The graph of a radical function

A radical as you might remember is something that is under a radical sign e.g. a square root. A radical function contains a radical expression with the independent variable (usually x) in the radicand. Usually radical equations where the radical is a square root is called square root functions.

An example of a radical function would be


This is the parent square root function and its graph looks like


If we compare this to the square root function


We will notice that the graph stretches or shrinks vertically when we vary a

$$\begin{matrix} \left | a \right | >0\: \: \: \: \: \: \: & &\Rightarrow vertical\: stretch \\ 0<\left | a \right |<1 & & \Rightarrow vertical\: shrink\: \: \: \\ \end{matrix}$$

In the graph below we have radical functions with different values of a


If a < 0 the graph


Is the reflection in the x-axis of the graph

$$y=\left |a \right |\sqrt{x}$$


Another square root equation would be


If you look at the graphs above which all have c = 0 you can see that they all have a range ≥ 0 (all of the graphs start at x=0 since there are no real solutions to the square root of a negative number). If you have a c ≠ 0 you'll have a radical function that starts in (0, c). An example of this can be seen in the graph below


The value of b tells us where the domain of the radical function begins. Again if you look at the parent function it has a b = 0 and thus begin in (0, 0) If you have a b ≠ 0 then the radical function starts in (b, 0).


If both b ≠ 0 and c ≠ 0 then the radical function starts in (b, c)

Video lesson

Compare the radical functions

$$y_{1} =\sqrt{ x}$$

$$y_{2} = 3\sqrt{x}$$

$$y_{3} = \sqrt{x} + 2$$

$$y_{4} = \sqrt{x- 1}$$

$$y_{5} = \sqrt{x(x-2)} + 1$$