Graphing Modulus Functions
The modulus function, also known as the absolute value function, is defined as:
$$|x| = \begin{cases} x & \text{ if } x \geq 0 \\ -x & \text{ if } x< 0 \end{cases} $$
This function changes the shape of graphs when applied to various functions.
y = |f(x)|
When the modulus is applied to a function f(x), it reflects the negative part of the function above the x-axis.
ExampleConsider the function y = x^2 - 4. The graph of y = |x^2 - 4| will look like a parabola, but the part below the x-axis will be reflected above it.
TipTo sketch y = |f(x)|, first graph f(x) normally, then reflect any parts below the x-axis above it.
y = f(|x|)
In this case, the modulus is applied to the input of the function. This results in the right side of the graph being reflected to the left side, creating an even function.
ExampleFor y = √(|x|), the left side of the square root function is reflected to create a V-shape.
NoteFunctions of the form y = f(|x|) always have symmetry about the y-axis.
