Problem 58 - College Algebra

Question

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function \(f\). $$ g(x)=f(2 x) $$

Step-by-Step Solution

Answer

The graph of \(g(x) = f(2x)\) is a horizontal compression by a factor of \(\frac{1}{2}\).

Step 1: Recognize the Transformation Type

The given function is \( g(x) = f(2x) \), which represents a horizontal transformation of the original function \( f(x) \). This transformation is known as a horizontal compression or horizontal shrink.

Step 2: Understand Horizontal Compression

A horizontal compression occurs when the input variable \( x \) is multiplied by a factor greater than 1. In the function \( g(x) = f(2x) \), the term \( 2x \) means that the \( x \)-values are divided by 2, effectively compressing the graph horizontally by a factor of \( \frac{1}{2} \).

Step 3: Describe the Effect on the Graph

Each point \((x, f(x))\) on the graph of \(f(x)\) will be transformed to the point \((x/2, f(x/2))\) on the graph of \(g(x)\). This means that the graph of \(g(x)\) will appear to be squished in the horizontal direction towards the \(y\)-axis.

Key Concepts

Horizontal Compression

Horizontal compression is a type of transformation that affects the width of a graph along the horizontal axis. Imagine taking a rubber band stretched from left to right; when you squeeze the ends closer together, that's similar to what's happening with horizontal compression.
In the context of functions, when you see something like \( g(x) = f(2x) \), it's an indication that a horizontal compression is applied. Here, the input \( x \) is multiplied by 2, signaling that each \( x \)-value in the function \( f(x) \) is divided by 2 in \( g(x) \).

• If the number in front of \( x \) is greater than 1, the graph becomes compressed.
• If the number is less than 1, the graph would be stretched instead.

So, for \( g(x) = f(2x) \), the graph of the original function becomes twice as narrow in comparison, continuing to retain its original shape.

Graph Transformation

Graph transformations are operations that move or change the shape of a function's graph in some way. These transformations can be either vertical or horizontal, affecting either the length or width.
We are focusing on horizontal transformations, such as what happens with \( g(x) = f(2x) \). Here are some key points:

• A horizontal compression effectively reduces the distance between points along the \( x \)-axis.
• The graph looks as if it is squeezed towards the \( y \)-axis.

This means that if you start with a graph of \( f(x) \) and then apply the transformation to get \( g(x) = f(2x) \), you will see the graph getting tightly compressed along the horizontal axis, maintaining its height but getting narrower.

Function Transformation

Function transformation involves changing a function's graph through various manipulations. This includes shifting, stretching, compressing, or reflecting. In the example \( g(x) = f(2x) \), we specifically look at the effect of compressions on the function.
Understanding how transformations work helps in predicting and sketching the new graph after the transformation:

• Each point on \( f(x) \) shifts to a new position based on the transformation rule.
• For \( g(x) = f(2x) \), the point \((x, f(x))\) shifts to \((x/2, f(x/2))\).

This compression doesn't affect the function's range but condenses the domain, effectively shaping the graph into a narrower form while preserving the overall qualities like peaks and valleys.