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1. Let f(x)=x-x-6x². Sketch a graph of the function. Label all zeros and asymptotes on the graph, if they exist.

1. Let f(x)=x-x-6x². Sketch a graph of the function. Label all zeros and asymptotes on the graph, if they exist.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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### Problem 1: Graphing a Polynomial Function **Problem Statement:** Let \( f(x) = x^4 - x^3 - 6x^2 \). Sketch a graph of the function. Label all zeros and asymptotes on the graph, if they exist. **Steps to Solve:** 1. **Determine the Zeros of the Function:** - To find the zeros of \( f(x) \), set \( f(x) = 0 \): \[ x^4 - x^3 - 6x^2 = 0 \] - Factor the equation: \[ x^2 (x^2 - x - 6) = 0 \] - Further factor \( x^2 - x - 6 \): \[ x^2 (x - 3)(x + 2) = 0 \] - The zeros are \( x = 0 \), \( x = 3 \), and \( x = -2 \). 2. **Determine the Behavior at Infinity:** - As \( x \to \infty \), the highest degree term \( x^4 \) dictates that \( f(x) \to \infty \). - As \( x \to -\infty \), \( f(x) \to \infty \) because \( x^4 \) is an even degree polynomial. 3. **Identify and Label Asymptotes:** - Polynomial functions do not have vertical or horizontal asymptotes, so there are none to label. 4. **Sketch the Graph:** - Plot the zeros: \( (0,0) \), \( (3,0) \), \( (-2,0) \). - Determine the function's behavior at these points and in between them, considering the factors and higher-order terms. By following these steps, you can create an accurate sketch of the quadratic polynomial function and appropriately label any critical points. > **Note for Educators:** > Ensure that students understand the process of finding zeros and their significance in the function. Encourage the use of graphing tools or software to visualize the function.
Transcribed Image Text:### Problem 1: Graphing a Polynomial Function **Problem Statement:** Let \( f(x) = x^4 - x^3 - 6x^2 \). Sketch a graph of the function. Label all zeros and asymptotes on the graph, if they exist. **Steps to Solve:** 1. **Determine the Zeros of the Function:** - To find the zeros of \( f(x) \), set \( f(x) = 0 \): \[ x^4 - x^3 - 6x^2 = 0 \] - Factor the equation: \[ x^2 (x^2 - x - 6) = 0 \] - Further factor \( x^2 - x - 6 \): \[ x^2 (x - 3)(x + 2) = 0 \] - The zeros are \( x = 0 \), \( x = 3 \), and \( x = -2 \). 2. **Determine the Behavior at Infinity:** - As \( x \to \infty \), the highest degree term \( x^4 \) dictates that \( f(x) \to \infty \). - As \( x \to -\infty \), \( f(x) \to \infty \) because \( x^4 \) is an even degree polynomial. 3. **Identify and Label Asymptotes:** - Polynomial functions do not have vertical or horizontal asymptotes, so there are none to label. 4. **Sketch the Graph:** - Plot the zeros: \( (0,0) \), \( (3,0) \), \( (-2,0) \). - Determine the function's behavior at these points and in between them, considering the factors and higher-order terms. By following these steps, you can create an accurate sketch of the quadratic polynomial function and appropriately label any critical points. > **Note for Educators:** > Ensure that students understand the process of finding zeros and their significance in the function. Encourage the use of graphing tools or software to visualize the function.
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