

Graph: Has multiple turns, but left side goes up, right side goes down
→ Similar to #12: odd degree, negative leading coefficient

- deg: odd
- coeff: negative
- justify: As $ x \to \infty $, $ f(x) \to -\infty $; as $ x \to -\infty $, $ f(x) \to \infty $. So odd degree, negative leading coefficient.

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#### 14.

Graph: Left side goes down, right side goes up
→ Odd degree, positive leading coefficient

- deg: odd
- coeff: positive
- justify: As $ x \to \infty $, $ f(x) \to \infty $; as $ x \to -\infty $, $ f(x) \to -\infty $. Odd degree, positive leading coefficient.

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#### 15.

Graph: Both ends go up
→ Even degree, positive leading coefficient

- deg: even
- coeff: positive
- justify: As $ x \to \pm\infty $, $ f(x) \to \infty $. This happens with even degree and positive leading coefficient.

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✔ Final Answers:

#### Part 1:

| Problem | Degree | Leading Coeff | End Behavior |
|--------|--------|----------------|--------------|
| 1 | 4 | 5 | Up on both ends |
| 2 | 4 | -2 | Down on both ends |
| 3 | 3 | 1 | Down on left, up on right |
| 4 | 3 | -7 | Up on left, down on right |
| 5 | 5 | -2 | Up on left, down on right |
| 6 | 7 | 4 | Down on left, up on right |
| 7 | 5 | 4 | Down on left, up on right |
| 8 | 6 | -1 | Down on both ends |
| 9 | 5 | 3 | Down on left, up on right |

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#### Part 2: Graph Analysis

| Problem | deg | coeff | justify |
|--------|-----|-------|---------|
| 10 | odd | positive | Ends go opposite directions: down-left, up-right → odd degree, positive lead coeff |
| 11 | even | negative | Both ends go down → even degree, negative lead coeff |
| 12 | odd | negative | Up on left, down on right → odd degree, negative lead coeff |
| 13 | odd | negative | Up on left, down on right → odd degree, negative lead coeff |
| 14 | odd | positive | Down on left, up on right → odd degree, positive lead coeff |
| 15 | even | positive | Both ends go up → even degree, positive lead coeff |

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