# 4.2 - Graphing polynomial functions.

- Goal: Draw a rough graph of polynomial function.

- We will take into account (1) the leading term test and (2) plot intercepts.

- Key fact with $x$ intercepts: if the multiplicity is odd, the graph crosses the axis. If the multiplicity is even, the graph bounces off the axis.

| Even Multiplicity (Bounce off x-axis) | Odd Multiplicity (Pass through x-axis) |
|---|---|
| | |
|| |

- The graph of a polynomial has generally one less turn than its degree. (Possibly less)

| | Even Degree | Odd Degree |
|---|---|---|
| Positive Leading Coefficient |  rises on both sides |  rises to the right, falls to the left|
| Negative Leading Coefficient |  falls on both sides |  rises to the left, falls to the right|

So :

- a quadratic has a graph with one turn (and up or down parabola).

- a degree $3$ polynomial has at most $2$ turns: and goes up-down-up or down-up-down (draw pictures).

- a degree $4$ polynomial has at most $3$ turns and goes up-down-up-down or down-up-down-up (draw pictures).

Etc....

---

## Example 1. $y=f(x)=4x^4-16x^2$

### Function: $f(x) = 4x^4 - 16x^2$ is expanded.

### End Behavior

#### Step 1: Identify the Leading Term
The leading term is:
$$4x^4$$

#### Step 2: Determine Degree and Leading Coefficient
- **Degree**: 4 → **Even Degree**
- **Leading Coefficient**: 4 → **Positive**

#### Step 3: Use the End Behavior Table
From the table:
- **Even Degree**
- **Positive Leading Coefficient**
→ The graph **rises on both sides**.

#### ✅ Final End Behavior:


#### End Behavior Conclusions:

1. The graph rises on the left.
2. The graph rises on the right.

### Root Behavior

To find the root behavior of $f(x)$ we need to factor it completely. The GCF is $4x^2$, so factor it out and then factor $x^2-4=(x+2)(x-2)$. Check by FOILing.

\begin{align}
f(x)&=4x^4-16x^2 \\
f(x)&=(4x^2)(x^2)+(4x^2)(-4) \\
f(x)&=4x^2(x^2-4) \\
f(x)&=4x^2(x+2)(x-2)
\end{align}

|Factors | Zeros | Multiplicities | Bounce or Cross |
|---|---|---|---|
|$4$ | NA | NA | NA |
|$x^2$ | $x=0$ | $2$ (even)| Bounce |
|$(x+2)$ | $x=-2$ | $1$ (odd)| Cross |
|$(x-2)$ | $x=2$ | $1$ (odd)| Cross |

#### Root Behavior Conclusions:

1. The graph crosses over the x-axis at $x=-2$.
2. The graph bounces off the x-axis at $x=0$.
2. The graph crosses over the x-axis at $x=2$.

#### Summary:

1. The graph rises on the left.
2. The graph rises on the right.
3. The graph crosses over the x-axis at $x=-2$.
4. The graph bounces off the x-axis at $x=0$.
5. The graph crosses over the x-axis at $x=2$.


---

## Example 2. $g(x) = -2x^3 + 32x$

### Function: $g(x) = -2x^3 + 32x$ is expanded.

### End Behavior

#### Step 1: Identify the Leading Term
The leading term is:
$$-2x^3$$

#### Step 2: Determine Degree and Leading Coefficient
- **Degree**: 3 → **Odd Degree**
- **Leading Coefficient**: $-2$ → **Negative**

#### Step 3: Use the End Behavior Table
From the table:
- **Odd Degree**
- **Negative Leading Coefficient**
→ The graph **rises to the left, falls to the right**.

