We want to use the function $f$ to graph $g$ and $h .$ We also want to describe the transformations from the graph of $f$ to the graphs of $g$ and $h .$ We will begin by graphing $f (x) = 3 x + 1 .$ To do it, let's use a table of values to find points on the graph of $f .$

$x$	$3 x + 1$	$f (x)$
$- 2$	$3 (- 2) + 1$	$- 5$
$0$	$3 (0) + 1$	$1$
$2$	$3 (2) + 1$	$7$

We can plot these points and connect them with a straight line to have the graph of $f (x) .$

Function $g (x)$

Now, let's look at how the function $g (x) = f (x) - 2$ differs from $f (x) .$ We will make a table of values again. This time to graph $g .$ We will use the same $x -$ values so that we can compare them.

$x$	$f (x)$	$f (x) - 2$	$g (x)$
$- 2$	$- 5$	$- 5 - 2$	$- 7$
$0$	$1$	$1 - 2$	$- 1$
$2$	$7$	$7 - 2$	$5$

If we plot these points on the same coordinate plane as $f (x),$ we can compare the two functions.

We see that each $y -$ value is being translated $2$ units down. This is a vertical translation.

Function $h (x)$

We can go through the same process with $h (x) = f (x - 2) .$

$x$	$f (x)$	$f (x - 2)$	$h (x)$
$- 2$	$- 5$	$3 (- 2 - 2) + 1$	$- 11$
$0$	$1$	$3 (0 - 2) + 1$	$- 5$
$2$	$7$	$3 (2 - 2) + 1$	$1$

If we plot these points on the same coordinate plane as $f (x),$ we can compare the two functions.

We see that each $y -$ value is being translated $2$ units to the right. This is a horizontal translation.

Function g(x)

Function h(x)

Function $g (x)$

Function $h (x)$

Exercise