Strategy -Graphing a Polynomial Function 1. Check for symmetry of any type. a. f(-x) = f(x) => symmetric about the y-axis b. f(-x) = -f(x) => symmetric about the origin c. quadratic function => symmetric about x = 2. Find all real zeros of the polynomial function. (where does it cross the x-axis) 3. Determine the behavior at the corresponding x-intercepts. 4. Determine the behavior as x -> 00 and as x -> -00. look at the degree of the polynomial and the leading coefficient b. if degree is odd, the graph approaches opposite oo if degree is even, the graph approaches the same oo 5. Calculate several ordered pairs including the y-intercept to verify your suspicions about the slope of the graph. 6. Draw a smooth curve through the points to make the graph. a. C. Graph the following function: f(x) = 6x – x³ – 25x² +4x+4
Strategy -Graphing a Polynomial Function 1. Check for symmetry of any type. a. f(-x) = f(x) => symmetric about the y-axis b. f(-x) = -f(x) => symmetric about the origin c. quadratic function => symmetric about x = 2. Find all real zeros of the polynomial function. (where does it cross the x-axis) 3. Determine the behavior at the corresponding x-intercepts. 4. Determine the behavior as x -> 00 and as x -> -00. look at the degree of the polynomial and the leading coefficient b. if degree is odd, the graph approaches opposite oo if degree is even, the graph approaches the same oo 5. Calculate several ordered pairs including the y-intercept to verify your suspicions about the slope of the graph. 6. Draw a smooth curve through the points to make the graph. a. C. Graph the following function: f(x) = 6x – x³ – 25x² +4x+4
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Strategy - Graphing a Polynomial Function**
1. **Check for symmetry of any type.**
- \( f(-x) = f(x) \) ⇒ symmetric about the y-axis
- \( f(-x) = -f(x) \) ⇒ symmetric about the origin
- Quadratic function ⇒ symmetric about \( x = -b/(2a) \)
2. **Find all real zeros of the polynomial function** (where does it cross the x-axis).
3. **Determine the behavior at the corresponding x-intercepts.**
4. **Determine the behavior as \( x \to \infty \) and as \( x \to -\infty \).**
- a. Look at the degree of the polynomial and the leading coefficient.
- b. If the degree is odd, the graph approaches opposite \(\infty\).
- c. If the degree is even, the graph approaches the same \(\infty\).
5. **Calculate several ordered pairs including the y-intercept to verify your suspicions about the slope of the graph.**
6. **Draw a smooth curve through the points to make the graph.**
---
**Graph the following function:**
\( f(x) = 6x^4 - x^3 - 25x^2 + 4x + 4 \)
---
**Graph Description:**
The graph is presented on a grid with axes ranging from -10 to 10 on both the x and y axes. The units are marked and divided into a decimal system for accurate plotting. Use this grid to plot the described polynomial function by following the six-step strategy mentioned above.
The function \( f(x) = 6x^4 - x^3 - 25x^2 + 4x + 4 \) is a fourth-degree polynomial, which means its behavior at \( x \to \infty \) and \( x \to -\infty \) will approach the same infinity, due to the even degree. Start by finding real zeros and plot the intercepts, then sketch the curve by determining the behavior and connecting the points smoothly.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3899185f-2e74-4a69-b98e-bc78fbd80797%2F4ec39b49-d20c-47af-8bab-d3c6f09711bf%2Fykztb9h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Strategy - Graphing a Polynomial Function**
1. **Check for symmetry of any type.**
- \( f(-x) = f(x) \) ⇒ symmetric about the y-axis
- \( f(-x) = -f(x) \) ⇒ symmetric about the origin
- Quadratic function ⇒ symmetric about \( x = -b/(2a) \)
2. **Find all real zeros of the polynomial function** (where does it cross the x-axis).
3. **Determine the behavior at the corresponding x-intercepts.**
4. **Determine the behavior as \( x \to \infty \) and as \( x \to -\infty \).**
- a. Look at the degree of the polynomial and the leading coefficient.
- b. If the degree is odd, the graph approaches opposite \(\infty\).
- c. If the degree is even, the graph approaches the same \(\infty\).
5. **Calculate several ordered pairs including the y-intercept to verify your suspicions about the slope of the graph.**
6. **Draw a smooth curve through the points to make the graph.**
---
**Graph the following function:**
\( f(x) = 6x^4 - x^3 - 25x^2 + 4x + 4 \)
---
**Graph Description:**
The graph is presented on a grid with axes ranging from -10 to 10 on both the x and y axes. The units are marked and divided into a decimal system for accurate plotting. Use this grid to plot the described polynomial function by following the six-step strategy mentioned above.
The function \( f(x) = 6x^4 - x^3 - 25x^2 + 4x + 4 \) is a fourth-degree polynomial, which means its behavior at \( x \to \infty \) and \( x \to -\infty \) will approach the same infinity, due to the even degree. Start by finding real zeros and plot the intercepts, then sketch the curve by determining the behavior and connecting the points smoothly.
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