Polynomial Functions | Systry - Free Printable

Let's solve this step-by-step based on the Polynomial Functions worksheet you've uploaded. The goal is to:

1. Graph each function (conceptually or using Desmos),
2. Sketch the end behavior,
3. Determine whether the graph rises or falls on the right,
4. Identify patterns in end behavior based on degree and leading coefficient.

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🔍 Understanding End Behavior

The end behavior of a polynomial depends on:
- The degree (even or odd)
- The leading coefficient (positive or negative)

| Degree | Leading Coeff. | End Behavior |
|--------|----------------|--------------|
| Even | Positive | Rises on both ends (U-shaped) |
| Even | Negative | Falls on both ends (∩-shaped) |
| Odd | Positive | Falls on left, rises on right (↗) |
| Odd | Negative | Rises on left, falls on right (↘) |

We'll use this to analyze each function.

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## ✔ Step 1: Fill in the Table

We’ll go row by row.

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🟦 Linear (Degree 1) – Odd degree

#### 1. $ f(x) = \frac{3}{4}x + 2 $
- Leading coeff: $ \frac{3}{4} > 0 $
- Odd degree → falls left, rises on the right
✔ Rising on the right

#### 2. $ f(x) = 4x - 3 $
- Leading coeff: $ 4 > 0 $
- Odd degree → rises on the right
✔ Rising on the right

#### 3. $ f(x) = -\frac{1}{5}x - 5 $
- Leading coeff: $ -\frac{1}{5} < 0 $
- Odd degree → falls on the right
✔ Falling on the right

#### 4. $ f(x) = -x + 4 $
- Leading coeff: $ -1 < 0 $
- Odd degree → falls on the right
✔ Falling on the right

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🟨 Quadratic (Degree 2) – Even degree

#### 5. $ f(x) = x^2 $
- Leading coeff: $ 1 > 0 $
- Even degree → rises on both ends
✔ Rising on the right

#### 6. $ f(x) = -2(x - 1)^2 + 3 $
- Leading coeff: $ -2 < 0 $
- Even degree → falls on both ends
✔ Falling on the right

#### 7. $ f(x) = (x + 2)^2 $
- Leading coeff: $ 1 > 0 $
- Even degree → rises on both ends
✔ Rising on the right

#### 8. $ f(x) = -3x^2 - 4x - 1 $
- Leading coeff: $ -3 < 0 $
- Even degree → falls on both ends
✔ Falling on the right

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🟩 Cubic (Degree 3) – Odd degree

#### 9. $ f(x) = x^3 $
- Leading coeff: $ 1 > 0 $
- Odd degree → rises on the right
✔ Rising on the right

#### 10. $ f(x) = -3x^3 + 2x^2 + 2x - 1 $
- Leading coeff: $ -3 < 0 $
- Odd degree → falls on the right
✔ Falling on the right

#### 11. $ f(x) = 2x^3 + 5 $
- Leading coeff: $ 2 > 0 $
- Odd degree → rises on the right
✔ Rising on the right

#### 12. $ f(x) = -2x^3 + x^2 + 5x + 3 $
- Leading coeff: $ -2 < 0 $
- Odd degree → falls on the right
✔ Falling on the right

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🟪 Quartic (Degree 4) – Even degree

#### 13. $ f(x) = x^4 $
- Leading coeff: $ 1 > 0 $
- Even degree → rises on both ends
✔ Rising on the right

#### 14. $ f(x) = x^4 - 4 $
- Leading coeff: $ 1 > 0 $
- Even degree → rises on both ends
✔ Rising on the right

#### 15. $ f(x) = -x^4 + 2x^3 - x + 4 $
- Leading coeff: $ -1 < 0 $
- Even degree → falls on both ends
✔ Falling on the right

#### 16. $ f(x) = -2x^4 + 8x^2 - 3 $
- Leading coeff: $ -2 < 0 $
- Even degree → falls on both ends
✔ Falling on the right

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✔ Completed Table (Answers):

| Function | End Behavior on Right |
|---------|-----------------------|
| $ f(x) = \frac{3}{4}x + 2 $ | Rising |
| $ f(x) = 4x - 3 $ | Rising |
| $ f(x) = -\frac{1}{5}x - 5 $ | Falling |
| $ f(x) = -x + 4 $ | Falling |
| $ f(x) = x^2 $ | Rising |
| $ f(x) = -2(x - 1)^2 + 3 $ | Falling |
| $ f(x) = (x + 2)^2 $ | Rising |
| $ f(x) = -3x^2 - 4x - 1 $ | Falling |
| $ f(x) = x^3 $ | Rising |
| $ f(x) = -3x^3 + 2x^2 + 2x - 1 $ | Falling |
| $ f(x) = 2x^3 + 5 $ | Rising |
| $ f(x) = -2x^3 + x^2 + 5x + 3 $ | Falling |
| $ f(x) = x^4 $ | Rising |
| $ f(x) = x^4 - 4 $ | Rising |
| $ f(x) = -x^4 + 2x^3 - x + 4 $ | Falling |
| $ f(x) = -2x^4 + 8x^2 - 3 $ | Falling |

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## 🔎 Look for a Pattern – Answers

Now answer the questions at the bottom:

❓ What do all of the polynomials that rise on the right have in common?

👉 They all have an odd degree with positive leading coefficient, or an even degree with positive leading coefficient.

But more precisely:
- Odd degree and positive leading coefficient → rises on the right
- Even degree and positive leading coefficient → rises on the right

✔ So: They all have a positive leading coefficient.

> ✔ Answer: All polynomials that rise on the right have a positive leading coefficient.

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❓ What do all of the polynomials that fall on the right have in common?

👉 They all have:
- Odd degree and negative leading coefficient, or
- Even degree and negative leading coefficient

So, regardless of degree, if the leading coefficient is negative, the graph falls on the right.

✔ Answer: All polynomials that fall on the right have a negative leading coefficient.

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❓ What do you notice about the polynomials that do the same on the left and the right?

These are even-degree polynomials.

- If even degree and positive leading coefficient → rises on both sides
- If even degree and negative leading coefficient → falls on both sides

So they behave the same on both ends.

✔ Answer: Polynomials that do the same on the left and right are those with even degree.

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❓ What do polynomials that do the opposite on the left and right have in common?

These are odd-degree polynomials.

- Odd degree → one end goes up, the other down
- Example: $ x^3 $: falls on left, rises on right

✔ Answer: Polynomials that do the opposite on the left and right have odd degree.

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## ✔ Final Summary of Patterns:

| Behavior | Common Feature |
|--------|----------------|
| Rise on the right | Positive leading coefficient |
| Fall on the right | Negative leading coefficient |
| Same on left and right | Even degree |
| Opposite on left and right | Odd degree |

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## 📌 Conclusion

This activity helps students see how the degree and leading coefficient determine the end behavior of polynomial functions. It’s foundational for understanding graphs without plotting every point.

Let me know if you’d like a printable version or a sketch guide!