We want to graph the given exponential function as a transformation of its parent function.

g (x) = 4^{0.5 x - 5}

Notice that the function is of the form $g (x) = 4^{a x - h},$ where $a = 0.5,$ or $\frac{1}{2},$ and $h = 5 .$ Therefore, the graph of $g$ is a translation $5$ units to the right, followed by a horizontal stretch by a factor of $2$ of the graph of its parent function. Let's show the transformations one at a time.

Translation

We will first consider the function $y = 4^{x - 5} .$ This is a horizontal translation of $f (x) = 4^{x}$ to the right by $5$ units.

Horizontal Stretch

Be aware that $0.5 = \frac{1}{2} .$ Therefore, if in $y = 4^{x - 5}$ we multiply the $x -$ variable by $0.5,$ we obtain a horizontal stretch by a factor of $2 .$ The resulting function is $g (x) = 4^{0.5 x - 5} .$

Extra

Possible Transformations

The following table illustrates the general form for all possible transformations of functions.

Transformations of $f (x)$
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 Translations	$Translation right h units, h > 0 y = f (x - h)$
Horizontal Translations	$Translation left h units, h > 0 y = f (x + h)$
Vertical Stretch or Compression	$Vertical stretch, a > 1 y = a f (x)$
Vertical Stretch or Compression	$Vertical compression, 0 < a < 1 y = a f (x)$
Horizontal Stretch or Compression	$Horizontal stretch, b > 1 y = f (\frac{1}{b} x)$
Horizontal Stretch or Compression	$Horizontal compression, 0 < b < 1 y = f (\frac{1}{b} x)$
Reflections	$In the x - axis y = - f (x)$
Reflections	$In the y - axis y = f (- x)$

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