Problem 57 - 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 \(f\). \(g(x)=f(5 x)\)

Step-by-Step Solution

Answer

The graph is horizontally compressed by a factor of 5.

Step 1: Identify the Type of Transformation

The given function is a transformation of the original function \(f(x)\). Specifically, the function \(g(x) = f(5x)\) involves modifying the independent variable \(x\) by multiplying it by 5. This change corresponds to a transformation involving horizontal scaling.

Step 2: Determine the Direction of the Transformation

Since \(g(x) = f(5x)\) involves the multiplication of \(x\) by a value greater than 1 (in this case, 5), the graph of the original function \(f(x)\) will be horizontally compressed or shrunk.

Step 3: Describe the Specific Nature of the Compression

A horizontal compression factor of 5 means that the graph of \(f(x)\) is squeezed towards the y-axis by a factor of 5. For every point \((a, b)\) on the graph of \(f(x)\), the corresponding point on the graph of \(g(x)\) is \((a/5, b)\). This indicates that the graph of \(g(x)\) moves towards the y-axis, reducing in width.

Step 4: Summarize the Transformation Effect

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

Key Concepts

Function Transformations

Function transformations are a fascinating area of mathematics where we modify the function's graph in various ways. By altering different parts of a function, such as its input or output, we can transform its graphical representation.

• Vertical transformations alter the graph in the up or down direction.
• Horizontal transformations, like in our example, change the x-values and shift the graph left or right.

Function transformations allow us to visualize how equations can modify shapes and directions of graphs. When we apply a transformation such as multiplying, adding, or scaling, it provides insight into how basic functions can evolve.
This makes understanding their effects crucial for graphing functions and analyzing equations.

Horizontal Scaling

Horizontal scaling is one form of transformation applied to functions. It adjusts how wide or narrow a function appears on a graph.
In terms of a mathematical concept, this involves multiplying the independent variable by a constant factor, often represented as the function's argument (inside the parentheses). For instance, if you have a function defined as

• g(x) = f(5x)

this implies a horizontal scaling. Depending on the factor:

• If you multiply by a number greater than 1, the graph gets closer together, often called compression.
• If the multiplying factor is between 0 and 1, it stretches out further from the y-axis.

These changes only affect the width without altering the height of the graph.

Compression of Functions

The compression of functions is a specific type of horizontal scaling where the graph becomes narrower.
For the function given as

• g(x) = f(5x)

the graph is compressed due to the multiplication by 5. Essentially, every x-value on the original function is divided by 5 when the function is transformed, which makes the graph 'squeeze' towards the y-axis.
A practical way to visualize this is to think about a rubber band stretched between two points. If you bring your hands closer together from its original position, the rubber band compresses. This compression in graph terms means the points on the new function are spaced less far apart horizontally.
Understanding compression helps in predicting how a graph will appear based on its equation and aids in making quick sketches of transformed functions.