Graphing Polynomial Functions Worksheets with Answer Key - Free Printable
Educational worksheet: Graphing Polynomial Functions Worksheets with Answer Key. Download and print for classroom or home learning activities.
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Step-by-step solution for: Graphing Polynomial Functions Worksheets with Answer Key
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Show Answer Key & Explanations
Step-by-step solution for: Graphing Polynomial Functions Worksheets with Answer Key
To determine the end behavior of a polynomial function, we analyze how the graph behaves as $ x \to \infty $ and $ x \to -\infty $. This depends on two things:
1. Degree of the polynomial (the highest power of $ x $)
2. Leading coefficient (the coefficient of the term with the highest degree)
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| Degree | Leading Coefficient | End Behavior |
|--------|----------------------|--------------|
| Even | Positive | $ f(x) \to \infty $ as $ x \to \pm\infty $ (up on both ends) |
| Even | Negative | $ f(x) \to -\infty $ as $ x \to \pm\infty $ (down on both ends) |
| Odd | Positive | $ f(x) \to \infty $ as $ x \to \infty $, $ f(x) \to -\infty $ as $ x \to -\infty $ (rises right, falls left) |
| Odd | Negative | $ f(x) \to -\infty $ as $ x \to \infty $, $ f(x) \to \infty $ as $ x \to -\infty $ (falls right, rises left) |
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Now let's go through each function.
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- Degree: 3 (odd)
- Leading coefficient: +1 (positive)
- → Rises to the right, falls to the left
- ✔ End behavior: As $ x \to \infty $, $ f(x) \to \infty $; as $ x \to -\infty $, $ f(x) \to -\infty $
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- Highest degree term: $ -2x^9 $
- Degree: 9 (odd)
- Leading coefficient: -2 (negative)
- → Falls to the right, rises to the left
- ✔ End behavior: As $ x \to \infty $, $ f(x) \to -\infty $; as $ x \to -\infty $, $ f(x) \to \infty $
---
- Degree: 3 (odd)
- Leading coefficient: -1 (negative)
- → Falls to the right, rises to the left
- ✔ End behavior: As $ x \to \infty $, $ f(x) \to -\infty $; as $ x \to -\infty $, $ f(x) \to \infty $
---
- Degree: 4 (even)
- Leading coefficient: +1 (positive)
- → Rises on both ends
- ✔ End behavior: As $ x \to \pm\infty $, $ f(x) \to \infty $
---
- Degree: 5 (odd)
- Leading coefficient: +1 (positive)
- → Rises to the right, falls to the left
- ✔ End behavior: As $ x \to \infty $, $ f(x) \to \infty $; as $ x \to -\infty $, $ f(x) \to -\infty $
---
- Degree: 5 (odd)
- Leading coefficient: -1 (negative)
- → Falls to the right, rises to the left
- ✔ End behavior: As $ x \to \infty $, $ f(x) \to -\infty $; as $ x \to -\infty $, $ f(x) \to \infty $
---
- Degree: 2 (even)
- Leading coefficient: -1 (negative)
- → Falls on both ends
- ✔ End behavior: As $ x \to \pm\infty $, $ f(x) \to -\infty $
---
- Degree: 3 (odd)
- Leading coefficient: +1 (positive)
- → Rises to the right, falls to the left
- ✔ End behavior: As $ x \to \infty $, $ f(x) \to \infty $; as $ x \to -\infty $, $ f(x) \to -\infty $
---
- Degree: 2 (even)
- Leading coefficient: -2 (negative)
- → Falls on both ends
- ✔ End behavior: As $ x \to \pm\infty $, $ f(x) \to -\infty $
---
- Degree: 4 (even)
- Leading coefficient: -1 (negative)
