Analyze the graph of the following function as follows:`f(x)=-(x-1)(x+2)^3`

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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` `...

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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:

 

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