A cube root function is a type of radical function with an index of $3 .$ We want to identify some of the characteristics of the graph of a cube root function. Let's first draw the graph of the parent function $y = 3 x .$

Graph of the Parent Function

We will first find some points on the graph. Unlike a square root, the cube root of a negative number is defined. Therefore, we will randomly choose some numbers that are both positive and negative.

$x$	$y = 3 x$	Point
$- 8$	$3 - 8 = - 2$	$(- 8, - 2)$
$- 1$	$3 - 1 = - 1$	$(- 1, - 1)$
$0$	$30 = 0$	$(0, 0)$
$1$	$31 = 1$	$(1, 1)$
$8$	$38 = 2$	$(8, 2)$

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

As we can see, the domain of the parent function is all real numbers and its range is also all real numbers because there are no undefined values.

Domain : Range : All real numbers All real numbers

Moreover, the graph increases on the entire domain and it is symmetric with respect to the origin.

Graph of $y = a 3 x - h + k$

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

y = a 3 x - h + k

Let's observe how the graph of the cube 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=a3x−h​+k

Graph of $y = a 3 x - h + k$

Exercise