Problem 63 - College Algebra

Question

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 transformed function is \( g(x) = |-4x| \).

Step 1: Identify the Base Function

The base function given is \( f(x) = |x| \). This represents the graph of the absolute value function.

Step 2: Reflect Over the Y-Axis

To reflect the graph of \( f(x) = |x| \) over the \( y \)-axis, we change \( x \) to \( -x \). This gives us the new function \( f(-x) = |-x| = |x| \). Since the absolute value function is symmetric, reflecting over the \( y \)-axis does not change it.

Step 3: Apply Horizontal Compression

Next, we apply a horizontal compression by a factor of \( \frac{1}{4} \). This involves multiplying the input \( x \) by 4 in the function. Therefore, \( f(-x) \) becomes \( f(-4x) = |-4x| \). The compression means that each point on the graph is four times closer to the \( y \)-axis than before.

Step 4: Write the Final Transformed Function

The final function \( g(x) \) is the result of reflecting the base function over the \( y \)-axis and then compressing it horizontally by a factor of \( \frac{1}{4} \). This function is \( g(x) = |-4x| \).

Key Concepts

Absolute Value Function

The absolute value function is a fundamental function in mathematics, often represented as \( f(x) = |x| \). This function is characterized by its simple V-shape when graphed in a coordinate plane.
Understanding the behavior of this graph is crucial as it acts as a base for many transformations.

• The absolute value operation turns negative inputs into their positive counterparts, which is why the graph lies in the positive part of the \( y \)-axis.
• At \( x = 0 \), the absolute value function hits its vertex, which is both the minimum point and a point of symmetry.
• The left side of the function is a mirror image of the right side, due to its symmetry about the \( y \)-axis.

These properties make it relatively easy to manipulate using transformations like reflections or compressions.

Reflection Over Y-Axis

A reflection over the \( y \)-axis is a geometric transformation that flips a graph horizontally. It’s akin to flipping a page to see its mirror image.
For functions, this means replacing \( x \) with \( -x \), resulting in \( f(-x) \).

• For symmetric functions like the absolute value function, this transformation does not visibly alter the graph, because \( |x| \) is the same as \( |-x| \).
• However, for other functions, such a transformation can significantly change their appearance.

Imagine the reflection as looking at the function in a horizontal mirror. Everything that was on the left moves to the right and vice versa, making it a neat transformation to visualize.

Horizontal Compression

Horizontal compression is a type of function transformation that alters the spacing of a graph’s points along the \( x \)-axis.
This is achieved by multiplying the input \( x \) by a factor greater than 1. For instance, compressing by \( \frac{1}{4} \) means using \( 4x \) in place of \( x \).

• This transformation brings all the points on the graph inwards towards the \( y \)-axis, effectively squeezing the graph horizontally.
• If you picture the graph being "squashed," you're on the right track!
• Pairing this with other transformations can lead to more complex and compound adjustments to the function's look.

Horizontal compression is crucial for squeezing broader graphs into tighter spaces, which can help in analyzing patterns and behaviors more clearly in functions.