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#

2 Answers
Dec 10, 2015

#(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)#

Dec 11, 2015

#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]}