Problem 63 - College Algebra with Corequisite Support

Question

For the following exercises, write a formula for the function g that results when the graph of a given toolkit function is transformed as described. The graph of \(f(x)=|x|\) is reflected over the \(y\) -axis and horizontally compressed by a factor of \(\frac{1}{4}\).

Step-by-Step Solution

Answer

The function is \(g(x) = |4x|\).

Step 1: Identify the Toolkit Function

The given function is \(f(x) = |x|\). This is the absolute value function, which is a V-shaped graph centered at the origin.

Step 2: Understanding Reflection Over the y-axis

Reflecting over the \(y\)-axis affects the \(x\) component of the function. For any function \(f(x)\), reflecting over the \(y\)-axis results in \(f(-x)\). Thus, the reflection of \(|x|\) is \(|-x|\), which simplifies back to \(|x|\) because absolute value removes any negative signs.

Step 3: Horizontal Compression by a Factor of \(\frac{1}{4}\)

To horizontally compress a function by a factor of \(\frac{1}{4}\), replace \(x\) with \(4x\) in the function. Therefore, \(g(x) = |4x|\) applies the horizontal compression to the previously reflected function.

Step 4: Write the Transformed Function

Combine the transformations. Given the reflection steps didn't change the verbal expression (\(|x|\) became \(|x|\)), and the compression step resulted in \(|4x|\), the formula for the transformed function is \(g(x) = |4x|\).

Key Concepts

Understanding the Absolute Value Function

The absolute value function, represented as \( f(x) = |x| \), is one of the fundamental toolkit functions in mathematics.This function creates a V-shaped graph that is symmetrical about the \( y \)-axis and centered at the origin.

• It essentially measures the distance of a number \( x \) from zero on a number line, always yielding a non-negative output.
• The absolute value of both positive and negative numbers results in a positive number: for example, \(|5| = 5\) and \(|-5| = 5\).
• In graphical terms, the graph cuts through the origin and arches upwards, making a perfect "V" shape.

This simple operation of taking the magnitude without regard to sign has significant implications for graph transformations.

Exploring Reflection Over the y-axis

Reflecting a function over the \( y \)-axis involves altering its appearance by flipping it horizontally across the \( y \)-axis.For any function \( f(x) \), this transformation is achieved by substituting \( x \) with \(-x\), resulting in \( f(-x) \).

• In the context of the absolute value function, \(|x|\) and \(|-x|\) produce identical results since the absolute value neutralizes sign changes.
• This means that graphically, reflecting \(|x|\) over the \( y \)-axis does not alter its appearance.
• The graph remains symmetric, maintaining the same shape and position as before the transformation.

Understanding this symmetry is crucial as it affects how we interact with additional transformations like compressions and stretches.

Grasping Horizontal Compression

Horizontal compression refers to the process of "squeezing" the graph of a function towards the vertical \( y \)-axis.It effectively alters the function so that events on the \( x \)-axis happen faster, creating a visual effect akin to stretching a rubber band horizontally.

• To compress horizontally by a factor of \( \frac{1}{4} \), every \( x \) in the function is substituted by \( 4x \).
• In the given context of \( |x| \), this results in \( g(x) = |4x| \), which means the graph changes more rapidly near the \( y \)-axis.
• This compression reduces the "width" of the V-shape of the graph by a quarter, making it steeper.

Horizontal compression is an intuitive concept that requires replacing \( x \) with its factor-modified equivalent to visualize rapid alterations along the horizontal axis.