The transformed function is:
[tex]g(x) = ( (x + 1)/3 - 2)^2 + 2*( (x + 1)/3 - 2)[/tex]
The parent function is:
[tex]f(x) = x^2 + 2x[/tex]
Here the axis of symmetry is at:
[tex]x = -2/2*1 = -1[/tex]
So first we need to apply a horizontal compression by a factor of 3 with respect to the line x = -1.
Here we can, for the moment, define a new variable that is zero when x = -1, let's define:
z = x + 1.
x = z - 1
Writing our function in terms of z, we get:
[tex]f(z) = (z - 1)^2 + 2*(z - 1)[/tex]
Now we can apply a compression by a factor of 3 around the origin. Then we have:
[tex]f(z/3) = (z/3 - 1)^2 + 2*(z/3 - 1)[/tex]
Returning to the original variable, we have:
[tex]f((x+1)/3) = ( (x + 1)/3 - 1)^2 + 2*( (x + 1)/3 - 1)[/tex]
Now we want to shift it one unit to the right, then we have:
g(x) = f( (x + 1)/3 - 1)
Replacing the actual function we get:
[tex]g(x) = f((x+1)/3 - 1) = ( (x + 1)/3 - 1 - 1)^2 + 2*( (x + 1)/3 - 1 - 1)\\\\g(x) = ( (x + 1)/3 - 2)^2 + 2*( (x + 1)/3 - 2)[/tex]
If you want to learn more about transformations:
https://brainly.com/question/4289712
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