HW 7.2 .2 - END BEHAVIOR AND OTHER CHARACTERISTICS OF POLYNOMIAL FUNCTIONS Identify the degree, leading coefficient, zeros, end behavior, and create a sketch of the function without the use of technology. (An accurate $$ y $$-intercept is not needed for the sketch). Part 1 $$ \begin{array}{ll} ext { (1) } p(r)=-2(r-1)(r+2)(r-2) & ext { (2) } f(t)=3 t(t-3)(t+4)\end{array} $$

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HW 7.2 .2 - END BEHAVIOR AND OTHER CHARACTERISTICS OF POLYNOMIAL FUNCTIONS Identify the degree, leading coefficient, zeros, end behavior, and create a sketch of the function without the use of technology. (An accurate $$ y $$-intercept is not needed for the sketch). Part 1 $$ \begin{array}{ll}	ext { (1) } p(r)=-2(r-1)(r+2)(r-2) & 	ext { (2) } f(t)=3 t(t-3)(t+4)\end{array} $$

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Sure, let's analyze each polynomial function step by step. We'll determine the **degree**, **leading coefficient**, **zeros**, **end behavior**, and provide a description for sketching the graph without using technology.

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## Part 1

### (1) $$ p(r) = -2(r - 1)(r + 2)(r - 2) $$

#### **1. Degree**
- **Definition:** The degree of a polynomial is the highest power of the variable $$ r $$ when the polynomial is expanded.
- **Analysis:** The polynomial is presented in factored form with three linear factors $$(r - 1)$$, $$(r + 2)$$, and $$(r - 2)$$.
- **Degree:** Since there are three factors, each of degree 1, the total degree is $$ 1 + 1 + 1 = 3 $$.

#### **2. Leading Coefficient**
- **Definition:** The leading coefficient is the coefficient of the term with the highest power.
- **Analysis:** The polynomial is $$ p(r) = -2(r - 1)(r + 2)(r - 2) $$. When expanded, the highest degree term comes from multiplying the leading terms of each factor:
  $$
  -2 \times r \times r \times r = -2r^3
  $$
- **Leading Coefficient:** $$ -2 $$

#### **3. Zeros**
- **Definition:** Zeros (or roots) are the values of $$ r $$ that make $$ p(r) = 0 $$.
- **Analysis:** Set each factor equal to zero:
  $$
  r - 1 = 0 \Rightarrow r = 1
  $$
  $$
  r + 2 = 0 \Rightarrow r = -2
  $$
  $$
  r - 2 = 0 \Rightarrow r = 2
  $$
- **Zeros:** $$ r = 1 $$, $$ r = -2 $$, and $$ r = 2 $$

#### **4. End Behavior**
- **Definition:** Describes how the graph behaves as $$ r $$ approaches positive or negative infinity.
- **Analysis:** For a cubic polynomial with a negative leading coefficient:
  - As $$ r \to +\infty $$, $$ p(r) \to -\infty $$
  - As $$ r \to -\infty $$, $$ p(r) \to +\infty $$
- **End Behavior:**
  - **Right End ( $$ r \to +\infty $$ ):** Downward
  - **Left End ( $$ r \to -\infty $$ ):** Upward

#### **5. Sketch Description**
- **Shape:** Since it's a cubic function, the graph will have an "S" shape.
- **Crossing Points:** The graph crosses the $$ r $$-axis at $$ r = -2 $$, $$ r = 1 $$, and $$ r = 2 $$.
- **Behavior:**
  - Starts from the top left ( $$ r \to -\infty $$, $$ p(r) \to +\infty $$ )
  - Descends to cross at $$ r = -2 $$
  - Rises to a local maximum between $$ r = -2 $$ and $$ r = 1 $$
  - Descends to cross at $$ r = 1 $$
  - Rises to a local minimum between $$ r = 1 $$ and $$ r = 2 $$
  - Finally, descends to cross at $$ r = 2 $$ and continues downward to the right.

---

### (2) $$ f(t) = 3t(t - 3)(t + 4) $$

#### **1. Degree**
- **Analysis:** The polynomial has three factors: $$ t $$, $$ (t - 3) $$, and $$ (t + 4) $$.
- **Degree:** Each factor is linear (degree 1), so total degree is $$ 1 + 1 + 1 = 3 $$.

