Problem 54 - College Algebra with Corequisite Support

Question

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function \(f\). \(g(x)=f(-x)\)

Step-by-Step Solution

Answer

The graph of \( g(x) = f(-x) \) is a reflection of the graph of \( f(x) \) across the y-axis.

Step 1: Understand the Original Function

Consider that the original function is given as \( f(x) \). The graph of this function is based on the general behavior of \( f \), such as its shape, direction, and position on the coordinate plane.

Step 2: Recognize the Transformation

The function \( g(x) = f(-x) \) tells us that the transformation involves \( f \) applied to the negative of the input variable \( x \).

Step 3: Identify the Type of Transformation

The negative sign in front of \( x \) suggests a reflection. In this case, \( g(x) = f(-x) \) results in a reflection of the graph of \( f(x) \) across the y-axis.

Step 4: Visualize the Transformed Graph

To understand the visualization, imagine each point \((a, b)\) on the graph of \( f(x) \). In the function \( g(x) = f(-x) \), the point \((a, b)\) is transformed to \((-a, b)\). This results in a mirror image of \( f(x) \), reflected over the y-axis.

Key Concepts

Function Reflection

In mathematical graph transformations, **function reflection** occurs when every point on a graph is mirrored over a particular line. Specifically, reflecting the function can be achieved by modifying the function equation itself.
When we look at the function transformation given by \( g(x) = f(-x) \), it signifies that each input \( x \) is replaced by \(-x\). This change results in the reflection of the graph across the y-axis. The process of reflection alters the position of every point on the graph, creating a mirror-like effect.

• If a point originally located on the graph of \( f(x) \) is \((a, b)\), then after reflecting, it will move to \((-a, b)\) in the new graph of \( g(x) \).
• This transformation doesn't alter the y-values; it only influences the x-coordinates to achieve a mirrored graph.

Function reflections are crucial for visualizing and understanding the symmetry and behavior of different mathematical functions.

Y-axis Reflection

The **y-axis reflection** is a specific type of reflection transformation occurring when a graph is flipped over the y-axis. Essentially, this transformation changes the direction of the graph without distorting its original shape.
The equation \( g(x) = f(-x) \) encapsulates this transformation. Here, the negative sign before \( x \) means that every x-coordinate from the original function's graph is negated.

• Imagine a graph with points such as \((2, 3)\), \((3, 5)\), and \((4, 7)\). After reflecting over the y-axis, these points would transform to \((-2, 3)\), \((-3, 5)\), and \((-4, 7)\) respectively.
• It's important to note that during a y-axis reflection, the graph's general structure remains intact, indicating true mirror symmetry about the y-axis.

Y-axis reflections help us to find equivalencies and symmetries in functions, which can simplify complex mathematical problems.

Coordinate Plane

When discussing transformations like reflections, it's essential to understand their context on the **coordinate plane**. The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface containing a horizontal axis (x-axis) and a vertical axis (y-axis).
Every point on this plane is represented by a pair of coordinates: \((x, y)\). It's here that transformations such as reflections, translations, and rotations take place. For instance, when reflecting a graph over the y-axis, the locations of the x-coordinates change signs while the y-coordinates remain unchanged.

• The origin, denoted by \((0, 0)\), is the intersection of the x-axis and y-axis, serving as a reference point for plotting any graph or transformation.
• Reflections, like the y-axis reflection in \( g(x) = f(-x) \), illustrate how the plane coordinates are manipulated to achieve the desired effect.

Understanding the coordinate plane and its axes helps us predict the outcomes of various function transformations, making it a foundational concept in graphing mathematics.