Question

For `y = 2(2/3)^x`, what is the y-intercept, growth rate, horizontal asymptote, domain and range?

Answer 1

See below.

Explanation:

`y=2(2/3)^x`

`y` axis intercept occurs where `x =0`

`y=2(2/3)^(0)=2`

y axis intercept at `color(blue)(( 0,2)`

as `x->oo` , `y=2(2/3)^x->0`

as `x->-oo` , `y=2(2/3)^x->oo`

The `x` axis is a horizontal asymptote.

There are no restrictions on `x` so domain is:

`color(blue)({x in RR })`

Range is:

`color(blue)({y in RR: 0< y < oo})`

`2/3<1` so this function represents decay:

This is represented by:

`y=a(1-r)^x`

Where `r` is the decay rate, which can be expressed as a decimal percentage.

From example

`1-r=2/3`

`r=1/3` or 33.3% as a decimal this is 0.333.

Decay rate 0.333 percent.

We can test this.

`y=2(2/3)^(1)=4/3`

`y=2(2/3)^(2)=8/9`

`(8/9)/(4/3)xx100%=66.6%`

This is a decay of `33.3%`

GRAPH:

graph{y=2(2/3)^x [-16.02, 16.02, -8.01, 8.01]}