Question

How do you graph `y>2^x-4`?

Answer 1

Plot `y=2^x-4` but with a dotted line.

The dotted line signifies that the value of y can never actually be `2^x-4`

Explanation:

`color(blue)("Determine the key points")`

When `x=0" "ul("the dotted line")` goes through `y=1-4=-3`

When `y=0" "ul("the dotted line")` goes through `2^x=4=>x=2`

As `x` increases a lot `2^x` increases a lot and the influence of -4 from `y>2^x-4` becomes less and less until it becomes isignificant.

`" "lim_(x-> +oo) y >lim_(x->oo)2^x -> oo`

As `x` decreases it becomes negative so we have

`y>2^(-x)-4" "->" "y>1/(2^(+x))-4` giving:

`" "lim_(x->-oo) y> lim_(x->oo) 1/(2^x) -4 ->-4`