#### **2. Leading Coefficient**
- **Analysis:** Multiply the leading terms of each factor with the coefficient outside:
  $$
  3 \times t \times t \times t = 3t^3
  $$
- **Leading Coefficient:** $$ 3 $$

#### **3. Zeros**
- **Analysis:** Set each factor equal to zero:
  $$
  t = 0
  $$
  $$
  t - 3 = 0 \Rightarrow t = 3
  $$
  $$
  t + 4 = 0 \Rightarrow t = -4
  $$
- **Zeros:** $$ t = 0 $$, $$ t = 3 $$, and $$ t = -4 $$

#### **4. End Behavior**
- **Analysis:** For a cubic polynomial with a positive leading coefficient:
  - As $$ t \to +\infty $$, $$ f(t) \to +\infty $$
  - As $$ t \to -\infty $$, $$ f(t) \to -\infty $$
- **End Behavior:**
  - **Right End ( $$ t \to +\infty $$ ):** Upward
  - **Left End ( $$ t \to -\infty $$ ):** Downward

#### **5. Sketch Description**
- **Shape:** Similar to other cubic functions with an "S" shape.
- **Crossing Points:** The graph crosses the $$ t $$-axis at $$ t = -4 $$, $$ t = 0 $$, and $$ t = 3 $$.
- **Behavior:**
  - Starts from the bottom left ( $$ t \to -\infty $$, $$ f(t) \to -\infty $$ )
  - Rises to cross at $$ t = -4 $$
  - Continues rising to a local maximum between $$ t = -4 $$ and $$ t = 0 $$
  - Descends to cross at $$ t = 0 $$
  - Continues descending to a local minimum between $$ t = 0 $$ and $$ t = 3 $$
  - Finally, rises to cross at $$ t = 3 $$ and continues upward to the right.

---

## Summary Table

| Function          | Degree | Leading Coefficient | Zeros           | End Behavior                                   |
|-------------------|--------|---------------------|-----------------|------------------------------------------------|
| $$ p(r) $$        | 3      | -2                  | r = -2, 1, 2     | Left: ↑, Right: ↓                              |
| $$ f(t) $$        | 3      | 3                   | t = -4, 0, 3     | Left: ↓, Right: ↑                              |

---

## Additional Tips for Sketching

1. **Plot the Zeros:** Mark the points where the graph crosses the axis.
2. **Determine Multiplicities:** Since all factors are to the first power, each zero has a multiplicity of 1, meaning the graph crosses the axis at each zero.
3. **Identify Turning Points:** A cubic function can have up to two turning points (local maximum and minimum). Estimate their positions based on the graph's progression.
4. **Draw the End Behavior Arrows:** Indicate the direction in which the graph heads as $$ r $$ or $$ t $$ approaches infinity.
5. **Connect the Points Smoothly:** Ensure the "S" shape characteristic of cubic polynomials is reflected.

---

By following these steps, you can accurately determine the key characteristics of each polynomial function and sketch their graphs without the need for technology.
**Polynomial Function Analysis** 1. **Degree and Leading Coefficient:** - **$$ p(r) = -2(r - 1)(r + 2)(r - 2) $$:** - Degree: 3 - Leading Coefficient: -2 - **$$ f(t) = 3t(t - 3)(t + 4) $$:** - Degree: 3 - Leading Coefficient: 3 2. **Zeros:** - **$$ p(r) $$:** $$ r = -2, 1, 2 $$ - **$$ f(t) $$:** $$ t = -4, 0, 3 $$ 3. **End Behavior:** - **$$ p(r) $$:** - As $$ r \to +\infty $$: Downward - As $$ r \to -\infty $$: Upward - **$$ f(t) $$:** - As $$ t \to +\infty $$: Upward - As $$ t \to -\infty $$: Downward 4. **Sketch Description:** - Both functions are cubic with an "S" shape. - **$$ p(r) $$:** - Crosses at $$ r = -2, 1, 2 $$ - Starts from the top left, descends, rises, descends, and continues downward. - **$$ f(t) $$:** - Crosses at $$ t = -4, 0, 3 $$ - Starts from the bottom left, rises, descends, rises, and continues upward. **Summary:** - Both polynomials are degree 3 with negative and positive leading coefficients, resulting in different end behaviors. - They share the same number of zeros but differ in their specific values. - The sketches will reflect the "S" shape with the respective end behaviors and crossing points.

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