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	$Translation right h units, h > 0 y = f (x - h)$
Horizontal Translations	$Translation left h units, h > 0 y = f (x + h)$
Vertical Translations	$Translation up k units, k > 0 y = f (x) + k$
Vertical Translations	$Translation down k units, k > 0 y = f (x) - k$
Horizontal Stretch or Shrink	$Horizontal stretch, b > 1 y = f (\frac{1}{b} x)$
Horizontal Stretch or Shrink	$Horizontal shrink, 0 < b < 1 y = f (\frac{1}{b} x)$

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

\frac{1}{8} .

g (x) = f (\frac{1}{8} x - 3) + 1

Using the rule for

f,

let's calculate

f (\frac{1}{8} x - 3) .

f (x) = ln x \Rightarrow f (\frac{1}{8} x - 3) = ln (\frac{1}{8} x - 3)

Finally, to find the rule for

g

we will substitute the above formula in

g (x) = f (\frac{1}{8} x - 3) + 1 .

g (x) = f (\frac{1}{8} x - 3) + 1

Substitute

$f (\frac{1}{8} x - 3) = l n (\frac{1}{8} x - 3)$

g (x) = l n (\frac{1}{8} x - 3) + 1

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