Kami Export - M3 U3 WS1 - End Behavior of Polynomial Functions ... - Free Printable

Let's go through the Math 3 Unit 3 Worksheet 1 on End Behavior of Polynomial Functions step by step, identifying and correcting any errors, and explaining each part clearly.

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🔹 Part 1: Identify the leading coefficient, degree, and end behavior

We need to determine:
- Degree: The highest power of $ x $
- Leading Coefficient: The coefficient of the term with the highest degree
- End Behavior: Based on degree (even/odd) and sign of leading coefficient

> ✔ Rules for End Behavior:
> - Even degree, positive leading coeff → up up
> - Even degree, negative leading coeff → down down
> - Odd degree, positive leading coeff → down up
> - Odd degree, negative leading coeff → up down

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#### 1. $ f(x) = 5x^2 + 7x - 3 $
- Degree: 2 ✘ (not 3)
- Leading Coeff: +5 → +
- End Behavior: up up ✔

✔ Correction: Degree is 2, not 3.

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#### 2. $ y = -2x^2 - 3x + 4 $
- Degree: 2 ✘ (not 3)
- Leading Coeff: -2 → -
- End Behavior: down down ✔

✔ Correction: Degree is 2, not 3.

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#### 3. $ g(x) = x^3 - 9x^2 + 2x + 6 $
- Degree: 3 ✘ (not 6)
- Leading Coeff: +1 → +
- End Behavior: down up ✘ (not "down down")

✔ Correction:
- Degree: 3 (odd)
- Leading Coeff: +
- End Behavior: down up

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#### 4. $ y = -7x^3 + 3x^2 + 12x - 1 $
- Degree: 3 ✘ (not 6)
- Leading Coeff: -7 → -
- End Behavior: up down ✔

✔ Correction: Degree is 3, not 6.

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#### 5. $ h(x) = -2x^7 + 5x^4 - 3x $
- Degree: 7 ✘ (not 12)
- Leading Coeff: -2 → -
- End Behavior: up down ✔

✔ Correction: Degree is 7, not 12.

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#### 6. $ g(x) = 8x^3 + 4x^2 + 7x^4 - 9x $
- First, rewrite in standard form: $ 7x^4 + 8x^3 + 4x^2 - 9x $
- Degree: 4 ✘ (not 10)
- Leading Coeff: +7 → +
- End Behavior: up up ✔

✔ Correction: Degree is 4, not 10.

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✔ Corrected Answers for Part 1:

| # | Function | Degree | Leading Coeff | End Behavior |
|---|--------|--------|----------------|---------------|
| 1 | $ f(x) = 5x^2 + 7x - 3 $ | 2 | + | up up |
| 2 | $ y = -2x^2 - 3x + 4 $ | 2 | - | down down |
| 3 | $ g(x) = x^3 - 9x^2 + 2x + 6 $ | 3 | + | down up |
| 4 | $ y = -7x^3 + 3x^2 + 12x - 1 $ | 3 | - | up down |
| 5 | $ h(x) = -2x^7 + 5x^4 - 3x $ | 7 | - | up down |
| 6 | $ g(x) = 8x^3 + 4x^2 + 7x^4 - 9x $ | 4 | + | up up |

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🔹 Part 2: Identify the end behavior. Justify your answer.

#### 7. $ f(x) = 4x^5 - 3x^4 + 2x^3 $
- Degree: 5 → odd
- Leading Coeff: +4 → positive
- End Behavior: down up ✔

✔️ Correct.

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#### 8. $ y = -x^4 + x^3 - x^2 + 1 - 1 $ → Simplify: $ -x^4 + x^3 - x^2 $
- Degree: 4 → even
- Leading Coeff: -1 → negative
- End Behavior: down down ✔

✔️ Correct.

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#### 9. $ h(x) = 3x^6 - 7x^4 - 2x^9 $
- Rewrite: $ -2x^9 + 3x^6 - 7x^4 $
- Degree: 9 → odd
- Leading Coeff: -2 → negative
- End Behavior: up down ✔

✔️ Correct.

