A square root function is a function that has its independent variable in the radicand of a square root. We want to identify some of the characteristics of the graph of a square root function. Let's first draw the graph of the parent function $y = x .$

Graph of the Parent Function

We will first find some points on the graph. Note that $x$ must be a non-negative number because the value inside a square root cannot be negative. Therefore, we will randomly choose $x = 0,$ $x = 1,$ and $x = 4 .$

$x$	$y = x$	Point
$0$	$0 = 0$	$(0, 0)$
$1$	$1 = 1$	$(1, 1)$
$4$	$4 = 2$	$(4, 2)$

Great! Now, we can draw a smooth curve through these points.

As we can see, the graph starts from

(0, 0),

increases quickly at first, and then slows its rate of increase. Since the inside of a square root cannot be negative, the domain of the parent function must be

x \geq 0 .

This means that the expression

x

produces only non-negative values. Therefore, the range of the function is

y \geq 0 .

Domain : Range : x \geq 0 y \geq 0

Graph of $y = a x - h + k$

Now, we will examine those same characteristics for the general form of a square root function.

y = a x - h + k

Let's observe how the graph of the square root function changes when the variables

a,

h,

and

k

change.

Here are some observations from the above graph.

The parent function stretches, shrinks, or reflects by the factor of $a .$
The parent function translates horizontally based on the value of $h .$
The parent function translates vertically based on the value of $k .$

Graph of the Parent Function

Graph of y=ax−h​+k

Graph of $y = a x - h + k$

Exercise

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