- → Falls on both ends
- ✔ End behavior: As $ x \to \pm\infty $, $ f(x) \to -\infty $
---
- Degree: 4 (even)
- Leading coefficient: +1 (positive)
- → Rises on both ends
- ✔ End behavior: As $ x \to \pm\infty $, $ f(x) \to \infty $
---
- Degree: 4 (even)
- Leading coefficient: -1 (negative)
- → Falls on both ends
- ✔ End behavior: As $ x \to \pm\infty $, $ f(x) \to -\infty $
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| # | Function | End Behavior |
|---|---------|--------------|
| ① | $ f(x) = x^3 - 2x^2 + 3 $ | As $ x \to \infty $, $ f(x) \to \infty $; as $ x \to -\infty $, $ f(x) \to -\infty $ |
| ② | $ f(x) = 3x^6 - 7x^4 - 2x^9 $ | As $ x \to \infty $, $ f(x) \to -\infty $; as $ x \to -\infty $, $ f(x) \to \infty $ |
| ③ | $ f(x) = -x^3 + 3x^2 - 4 $ | As $ x \to \infty $, $ f(x) \to -\infty $; as $ x \to -\infty $, $ f(x) \to \infty $ |
| ④ | $ f(x) = x^4 - 4x^2 + 2x + 4 $ | As $ x \to \pm\infty $, $ f(x) \to \infty $ |
| ⑤ | $ f(x) = x^5 + 3x^3 + 3 $ | As $ x \to \infty $, $ f(x) \to \infty $; as $ x \to -\infty $, $ f(x) \to -\infty $ |
| ⑥ | $ f(x) = -x^5 + 3x^3 + 3 $ | As $ x \to \infty $, $ f(x) \to -\infty $; as $ x \to -\infty $, $ f(x) \to \infty $ |
| ⑦ | $ f(x) = -x^2 - 8x - 15 $ | As $ x \to \pm\infty $, $ f(x) \to -\infty $ |
| ⑧ | $ f(x) = x^3 - 3x^2 + 1 $ | As $ x \to \infty $, $ f(x) \to \infty $; as $ x \to -\infty $, $ f(x) \to -\infty $ |
| ⑨ | $ f(x) = -2x^2 + 16x - 29 $ | As $ x \to \pm\infty $, $ f(x) \to -\infty $ |
| ⑩ | $ f(x) = -x^4 + x^3 - x^2 $ | As $ x \to \pm\infty $, $ f(x) \to -\infty $ |
| ⑪ | $ f(x) = x^4 - 6x^3 + 8x^2 $ | As $ x \to \pm\infty $, $ f(x) \to \infty $ |
| ⑫ | $ f(x) = -x^4 + 4x^3 - 4x^2 $ | As $ x \to \pm\infty $, $ f(x) \to -\infty $ |
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Let me know if you'd like this formatted into a printable answer sheet!
1. Degree of the polynomial (the highest power of $ x $)
2. Leading coefficient (the coefficient of the term with the highest degree)
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🔍 Rules for End Behavior:
| Degree | Leading Coefficient | End Behavior |
|--------|----------------------|--------------|
| Even | Positive | $ f(x) \to \infty $ as $ x \to \pm\infty $ (up on both ends) |
| Even | Negative | $ f(x) \to -\infty $ as $ x \to \pm\infty $ (down on both ends) |
| Odd | Positive | $ f(x) \to \infty $ as $ x \to \infty $, $ f(x) \to -\infty $ as $ x \to -\infty $ (rises right, falls left) |
| Odd | Negative | $ f(x) \to -\infty $ as $ x \to \infty $, $ f(x) \to \infty $ as $ x \to -\infty $ (falls right, rises left) |
---
Now let's go through each function.
---
① $ f(x) = x^3 - 2x^2 + 3 $
- Degree: 3 (odd)
- Leading coefficient: +1 (positive)
- → Rises to the right, falls to the left
- ✔ End behavior: As $ x \to \infty $, $ f(x) \to \infty $; as $ x \to -\infty $, $ f(x) \to -\infty $
---
② $ f(x) = 3x^6 - 7x^4 - 2x^9 $
- Highest degree term: $ -2x^9 $
- Degree: 9 (odd)
- Leading coefficient: -2 (negative)
- → Falls to the right, rises to the left
- ✔ End behavior: As $ x \to \infty $, $ f(x) \to -\infty $; as $ x \to -\infty $, $ f(x) \to \infty $
---
③ $ f(x) = -x^3 + 3x^2 - 4 $
- Degree: 3 (odd)
- Leading coefficient: -1 (negative)
- → Falls to the right, rises to the left
- ✔ End behavior: As $ x \to \infty $, $ f(x) \to -\infty $; as $ x \to -\infty $, $ f(x) \to \infty $