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✔ Part 2 Summary:

| # | Function | End Behavior | Justification |
|---|---------|--------------|---------------|
| 7 | $ f(x) = 4x^5 - \dots $ | down up | Odd degree, positive leading coeff |
| 8 | $ y = -x^4 + \dots $ | down down | Even degree, negative leading coeff |
| 9 | $ h(x) = \dots -2x^9 $ | up down | Odd degree, negative leading coeff |

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🔹 Part 3: Graphs – Identify degree (odd/even), leading coefficient (+/-), justify

We analyze based on end behavior:

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#### 10.
- Graph: Left goes down, right goes up
- So: down up
- This matches: odd degree, positive leading coeff
- ✔ deg: odd, coeff: +, justify: down up

✔️ Correct.

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#### 11.
- Graph: Both ends go down
- So: down down
- This means: even degree, negative leading coeff
- ✔ deg: even, coeff: -, justify: down down

✔️ Correct.

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#### 12.
- Graph: Left goes up, right goes down
- So: up down
- This means: odd degree, negative leading coeff
- ✔ deg: odd, coeff: -, justify: up down

✔️ Correct.

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#### 13.
- Graph: Both ends go up
- So: up up
- This means: even degree, positive leading coeff
- ✔ deg: even, coeff: +, justify: up up

✔️ Correct.

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#### 14.
- Graph: Left goes up, right goes down
- So: up down
- This means: odd degree, negative leading coeff
- ✔ deg: odd, coeff: -, justify: up down

✔️ Correct.

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#### 15.
- Graph: Left goes down, then turns up, then down again → ends go down on both sides?
- Wait: Let's look carefully.

The graph shows:
- As $ x \to -\infty $: $ y \to -\infty $
- As $ x \to +\infty $: $ y \to -\infty $

So: down down

But the justification says: "down up turn turn down" — this seems messy.

Actually, the end behavior is down down → so:
- Even degree
- Negative leading coefficient

But the worksheet says:
- deg: odd ✘
- coeff: - ✔
- justify: down up turn turn down ✘

✘ Incorrect.

Let’s correct it:

- Ends both go down → down down
- So: even degree, negative leading coefficient
- Justify: down down

✔ Correction:
- deg: even
- coeff: -
- justify: down down

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✔ Final Corrections for Part 3:

| # | deg | coeff | justify |
|----|-----|-------|--------|
| 10 | odd | + | down up |
| 11 | even | - | down down |
| 12 | odd | - | up down |
| 13 | even | + | up up |
| 14 | odd | - | up down |
| 15 | even | - | down down |

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📌 Summary of Errors & Corrections:

| Problem | Error | Correction |
|--------|------|------------|
| 1 | Degree = 3 | Degree = 2 |
| 2 | Degree = 3 | Degree = 2 |
| 3 | Degree = 6, End Behavior = down down | Degree = 3, End Behavior = down up |
| 4 | Degree = 6 | Degree = 3 |
| 5 | Degree = 12 | Degree = 7 |
| 6 | Degree = 10 | Degree = 4 |
| 15 | deg: odd, justify: down up turn turn down | deg: even, justify: down down |

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✔ Final Answer Key (Corrected)

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#### Part 1:
1. Degree: 2, LC: +, EB: up up
2. Degree: 2, LC: -, EB: down down
3. Degree: 3, LC: +, EB: down up
4. Degree: 3, LC: -, EB: up down
5. Degree: 7, LC: -, EB: up down
6. Degree: 4, LC: +, EB: up up

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#### Part 2:
7. down up → odd degree, positive LC
8. down down → even degree, negative LC
9. up down → odd degree, negative LC

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#### Part 3:
10. deg: odd, coeff: +, justify: down up
11. deg: even, coeff: -, justify: down down
12. deg: odd, coeff: -, justify: up down
13. deg: even, coeff: +, justify: up up
14. deg: odd, coeff: -, justify: up down
15. deg: even, coeff: -, justify: down down

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✔ Final Notes:

- Always identify the degree correctly (highest exponent).
- Write the polynomial in standard form (descending powers) to find the leading term.
- Use the degree and sign of leading coefficient to predict end behavior.
- For graphs: look at the far left and far right of the graph.

Let me know if you'd like a visual explanation or practice problems!