Problem 58 - Beginning and Intermediate Algebra

Question

If the following transformations are performed on the graph of \(f(x)\) to obtain the graph of \(g(x),\) write the equation of \(g(x)\). \(f(x)=|x|\) is reflected about the \(x\) -axis.

Step-by-Step Solution

Answer

The equation for g(x) is \(g(x) = -|x|\), after reflecting the given function \(f(x) = |x|\) about the x-axis.

Step 1: Identify the given function

The given function is \(f(x) = |x|\).

Step 2: Reflect the function about the x-axis

Reflecting the function about the x-axis means that every y-value (output) in the function will be replaced with its negative value. In other words, if a point on the graph has coordinates (x, y), after reflecting it about the x-axis, it will have coordinates (x,-y). To find the equation g(x), we can replace every output of f(x) with its negative value. Since the output of a function is denoted as 'y', the transformation can be written as: \(y = -f(x)\)

Step 3: Write the equation for g(x)

We have y = -f(x), so to find the equation of g(x), we can substitute the given function f(x): \(g(x) = - |x|\) So, the equation for g(x) is \(g(x) = -|x|\).

Key Concepts

Graph Transformations

Graph transformations are a way to manipulate the graph of a function to produce a new graph, often by transforming its position, shape, or orientation. These transformations include translations, reflections, stretches, and compressions.

For example, when dealing with absolute value functions like \(f(x) = |x|\), graph transformations can significantly change how the graph appears without altering the basic structure of an absolute value function. The graph of \(f(x) = |x|\) is a V-shaped graph that symmetrically divides at the origin (0,0).

Graph transformations help us move, flip, or resize this V-shape in various ways. When transformations are applied are:

• Vertical Translations: Moving the graph up or down, controlled by adding a constant to the function.
• Horizontal Translations: Moving the graph left or right, achieved by adding or subtracting a constant inside the function arguments.
• Reflections: Flipping the graph over an axis.
• Stretches and Compressions: Changing the graph's steepness by multiplying the function by a constant.

Each transformation can be combined with others to create complex movements and reorientations, enabling the creation of various graph shapes based on initial functions.

Reflection About the x-Axis

A reflection about the x-axis is a specific type of graph transformation where the graph of a function is flipped over the x-axis. This transformation alters the y-values (outputs) of the function by making them negative. Essentially, every point \((x, y)\) on the graph is transformed to the point \((x, -y)\).

This kind of reflection doesn't change the x-values of the graph. It's the y-values that switch signs. As a result, a point that was originally above the x-axis will move to a location equidistant below it, and vice versa. Regarding absolute value functions like \(f(x) = |x|\), reflecting over the x-axis changes the graph from opening upwards to opening downwards, resulting in the graph of \(g(x) = -|x|\).

The effect of this transformation can be visualized as flipping the V-shaped graph upside down, making it like an inverted V. This is particularly useful in numerous applications like physics where directions or signals are often inverted for better interpretation.

Function Notation

Function notation is a concise and standardized way to represent functions in mathematics. It is expressed in the form \(f(x)\), where \(f\) denotes the function and \(x\) represents the input variable. This notation efficiently communicates information about the relationships between the input \(x\) and the corresponding output values.

Understanding function notation is essential, especially when dealing with transformations and reflections. In our transformation process of \(f(x) = |x|\) to find \(g(x)\), notation simplifies explanations. Here, \(g(x)\) signifies a new function derived from \(f(x)\). By applying a transformation or reflection, we write \(g(x) = -f(x)\) to show that the new function \(g\) results by taking the negative of \(f(x)\).

This method simplifies changing the forms and behavior of functions based on defined operations. For this problem, recognizing how reflections affect function graphs leads to quickly writing \(g(x) = -|x|\). Consequently, understanding function notation allows you to accurately convey changes or manipulations performed on the function.