Question

How do you graph `y=2xx3^(x-2)-1`?

Answer 1

See explanation

Explanation:

`color(blue)("Determine the x-intercept")`

Set `y=0=2(3^(x-2))-1`

`1=2(3^(x-2))`

`3^(x-2)=1/2`

Lake logs

`(x-2)ln(3)=ln(1/2)`

`x=(ln(1)-ln(2))/ln(3) +2`

`x~~1.37` to 2 decimal places
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Determine the y-intercept")`

Set `y=2(3^(0-2))-1`

`y=2/(3^2)-1 = -7/9`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Determine the extremities")`

Set `y=k" "` where `" "k=lim_(x->+oo) 2(3^(x-2))-1`

`" "k->2oo-1 = oo`

Thus for `x ->oo" we have "y->oo`

Set `y=k" "` where `" "k=lim_(x->-oo) 2(3^(x-2))-1`

`" "k->2/oo-1 = 0-1 = -1`
This is an asymptote.