Visualizing Square Root and Absolute Value

square root

absolute value

constant a

constant b

function

linear

trig

quadratic

Taking the absolute value of a number leaves the number unchanged if the number is non-negative and changes the sign of the number otherwise. Thus, when you take the absolute value of a function, the effect on the graph is to leave portions above the

x

axis unchanged and to reflect portions below the

x

axis through the

x

axis.

You can only take (real) square roots of non-negative numbers, so when you take the square root of a function, the portion of the graph below the

x

axis disappears, while the portion above the

x

axis is distorted by the taking of square roots.

In this Demonstration, you can take the square root or absolute value of a function and see the effect. The original function is shown as a purple, dashed curve. If both boxes are checked, the graph shown will be that of

|x|

, because

|

x

|

is the same as

x

for real-valued

x

.