---
④ $ f(x) = x^4 - 4x^2 + 2x + 4 $
- Degree: 4 (even)
- Leading coefficient: +1 (positive)
- → Rises on both ends
- ✔ End behavior: As $ x \to \pm\infty $, $ f(x) \to \infty $
---
⑤ $ f(x) = x^5 + 3x^3 + 3 $
- Degree: 5 (odd)
- Leading coefficient: +1 (positive)
- → Rises to the right, falls to the left
- ✔ End behavior: As $ x \to \infty $, $ f(x) \to \infty $; as $ x \to -\infty $, $ f(x) \to -\infty $
---
⑥ $ f(x) = -x^5 + 3x^3 + 3 $
- Degree: 5 (odd)
- Leading coefficient: -1 (negative)
- → Falls to the right, rises to the left
- ✔ End behavior: As $ x \to \infty $, $ f(x) \to -\infty $; as $ x \to -\infty $, $ f(x) \to \infty $
---
⑦ $ f(x) = -x^2 - 8x - 15 $
- Degree: 2 (even)
- Leading coefficient: -1 (negative)
- → Falls on both ends
- ✔ End behavior: As $ x \to \pm\infty $, $ f(x) \to -\infty $
---
⑧ $ f(x) = x^3 - 3x^2 + 1 $
- Degree: 3 (odd)
- Leading coefficient: +1 (positive)
- → Rises to the right, falls to the left
- ✔ End behavior: As $ x \to \infty $, $ f(x) \to \infty $; as $ x \to -\infty $, $ f(x) \to -\infty $
---
⑨ $ f(x) = -2x^2 + 16x - 29 $
- Degree: 2 (even)
- Leading coefficient: -2 (negative)
- → Falls on both ends
- ✔ End behavior: As $ x \to \pm\infty $, $ f(x) \to -\infty $
---
⑩ $ f(x) = -x^4 + x^3 - x^2 $
- Degree: 4 (even)
- Leading coefficient: -1 (negative)
- → Falls on both ends
- ✔ End behavior: As $ x \to \pm\infty $, $ f(x) \to -\infty $
---
⑪ $ f(x) = x^4 - 6x^3 + 8x^2 $
- Degree: 4 (even)
- Leading coefficient: +1 (positive)
- → Rises on both ends
- ✔ End behavior: As $ x \to \pm\infty $, $ f(x) \to \infty $
---
⑫ $ f(x) = -x^4 + 4x^3 - 4x^2 $
- Degree: 4 (even)
- Leading coefficient: -1 (negative)
- → Falls on both ends
- ✔ End behavior: As $ x \to \pm\infty $, $ f(x) \to -\infty $
---
✔ Final Answers Summary:
| # | Function | End Behavior |
|---|---------|--------------|
| ① | $ f(x) = x^3 - 2x^2 + 3 $ | As $ x \to \infty $, $ f(x) \to \infty $; as $ x \to -\infty $, $ f(x) \to -\infty $ |
| ② | $ f(x) = 3x^6 - 7x^4 - 2x^9 $ | As $ x \to \infty $, $ f(x) \to -\infty $; as $ x \to -\infty $, $ f(x) \to \infty $ |
| ③ | $ f(x) = -x^3 + 3x^2 - 4 $ | As $ x \to \infty $, $ f(x) \to -\infty $; as $ x \to -\infty $, $ f(x) \to \infty $ |
| ④ | $ f(x) = x^4 - 4x^2 + 2x + 4 $ | As $ x \to \pm\infty $, $ f(x) \to \infty $ |
| ⑤ | $ f(x) = x^5 + 3x^3 + 3 $ | As $ x \to \infty $, $ f(x) \to \infty $; as $ x \to -\infty $, $ f(x) \to -\infty $ |
| ⑥ | $ f(x) = -x^5 + 3x^3 + 3 $ | As $ x \to \infty $, $ f(x) \to -\infty $; as $ x \to -\infty $, $ f(x) \to \infty $ |
| ⑦ | $ f(x) = -x^2 - 8x - 15 $ | As $ x \to \pm\infty $, $ f(x) \to -\infty $ |
| ⑧ | $ f(x) = x^3 - 3x^2 + 1 $ | As $ x \to \infty $, $ f(x) \to \infty $; as $ x \to -\infty $, $ f(x) \to -\infty $ |
| ⑨ | $ f(x) = -2x^2 + 16x - 29 $ | As $ x \to \pm\infty $, $ f(x) \to -\infty $ |
| ⑩ | $ f(x) = -x^4 + x^3 - x^2 $ | As $ x \to \pm\infty $, $ f(x) \to -\infty $ |
| ⑪ | $ f(x) = x^4 - 6x^3 + 8x^2 $ | As $ x \to \pm\infty $, $ f(x) \to \infty $ |
| ⑫ | $ f(x) = -x^4 + 4x^3 - 4x^2 $ | As $ x \to \pm\infty $, $ f(x) \to -\infty $ |
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Let me know if you'd like this formatted into a printable answer sheet!
Parent Tip: Review the logic above to help your child master the concept of end behavior worksheet.
