Problem 57 - College Algebra

Question

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function \(f\). $$ g(x)=f(5 x) $$

Step-by-Step Solution

Answer

The graph of \( g(x) = f(5x) \) is a horizontal compression of the graph of \( f(x) \) by a factor of 5.

Step 1: Understanding the Original Function

First, let's understand the original function. We start with \( f(x) \), which is given but unspecified. This function has a standard graph.

Step 2: Identifying the Transformation

The given function is \( g(x) = f(5x) \). This involves a transformation related to the input of the function. Specifically, the entire input of the function \( f \) is now scaled by a factor of 5.

Step 3: Recognizing Horizontal Compression

When we replace \( x \) with \( 5x \), it indicates a horizontal transformation. Normally, \( f(x) \) and \( f(5x) \) are compared at each \( x \), but here the factor \( 5 \) compresses the graph horizontally. This is because the value of \( x \) needs to be smaller to result in the same output. Thus, the graph is horizontally compressed by a factor of 5.

Step 4: Analyzing the Impact

The result is that every point on \( f(x) \) that lies at \( x \, ext{units} \) from the origin now lies at \( \frac{x}{5} \, ext{units} \) from the origin on \( g(x) \). This makes the graph appear squeezed towards the y-axis, affecting its width.

Key Concepts

Horizontal Compression

Horizontal compression in the transformation of functions is a key concept in understanding how changes to the input of a function affect its graph. When dealing with a function like \[ g(x) = f(5x) \],you perform a horizontal compression.What happens here is that the input variable \( x \) in the original function \( f(x) \) is replaced by \( 5x \). The value 5 indicates a horizontal compression factor.

• A number greater than 1, like 5, compresses the graph horizontally.
• This is because in order to produce the same output that \( f(x) \) would at \( x \), \( g(x) \) requires a smaller \( x \).

Overall, every point on the graph of \( f(x) \) that originally lay at \( x \) now shifts to \( \frac{x}{5} \). Therefore, the graph appears to be squished inward towards the y-axis.

Transformation of Functions

Transforming functions involves modifying the function's equation to achieve a desired shift, stretch, or compression of the graph. In the case of \( g(x) = f(5x) \),we witness a transformation through horizontal compression. This type of transformation occurs on the independent variable, the \( x \)-coordinate.

• The transformation is internal to the function, indicated by the modification inside the argument of \( f \).
• Specifically, multiplication by a factor like 5 inside the function's argument modifies how the input \( x \) maps to produce the output.
• This results not in direct vertical shifts, but rather repositions the points of the function horizontally along the x-axis.

By understanding transformations, we comprehend how changing the input directly influences the presentation of the graph in terms of size and position.

Function Graph Analysis

Function graph analysis is the study of how transformations affect the appearance and behavior of a function's graph.When analyzing \( g(x) = f(5x) \),the main focus is the horizontal compression transformation. To perform a thorough function graph analysis:

• Identify the key features of the original function \( f(x) \),which may include intercepts, asymptotic behavior, and periodicity.
• Evaluate how these features change due to the horizontal compression brought on by the factor of 5.
• Graphically, check how each point and characteristic move to their new places according to the transformation.

This analysis enables us to predict the new layout of the graph and maintain an understanding of the function's characteristics despite the transformation.