We want to write a rule for g that represents the indicated transformations of the graph of f. To do so, we will look at how the indicated transformations will affect the parent function. Then we can apply them.
Transformations of f(x)
Horizontal Translations
Translation right h units, h>0
y=f(x- h)
Translation left h units, h>0
y=f(x+ h)
Vertical Translations
Translation up k units, k>0
y=f(x)+ k
Translation down k units, k>0
y=f(x)- k
Horizontal Stretch or Shrink
Horizontal stretch, b>1
y=f(1/bx)
Horizontal shrink, 0< b<1
y=f(1/bx)
The transformations given in the exercise were a translation 3 units to the right and 1 unit up, followed by a horizontal stretch by a factor of 8. Using the table, we can write these as transformations of f. Remember that in order to perform a horizontal stretch by 8 we need to multiply the x-variable by 1 8.
g(x)=f( 1 8x- 3)+ 1
Using the rule for f, let's calculate f( 18x-3).
f(x)=ln x ⇒ f( 1 8x- 3)=ln ( 1 8x- 3)
Finally, to find the rule for g we will substitute the above formula in g(x)=f( 18x-3)+1.
