Remember that a reflection across the y-axis affects only the x-variable.
g(x)=log_()12 (- x+3)+2
Practice makes perfect
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
Reflections
In the x-axis
y=- f(x)
In the y-axis
y=f(- x)
The transformations given in the exercise were a translation 3 units to the left and 2 units up, followed by a reflection across the y-axis.
Using the table, we can write these as transformations of f.
g(x)=f(- x+ 3)+ 2
Using the rule for f, let's calculate f(- x+3).
f(x)=log_()12 x ⇒ f(- x+ 3)=log_()12 (- x+ 3)
Finally, to find the rule for g, let's substitute the above formula in g(x)=f(- x+3)+2.
