Question

Given a function $f(x)$, describe the graph transformation of function $\left| {f(x)} \right|$.

Answer 1

In the given problem, we are required to transform the graph of $f(x)$ to $\left| {f(x)} \right|$ using the graph transformation methods. There are various methods and logics to do the given problem. The best method is to go by the basic definition of modulus function, interpret the function as a composite function and plot the graph of $\left| {f(x)} \right|$.

Complete step by step answer:
In the given question, we have to trace the graph of $\left| {f(x)} \right|$, given the graph of $f(x)$.

So, firstly we have to understand the basic definition of modulus function.

Now, The definition of $\left| x \right|$ is \left| x \right| = \left\\{ \left( x \right);x \geqslant 0 \\\ \left( { - x} \right);x < 0 \\\ \right\\}.

Hence, the function $\left| {f(x)} \right|$ can defined as: \left| {f(x)} \right| = \left\\{ \left( {f(x)} \right);f(x) \geqslant 0 \\\ \- \left( {f(x)} \right);f(x) < 0 \\\ \right\\}.

So, to transform the graph of $f(x)$ into $\left| {f(x)} \right|$ and given the function $f(x)$ we have to first identify the range where the value of function $f(x)$ is positive and negative.

So, the range of values where the function $f(x)$ acquires positive value would be left unchanged and the range of values where the function $f(x)$ acquires negative value, the graph of $f(x)$ would be transformed by reflecting the graph about the x-axis to get the graph of $\left| {f(x)} \right|$.
Basically, we have to just reflect the graph of $f(x)$ above the x-axis.

So, the graph of $\left| {f(x)} \right|$ is just the reflection of the graph of $f(x)$ above the x-axis.

Note: Given question involves graphical transformations. It requires knowledge of composite functions and the basic definition of modulus function as well. When a modulus function is applied in front of a function, the function acquires positive value for any input value.