#### ✅ Final End Behavior:


#### End Behavior Conclusions:

1. The graph rises on the left.
2. The graph falls on the right.

### Root Behavior

To find the root behavior of $g(x)$, we factor it completely. The GCF is $-2x$, so factor it out and then factor the remaining quadratic.

$$
\begin{align}
g(x) &= -2x^3 + 32x \\
g(x) &= -2x(x^2 - 16) \\
g(x) &= -2x(x + 4)(x - 4)
\end{align}
$$

| Factors | Zeros | Multiplicities | Bounce or Cross |
|---------------|-----------|----------------|------------------|
| $-2$ | N/A | N/A | N/A |
| $x$ | $x = 0$ | 1 (odd) | Cross |
| $(x + 4)$ | $x = -4$ | 1 (odd) | Cross |
| $(x - 4)$ | $x = 4$ | 1 (odd) | Cross |

#### Root Behavior Conclusions:

1. The graph crosses over the x-axis at $x = -4$.
2. The graph crosses over the x-axis at $x = 0$.
3. The graph crosses over the x-axis at $x = 4$.

### Summary:

1. The graph rises on the left.
2. The graph falls on the right.
3. The graph crosses over the x-axis at $x = -4$.
4. The graph crosses over the x-axis at $x = 0$.
5. The graph crosses over the x-axis at $x = 4$.


---

## Example 3. $h(x) = (x+2)(x-1)^2(x-2)^2$

### Function: $h(x) = (x+2)(x-1)^2(x-2)^2$ is factored.

### End Behavior

#### Step 1: Identify the Leading Term
To find the leading term, multiply the leading terms of each factor:
$$
x \cdot x^2 \cdot x^2 = x^5
$$

So, the leading term is:
$$x^5$$

#### Step 2: Determine Degree and Leading Coefficient
- **Degree**: 5 → **Odd Degree**
- **Leading Coefficient**: 1 → **Positive**

#### Step 3: Use the End Behavior Table
From the table:
- **Odd Degree**
- **Positive Leading Coefficient**
→ The graph **falls to the left, rises to the right**.

#### ✅ Final End Behavior:


#### End Behavior Conclusions:

1. The graph falls on the left.
2. The graph rises on the right.

### Root Behavior

To find the root behavior of $h(x)$, we examine the factors and multiplicities:

$$
h(x) = (x+2)(x-1)^2(x-2)^2
$$

| Factors | Zeros | Multiplicities | Bounce or Cross |
|-----------------|-----------|----------------|------------------|
| $(x+2)$ | $x = -2$ | 1 (odd) | Cross |
| $(x-1)^2$ | $x = 1$ | 2 (even) | Bounce |
| $(x-2)^2$ | $x = 2$ | 2 (even) | Bounce |

#### Root Behavior Conclusions:

1. The graph **crosses** at $x = -2$.
2. The graph **bounces** at $x = 1$.
3. The graph **bounces** at $x = 2$.

### Summary:

1. The graph **falls on the left**.
2. The graph **rises on the right**.
3. The graph **crosses** the x-axis at $x = -2$.
4. The graph **bounces** off the x-axis at $x = 1$.
5. The graph **bounces** off the x-axis at $x = 2$.


Zoomed in near x=1 and x=2:


---

## Example 4. $f(x) = -(x+3)^2(x-1)^2(x-4)$

### Function: $f(x) = -(x+3)^2(x-1)^2(x-4)$ is factored.

### End Behavior

#### Step 1: Identify the Leading Term
Multiply the leading terms of each factor:
$$
-(x^2)(x^2)(x) = -x^5
$$

So, the leading term is:
$$-x^5$$

#### Step 2: Determine Degree and Leading Coefficient
- **Degree**: 5 → **Odd Degree**
- **Leading Coefficient**: $-1$ → **Negative**

#### Step 3: Use the End Behavior Table
From the table:
- **Odd Degree**
- **Negative Leading Coefficient**
→ The graph **rises on the left, falls on the right**.

