Analyze the graph of the following function as follows:


Expert Answers

An illustration of the letter 'A' in a speech bubbles

Given `f(x)=-(x-1)(x+2)^3` :

(1) The y-intercept is found when x=0, thus y=-(-1)(2)^3=8 and the y-intercept is 8

The x-intercept(s) are where y=0: 0=(x-1)(x+2)^3

By the zero product property, either x-1=0 and/or (x+2)^3=0. Thus the x-intercepts are x=1 and x=-2.

(2) The e` `nd behavior is `f(x)=-x^4`` ` (See graph below). Expanding the binomials and multiplying, we see that this is a polynomial with highest degree 4 and leading coefficient -1. (Expands as `-x^4-5x^3-6x^2+4x+8` )

(3) Since this is a fourth degree polynomial, the maximum number of turning points is 4-1=3.

(4) Graph:

Graphed with y=x^4 on larger scale:

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial