Problem 54 - College Algebra

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 the reflection of \( f(x) \) across the y-axis.

Step 1: Understand the Original Function

The original function is denoted as \( f(x) \). This function could be any basic or complex function like linear, quadratic, etc. To solve this problem, the specific type of \( f(x) \) is not necessary, but understanding that it represents a graph is essential.

Step 2: Analyze the Transformation

The function \( g(x) = f(-x) \) represents a transformation of \( f(x) \). The transformation that occurs is known as a reflection over the y-axis. This means every point on the graph of \( f(x) \) that was originally at \((x, y)\) will be moved to \((-x, y)\).

Step 3: Describing the Transformation

The graph of \( g(x) \) is a mirror image of the graph of \( f(x) \) with respect to the y-axis. This indicates that if you fold the graph along the y-axis, the right side will perfectly match the left side.

Key Concepts

Reflection Over the Y-Axis

When we talk about the reflection over the y-axis, we're referring to a specific transformation that flips the graph of a function across the y-axis. Imagine holding up a mirror along the y-axis; the reflection you see in the mirror is how the transformed function would look. In mathematical terms, if you have a point \(x, y\) on the graph of a function \(f(x)\), after a reflection over the y-axis, that point moves to \(-x, y\)\.

For example, if point \(3, 5\) is on the graph of \(f(x)\), then \(-3, 5\) will be on the graph of \(g(x) = f(-x)\). Since only the x-coordinate changes its sign, the y-coordinate remains unchanged, resulting in this mirror effect. It’s like flipping the graph from side to side without altering its vertical components.

Graph of a Function

A graph of a function is a visual representation of all the possible outputs (y-values) for each input (x-value) within a given domain. This graph helps us see patterns and behaviors of the function. For example, a simple linear function \(f(x) = 2x + 3\) results in a straight line when graphed.

Key points to understand when analyzing a graph include:

• The x-axis and y-axis form the coordinate system where the graph is drawn.
• Every point on this graph corresponds to an ordered pair \(x, y\), where y is the result of applying the function to x.
• Graphs can show intercepts with the axes, slopes, curvature, and other critical characteristics that reveal how the function behaves across its domain.

Understanding these elements can help you identify transformations, as they are often visually distinct on the graph itself.

Transformation of Functions

Transformations of functions can change the position or shape of a graph in various ways. Common transformations include translations, stretches, compressions, and reflections. Each type of transformation alters the function's graph in a specific manner.

Regarding reflections, specifically, they are a type of transformation where the graph is flipped along an axis.
To recognize and describe transformations, consider the following:

• Translation: Moves the graph horizontally or vertically without changing its shape.
• Stretch/Compression: Alters the height or width of the graph by scaling it vertically or horizontally.
• Reflection: Flips the graph over a specified axis, either the x-axis, y-axis, or another line.

In our specific exercise, reflecting over the y-axis is a common transformation that alters the horizontal symmetry but keeps the vertical alignment intact. This means the original behavior of the function is preserved but appears flipped about the y-axis.