How do you sketch the general shape of #f(x)=x^3-x^2+4# using end behavior?

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Answer 1 · 2016-11-01T12:00:47

See Explanation

Consider end behaviour. That is when #x# tends to extreme negative and extreme positive for this function.

#color(brown)("Consider the case "x<0)#

#x^2>0 " so "-x^2<0#

In this context #x<0" so "x^3<0#

But in this case

#|x^3|>|x^2|" so "x^3" has the greater influence "->x^3-x^2<0#

#color(blue)(lim_(x->-oo) x^3-x^2->-oo )#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(brown)("Consider the case "x>0)#

#x^2>0 " so "-x^2<0#

In this context #x>0" so "x^3>0#

#|x^3|>|x^2|" so "x^3" has the greater influence "->x^3-x^2>0#

#color(blue)(lim_(x->+oo) x^3-x^2->+oo) #
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(brown)("General shape y-intercept point")#

y-intercept at #x=0 -> y=(0)^3-(0)^2+4=4#

The x-intercept is harder to determine but as you are only asked to sketch the 'general case' it is not important that you find it.

Tony B