Explanation 1) Approach: For draw the graph of the polynomial, Step 1: Find the x -and y -intercepts of the graph of the polynomial. Step 2: Determine the end behavior. Step 3: Make a sign chart to determine where the graph is above and below the x -axis. Step 4: Test for symmetry about y -axis, and the origin Step 5: Find an additional point and table the co-ordinates. Step 6: Draw the graph of the polynomial function. 2) Calculation: Given, f ( x ) = − x 4 + 2 x 3 + 8 x 2 Step 1: Find the x - and y -intercepts of the graph. To find the x -intercepts, we substitute 0 for f ( x ) and solve for x . 0 = − x 4 + 2 x 3 + 8 x 2 (i.e.) − x 4 + 2 x 3 + 8 x 2 = 0 Factorize x 4 + 2 x 3 + 8 x 2 as − x 2 ( x 2 − 2 x − 8 ) By the zero factor property, − x 2 = 0 and x 2 − 2 x − 8 = 0 Taking, − x 2 = 0 Multiply − 1 on both sides, x 2 = 0 Taking, x 2 − 2 x − 8 = 0 Factorize x 2 − 2 x − 8 = 0 ( x − 4 ) ( x + 2 ) = 0 By the zero factor property, x − 4 = 0 , x + 2 = 0 Taking x − 4 = 0 Add 4 on both sides, x − 4 + 4 = 0 + 4 x = 4 Taking x + 2 = 0 Add − 2 on both sides, x + 2 − 2 = 0 − 2 x = − 2 Thus, the x -intercepts are ( − 2 , 0 ) ( 0 , 0 ) and ( 4 , 0 ) To find the y -intercept, we substitute 0 for x and solve for y . Here, f ( x ) = y y = − x 4 + 2 x 3 + 8 x 2 y = − ( 0 ) 4 + 2 ( 0 ) 3 + 8 ( 0 ) 2 y = 0 + 0 + 0 y = 0 Thus, the y -intercept is ( 0 , 0 ) . Step 2: Determine the end behavior. The degree of the function f ( x ) = − x 4 + 2 x 3 + 8 x 2 is 4 which is even, and the leading co-efficient is − 1 which is negative. The graph falls on the left and falls on the right. Step 3: Make a sign chart to determine where the graph is above and below the x -axis. We will make a sign chart to determine the behavior of the function between the x -intercepts. The zero of the polynomial function divides the x -axis into four intervals. ( − ∞ , − 2 ) , ( − 2 , 0 ) , ( 0 , 4 ) and ( 4 , ∞ ) Table the test value of x from each interval and determine the sign. f ( x ) = − x 4 + 2 x 3 + 8 x 2 Interval Test point of f ( x ) Sign of f ( x ) Graph f ( x ) ( − ∞ , − 2 ) f ( − 3 ) = − ( − 3 ) 4 + 2 ( − 3 ) 3 + 8 ( − 3 ) 2 = − 81 + 2 ( − 27 ) + 8 ( 9 ) = − 81 − 54 + 72 f ( − 3 ) = − <

WebAssign with Corequisite Support for Gustafson/Hughes' College Algebra, 12th Edition [Instant Access], Single-Term

12th Edition

ISBN: 9780357422533

Author: Gustafson; R. David; Hughes; Jeff

Publisher: Cengage Learning US

See similar textbooks

Concept explainers

Transformation of Graphs

The word ‘transformation’ means modification. Transformation of the graph of a function is a process by which we modify or change the original graph and make a new graph.

Exponential Functions

The exponential function is a type of mathematical function which is used in real-world contexts. It helps to find out the exponential decay model or exponential growth model, in mathematical models. In this topic, we will understand descriptive rules, c…

Videos

Question

Chapter 4.2, Problem 64E

To determine

To graph:

The polynomial function

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.2, Problem 63E

Chapter 4.2, Problem 65E

Chapter 4 Solutions

WebAssign with Corequisite Support for Gustafson/Hughes' College Algebra, 12th Edition [Instant Access], Single-Term

Ch. 4.1 - Find the vertex of the graph of the quadratic...Ch. 4.1 - Prob. 2SC Ch. 4.1 - Prob. 3SC Ch. 4.1 - Prob. 4SC Ch. 4.1 - Find the largest area possible if the association...Ch. 4.1 - A company that makes and sells baseball caps has...Ch. 4.1 - Fill in the blanks. A quadratic function is...Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E

Knowledge Booster

$Algebra$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.

To graph: The polynomial function

Solution Summary: The author explains how the polynomial f(x) is graphed.