Let's recall how to write each of the transformations of the graph of $f (x) .$

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	$Horizontal stretch, 0 < b < 1 y = f (b x)$
Horizontal Stretch or Compression	$Horizontal compression, b > 1 y = f (b x)$
Reflections	$In the x - axis y = - f (x)$
Reflections	$In the y - axis y = f (- x)$

Let's create $g (x)$ by applying these three transformations one at a time.

First, we choose

h .

Since we have a translation to the right, we need to choose a positive number for

h .

Let's choose

1,

since it is one of the options we have.

g (x) = f (x - 1)

Next, we need to choose

a

which is going to multiply all the outputs of

f (x - 1) .

Since this is a vertical stretch,

a

should be a positive number greater than

1 .

Let's choose

3,

as it is one of the options we have.

g (x) = 3 \times f (x - 1)

Finally, we add

k .

Since we need a translation downwards, we have to add a negative number. Let's choose

- 3,

as it is one of the options.

g (x) = 3 \times f (x - 1) - 3

We know that

f (x) = x,

so let's write

g (x) .

g (x) = 3 \times (x - 1) - 3

Exercise