Visualizing Square Root and Absolute Value

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 axis unchanged and to reflect portions below the axis through the 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 axis disappears, while the portion above the 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 , because is the same as for real-valued .

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Related Curriculum Standards

US Common Core State Standards, Mathematics