Problem 58 - College Algebra

Question

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 of \(f(x)\) by a factor of 2.

Step 1: Understanding the Function Transformation

The function given is \(g(x) = f(2x)\). This is a transformation of the original function \(f(x)\). It is important to identify what happens to the graph of \(f(x)\) when \(x\) is replaced by \(2x\).

Step 2: Determine the Type of Transformation

The transformation \(g(x) = f(2x)\) involves replacing \(x\) with \(2x\), which indicates a horizontal scaling of the graph of the function \(f(x)\). This is a horizontal compression.

Step 3: Calculate the Effect of Transformation

In the transformation \(g(x) = f(2x)\), each \(x\) value of the function \(f(x)\) is multiplied by \(\frac{1}{2}\). Therefore, the graph of \(f(x)\) is compressed horizontally by a factor of 2.

Key Concepts

Horizontal Scaling

Horizontal scaling is a concept where the stretching or compressing of the graph of a function occurs along the horizontal axis. This happens when the input variable (usually represented as \(x\)) is multiplied by a constant. In the function transformation \(g(x) = f(2x)\), the multiplication by 2 indicates a horizontal scaling factor.

Horizontal scaling can either stretch or compress the graph, depending on the value of the constant:

• If the constant is greater than 1, the graph experiences compression, bringing the points closer together.
• If the constant is between 0 and 1, the graph stretches, making it appear wider.

In the example \(g(x) = f(2x)\), the horizontal scaling factor is 2, causing a compression. As we see, each \(x\)-value in the original function \(f(x)\) is effectively scaled by multiplying by \(\frac{1}{2}\).

Function Transformation

Function transformation refers to altering the fundamental graph of a function in some manner. These transformations can include translating, scaling, reflecting, or rotating the graph. With the given function \(g(x) = f(2x)\), the type of transformation in question is horizontal scaling, leading to a horizontal compression.

When understanding function transformations, it's helpful to focus on:

• What changes in the function equation? Here, the \(x\) is changed to \(2x\).
• How does this change affect the graph? This particular change compresses the graph horizontally.
• Are there specific values that indicate specific transformation types? The factor that replaces \(x\) plays a crucial role.

Function transformations are powerful tools that allow us to visualize and manipulate a graph to better understand the relationships within the function. In cases like \(g(x) = f(2x)\), the transformation quickly and effectively changes the dynamics of how the graph is presented by compressing it horizontally by a factor of 2.

Horizontal Compression

Horizontal compression is a specific type of horizontal scaling where the graph of a function is pressed closer together along the x-axis. When we come across the function \(g(x) = f(2x)\), we're observing a horizontal compression because each \(x\)-coordinate of the graph is effectively halved.

Here's what to remember about horizontal compression:

• The constant multiplying \(x\) (in this case, 2) indicates the compression factor.
• A horizontal compression factor of 2 means that the width of the graph is reduced to half its original size.
• This causes the points on the graph to move towards the y-axis, making peaks and valleys appear narrower.

In practical terms, if every x-value is halved, then the function's behavior in terms of where it increases or decreases happens twice as fast compared to the original graph of \(f(x)\). Understanding this type of compression can be essential for analyzing how functions can be modified simply by adjusting the input variable.