#### ✅ Final End Behavior:


#### End Behavior Conclusions:

1. The graph rises on the left.
2. The graph falls on the right.

### Root Behavior

We analyze the factors and multiplicities of the function:
$$
f(x) = -(x+3)^2(x-1)^2(x-4)
$$

| Factors | Zeros | Multiplicities | Bounce or Cross |
|-----------------|-----------|----------------|------------------|
| $-1$ | NA | NA | NA |
| $(x+3)^2$ | $x = -3$ | 2 (even) | Bounce |
| $(x-1)^2$ | $x = 1$ | 2 (even) | Bounce |
| $(x-4)$ | $x = 4$ | 1 (odd) | Cross |

#### Root Behavior Conclusions:

1. The graph **bounces** at $x = -3$.
2. The graph **bounces** at $x = 1$.
3. The graph **crosses** at $x = 4$.

### Summary:

1. The graph **rises on the left**.
2. The graph **falls on the right**.
3. The graph **bounces** at $x = -3$.
4. The graph **bounces** at $x = 1$.
5. The graph **crosses** the x-axis at $x = 4$.


---

## Example 5. $g(x) = -4(x+2)(x-1)^2(x-3)$

### Function: $g(x) = -4(x+2)(x-1)^2(x-3)$ is factored.

### End Behavior

#### Step 1: Identify the Leading Term
Multiply the leading terms of each factor:
$$
-4(x)(x^2)(x) = -4x^4
$$

So, the leading term is:
$$-4x^4$$

#### Step 2: Determine Degree and Leading Coefficient
- **Degree**: 4 → **Even Degree**
- **Leading Coefficient**: $-4$ → **Negative**

#### Step 3: Use the End Behavior Table
From the table:
- **Even Degree**
- **Negative Leading Coefficient**
→ The graph **falls on both sides**.

#### ✅ Final End Behavior:


#### End Behavior Conclusions:

1. The graph falls on the left.
2. The graph falls on the right.

### Root Behavior

We analyze the factored form of the function:
$$
g(x) = -4(x+2)(x-1)^2(x-3)
$$

| Factors | Zeros | Multiplicities | Bounce or Cross |
|-----------------|-----------|----------------|------------------|
| $-4$ | NA | NA | NA |
| $(x+2)$ | $x = -2$ | 1 (odd) | Cross |
| $(x-1)^2$ | $x = 1$ | 2 (even) | Bounce |
| $(x-3)$ | $x = 3$ | 1 (odd) | Cross |

#### Root Behavior Conclusions:

1. The graph **crosses** at $x = -2$.
2. The graph **bounces** at $x = 1$.
3. The graph **crosses** at $x = 3$.

### Summary:

1. The graph **falls on the left**.
2. The graph **falls on the right**.
3. The graph **crosses** the x-axis at $x = -2$.
4. The graph **bounces** at $x = 1$.
5. The graph **crosses** the x-axis at $x = 3$.


---

## Example 6. $h(x) = 2x^3 - 4x^2$

### Function: $h(x) = 2x^3 - 4x^2$ is expanded.

### End Behavior

#### Step 1: Identify the Leading Term
The leading term is:
$$2x^3$$

#### Step 2: Determine Degree and Leading Coefficient
- **Degree**: 3 → **Odd Degree**
- **Leading Coefficient**: 2 → **Positive**

#### Step 3: Use the End Behavior Table
From the table:
- **Odd Degree**
- **Positive Leading Coefficient**
→ The graph **falls on the left, rises on the right**.

#### ✅ Final End Behavior:


#### End Behavior Conclusions:

1. The graph falls on the left.
2. The graph rises on the right.

### Root Behavior

To find the root behavior of $h(x)$, factor out the GCF $2x^2$:

$$
\begin{align}
h(x) &= 2x^3 - 4x^2 \\
&=(2x^2)(x)+(2x^2)(-2) \\
&= 2x^2(x - 2)
\end{align}
$$

| Factors | Zeros | Multiplicities | Bounce or Cross |
|------------|-----------|----------------|-----------------|
| $2$ | N/A | N/A | N/A |
| $x^2$ | $x = 0$ | 2 (even) | Bounce |
| $(x - 2)$ | $x = 2$ | 1 (odd) | Cross |

#### Root Behavior Conclusions:

1. The graph **bounces** at $x = 0$.
2. The graph **crosses** at $x = 2$.

### Summary:

1. The graph **falls on the left**.
2. The graph **rises on the right**.
3. The graph **bounces** off the x‑axis at $x = 0$.
4. The graph **crosses** the x‑axis at $x = 2$.