Multiple Transformations of Functions

[saucybadimo]

View attachment 9341

I have to transform the first function which is f(x)=x^3 to the second function. First, I have to find each shift then combine those to make a new function equation. I've used desmos and I know that there is a horizontal shift 3 units to the right. There is a vertical shift up but I don't know how many units. And I believe there is a stretch. There are only 3 transformations. PLEASE HELP!

 

[skeeter]

note the function center point $(0,0)$ is translated to $(3,1)$, a horizontal shift right 3 units and a vertical shift up 1 unit.

taking into account the horizontal & vertical shifts, we have ...

$f(x) = a(x-3)^3 + 1$

... where $a$ is the constant causing the stretch

using the point $(4,1.5)$, can you determine the value of $a$ ?