Question

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

Answer 1

See Explanation

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.