Polynomial End Behavior Worksheet | PDF - Free Printable

Let's solve each problem step by step using the Polynomial End Behavior Worksheet. For each polynomial function, we will analyze:

- A) Degree
- B) Even or odd degree
- C) Leading coefficient
- D) Sign of leading coefficient (positive/negative)
- E) End behavior in words
- F) End behavior using symbols (e.g., $ x \to \infty $, $ f(x) \to ? $)
- G) General shape (we'll describe it since no graphing calculator is used here)

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1) $ f(x) = -x^5 + 3x^3 - 2x - 1 $

- A) Degree: 5
- B) Odd
- C) Leading coefficient: $-1$
- D) Negative
- E) As $ x \to \infty $, $ f(x) \to -\infty $; as $ x \to -\infty $, $ f(x) \to \infty $
- F) $ \lim_{x \to \infty} f(x) = -\infty $, $ \lim_{x \to -\infty} f(x) = \infty $
- G) Falls to the right, rises to the left → "down on the right, up on the left"

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2) $ f(x) = -x^3 + 3x^3 - 3x $

Wait! Let's simplify this first:
$$
f(x) = (-x^3 + 3x^3) - 3x = 2x^3 - 3x
$$

Now analyze:

- A) Degree: 3
- B) Odd
- C) Leading coefficient: $2$
- D) Positive
- E) As $ x \to \infty $, $ f(x) \to \infty $; as $ x \to -\infty $, $ f(x) \to -\infty $
- F) $ \lim_{x \to \infty} f(x) = \infty $, $ \lim_{x \to -\infty} f(x) = -\infty $
- G) Rises to the right, falls to the left → "up on the right, down on the left"

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

- A) Degree: 2
- B) Even
- C) Leading coefficient: $1$
- D) Positive
- E) As $ x \to \infty $, $ f(x) \to \infty $; as $ x \to -\infty $, $ f(x) \to \infty $
- F) $ \lim_{x \to \infty} f(x) = \infty $, $ \lim_{x \to -\infty} f(x) = \infty $
- G) U-shaped parabola (opens upward)

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4) $ f(x) = x^3 - 2x^2 + 3 $

- A) Degree: 3
- B) Odd
- C) Leading coefficient: $1$
- D) Positive
- E) As $ x \to \infty $, $ f(x) \to \infty $; as $ x \to -\infty $, $ f(x) \to -\infty $
- F) $ \lim_{x \to \infty} f(x) = \infty $, $ \lim_{x \to -\infty} f(x) = -\infty $
- G) Rises to the right, falls to the left → "up on the right, down on the left"

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5) $ f(x) = -2x^2 + 16x - 26 $

- A) Degree: 2
- B) Even
- C) Leading coefficient: $-2$
- D) Negative
- E) As $ x \to \infty $, $ f(x) \to -\infty $; as $ x \to -\infty $, $ f(x) \to -\infty $
- F) $ \lim_{x \to \infty} f(x) = -\infty $, $ \lim_{x \to -\infty} f(x) = -\infty $
- G) Inverted U-shape (opens downward)

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6) $ f(x) = -x^3 + 2x^2 - 2 $

- A) Degree: 3
- B) Odd
- C) Leading coefficient: $-1$
- D) Negative
- E) As $ x \to \infty $, $ f(x) \to -\infty $; as $ x \to -\infty $, $ f(x) \to \infty $
- F) $ \lim_{x \to \infty} f(x) = -\infty $, $ \lim_{x \to -\infty} f(x) = \infty $
- G) Falls to the right, rises to the left → "down on the right, up on the left"

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

- A) Degree: 5
- B) Odd
- C) Leading coefficient: $-1$
- D) Negative
- E) As $ x \to \infty $, $ f(x) \to -\infty $; as $ x \to -\infty $, $ f(x) \to \infty $
- F) $ \lim_{x \to \infty} f(x) = -\infty $, $ \lim_{x \to -\infty} f(x) = \infty $
- G) Falls to the right, rises to the left → "down on the right, up on the left"

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8) $ f(x) = -x^2 + 4x - 3 $

- A) Degree: 2
- B) Even
- C) Leading coefficient: $-1$
- D) Negative
- E) As $ x \to \infty $, $ f(x) \to -\infty $; as $ x \to -\infty $, $ f(x) \to -\infty $
- F) $ \lim_{x \to \infty} f(x) = -\infty $, $ \lim_{x \to -\infty} f(x) = -\infty $
- G) Inverted U-shape (opens downward)

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9) $ f(x) = -x^5 + 4x^3 - 4x + 2 $

- A) Degree: 5
- B) Odd
- C) Leading coefficient: $-1$
- D) Negative
- E) As $ x \to \infty $, $ f(x) \to -\infty $; as $ x \to -\infty $, $ f(x) \to \infty $
- F) $ \lim_{x \to \infty} f(x) = -\infty $, $ \lim_{x \to -\infty} f(x) = \infty $
- G) Falls to the right, rises to the left → "down on the right, up on the left"

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✔ Summary Table for Quick Reference:

| Problem | Degree | Even/Odd | Leading Coeff | Sign | End Behavior (Words) | Symbolic | Shape |
|--------|--------|----------|----------------|------|------------------------|-----------|--------|
| 1 | 5 | Odd | -1 | Neg | Down right, up left | $ \infty \to -\infty $, $ -\infty \to \infty $ | ↘ on right, ↗ on left |
| 2 | 3 | Odd | 2 | Pos | Up right, down left | $ \infty \to \infty $, $ -\infty \to -\infty $ | ↗ on right, ↘ on left |
| 3 | 2 | Even | 1 | Pos | Up both ends | $ \pm\infty \to \infty $ | U-shaped |
| 4 | 3 | Odd | 1 | Pos | Up right, down left | $ \infty \to \infty $, $ -\infty \to -\infty $ | ↗ on right, ↘ on left |
| 5 | 2 | Even | -2 | Neg | Down both ends | $ \pm\infty \to -\infty $ | ∩-shaped |
| 6 | 3 | Odd | -1 | Neg | Down right, up left | $ \infty \to -\infty $, $ -\infty \to \infty $ | ↘ on right, ↗ on left |
| 7 | 5 | Odd | -1 | Neg | Down right, up left | $ \infty \to -\infty $, $ -\infty \to \infty $ | ↘ on right, ↗ on left |
| 8 | 2 | Even | -1 | Neg | Down both ends | $ \pm\infty \to -\infty $ | ∩-shaped |
| 9 | 5 | Odd | -1 | Neg | Down right, up left | $ \infty \to -\infty $, $ -\infty \to \infty $ | ↘ on right, ↗ on left |

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🔍 Key Concept Recap:

The end behavior of a polynomial is determined by:
- The degree: even or odd
- The sign of the leading coefficient

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

This helps you sketch the general shape without plotting points.

Let me know if you'd like hand-drawn sketches or graphs!