Question

How do you graph `y=2*8^x`? What is the `y`-intercept and the domain and range?

Answer 1

Graph as shown below

Explanation:

The first thing we can do is simplify `2*8^x`

We know `8 = 2^3`

So hence `y = 2*8^x => y= 2* (2^3)^x`

`=>y= 2*2^(3x)`

`=> y = 2^(3x+1)`

we know what `y=2^x` looks like:

graph{y = 2^x [-10.085, 9.915, -1.44, 8.56]}

But `2^(3x+1) = 2^(3(x+1/3 ))`

What is a stretch of `2^x`, by `1/3` in the x direction, and then a translation of `(-1/3,0)`

Hence we yield:

graph{2^(3x+1) [-10.21, 9.79, -1.52, 8.48]}

We can see from this graph that it has a domain of `x in RR` it is defined for all `x`

We can also see that it has a range of: `y>0` it's always positive, but never equalls `0`.

It hence has a `y`-intercept of `(0,2)` as at `x=0`: `y=2`, we can see this by simply substituing `x=0`