Question

Describe a sequence of transformations that transform the graph of f(x) into the graph of g(x)? `f(x)=sqrtx` and `g(x)=-3(sqrt(x+1))-4`

Answer 1

`(x, f(x)) rightarrow (x - 1, sqrt x) rightarrow (x - 1, -3 sqrt(x)) rightarrow (x - 1, g(x-1))`

Explanation:

1) Translation. `x' := x - 1 ; y' = y`.
Just send `(x,sqrt(x)) rightarrow (x - 1, sqrt(x))` therefore `(0, 0) rightarrow (-1, 0)`
And `(1, 1) rightarrow (0, 1)`

2) Stretch. `x'' = x' ; y'' := -3y'`
Just send `(x', y') rightarrow (x', -3y')` therefore `(0, 1) rightarrow (0, -3)`
This is a dilatation (multiply by 3)
followed by a refraction (multiply by -1). The mirror is x-axis.

3) Translation. `x''' := x'' ; y''' := y'' - 4`.
Just send `(x'', y'') rightarrow (x'', y'' - 4)`
Therefore `(-1,0) rightarrow (-1, -4)`
And `(0, -3) rightarrow (0, -7)`

Answer 2

`f(x)=sqrtx`
`(16,4)`
graph{sqrtx [-1.705, 18.295, -3.44, 6.56]}

`a(x)=sqrt(xcolor(red)(+1))`
Function shifts one to the left.
`(15,4)`
graph{sqrt(x+1) [-2.16, 17.84, -2.08, 7.92]}

`b(x)=color(red)3(sqrt(x+1))`
Function is stretched vertically by a factor of `"3"`.
`(15,12)`
graph{3sqrt(x+1) [-2.08, 33.48, -2.43, 15.35]}

`c(x)=color(red)-3(sqrt(x+1))`
Function is reflected across the `x`-axis.
`(15,-12)`
graph{-3sqrt(x+1) [-3.42, 32.14, -15.24, 2.54]}

`g(x)=-3(sqrt(x+1))color(red)(-4)`
Function is moved `"4"` units down.
`(15,-16)`
graph{-3sqrt(x+1)-4 [-4.75, 35.25, -16.85, 3.15]}