Let's start by recalling the vertex form of a quadratic function.

f (x) = a (x - h)^{2} + k

In this form the vertex of the parabola is the point

(h, k),

and the axis of symmetry is the vertical line

x = h .

Now consider the given function.

g (x) = x^{2} - 5 ⇕ g (x) = (x - 0)^{2} + (- 5)

We can see that

h = 0

and that

k = - 5 .

Therefore, the vertex is

(0, - 5),

and the axis of symmetry is

x = 0 .

To graph the function we will make a table of values. Make sure to include

x -

values to the left and to the right of the axis of symmetry.

$x$	$x^{2} - 5$	$g (x) = x^{2} - 5$
$- 2$	$(- 2)^{2} - 5$	$- 1$
$- 1$	$(- 1)^{2} - 5$	$- 4$
$1$	$1^{2} - 5$	$- 4$
$2$	$2^{2} - 5$	$- 1$

Let's now draw the parabola that connects the obtained points and the vertex. We will also draw the axis of symmetry $x = 0,$ and the parent function $f (x) = x^{2} .$

From the graph above, we can note the following.

Both graphs open up.
Both graphs have the same axis of symmetry, the $y -$ axis.
The vertex of the given function, $(0, - 5),$ is below the vertex of the parent function, $(0, 0) .$

From the graph and the observations above, we can conclude that the graph of $g$ is a vertical translation down $5$ units of the graph of $f .$

Exercise