How do you sketch the general shape of f(x)=x^3-x^2+4f(x)=x3−x2+4 using end behavior?

1 Answer
Tony B
Nov 1, 2016

See Explanation

Explanation:

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

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

x^2>0 " so "-x^2<0x2>0 so −x2<0

In this context x<0" so "x^3<0x<0 so x3<0

But in this case

|x^3|>|x^2|" so "x^3" has the greater influence "->x^3-x^2<0∣∣x3∣∣>∣∣x2∣∣ so x3 has the greater influence →x3−x2<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

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