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.
Consider 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.
To 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.
For y = √(|x|), the left side of the square root function is reflected to create a V-shape.
Functions of the form y = f(|x|) always have symmetry about the y-axis.
y = 1/f(x)
While not strictly a modulus function, the reciprocal of a function is often studied alongside modulus functions due to its similar transformative properties.
The graph of y = 1/x^2 is a reflection of y = x^2 about the line y = 1, with vertical asymptotes where x^2 = 0.
Students often forget to consider the domain restrictions when graphing reciprocal functions. Remember that the reciprocal is undefined when the original function equals zero.
Solving Modulus Equations and Inequalities
Solving equations and inequalities involving modulus functions requires both analytical and graphical approaches.
Analytical Method
When solving |f(x)| = a, where a is a non-negative constant, we consider two cases:
- f(x) = a
- f(x) = -a
It may be helpful to consider squaring values here since $|x|^2 =x^2$, which may simplify the problem for you
To solve |2x - 1| = 3:
- 2x - 1 = 3 → x = 2
- 2x - 1 = -3 → x = -1 Therefore, the solutions are x = 2 or x = -1.
