Question

In Exercises 33–38, a. Use the Leading Coefficient Test to determine the graph’s end behavior. b. Determine whether the graph has y@axis symmetry, origin symmetry, or neither. c. Graph the function. $$ f(x)=3 x^{4}-15 x^{3} $$

Name: Problem 38, Chapter 3: Precalculus
Uploaded: 2020-06-12T20:20:52-08:00
Duration: 2 min 1 s
Channel: Karly Williams
Description: In Exercises 33–38, a. Use the Leading Coefficient Test to determine the graph’s end behavior. b. Determine whether the graph has y@axis symmetry, origin symmetry, or neither. c. Graph the function. f(x)=3 x^4-15 x^3

   In Exercises 33–38,
a. Use the Leading Coefficient Test to determine the graph’s end behavior.
b. Determine whether the graph has y@axis symmetry, origin symmetry, or neither.
c. Graph the function.
$$
f(x)=3 x^{4}-15 x^{3}
$$

Precalculus

Robert Blitzer 5th Edition

Chapter 3, Problem 38

Instant Answer

Solved by Expert Karly Williams

06/12/2020

Step 1

The Leading Coefficient Test examines the highest degree term in a polynomial to predict the behavior of the function as $ x $ approaches positive or negative infinity. For the function $ f(x) = 3x^4 - 15x^3 $, the highest degree term is $ 3x^4 $. Since the Show more…

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In Exercises 33–38, a. Use the Leading Coefficient Test to determine the graph’s end behavior. b. Determine whether the graph has y@axis symmetry, origin symmetry, or neither. c. Graph the function. $$ f(x)=3 x^{4}-15 x^{3} $$

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Key Concepts

Leading Coefficient Test

This concept involves determining the end behavior of a polynomial function by focusing on the term with the highest power. The sign and the degree of this term indicate whether the graph rises or falls as x approaches positive or negative infinity.

Graph Symmetry

This concept involves testing a function for symmetry about the y-axis or the origin. For y-axis symmetry, the function must satisfy f(-x) = f(x) (commonly associated with even functions), while for origin symmetry, the function must satisfy f(-x) = -f(x) (a characteristic of odd functions).

Graphing Polynomial Functions

This concept entails constructing a visual representation of a polynomial function by identifying key characteristics such as intercepts, turning points, and end behavior. Techniques like the Leading Coefficient Test and symmetry analysis are used to outline the overall shape of the graph before plotting specific points.

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