Problem 63 - College Algebra

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 refl cted 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 provided is the absolute value function, which is \( f(x) = |x| \). This function has a characteristic V shape that opens upwards on the coordinate plane.

Step 2: Reflect over the y-axis

To reflect the function \( f(x) = |x| \) over the y-axis, we need to replace \( x \) with \(-x\) in the function. The reflected function becomes \( f(x) = |-x| \). However, since the absolute value function \(|-x|\) is the same as \(|x|\), the reflection does not change the expression here: it remains \( g(x) = |x| \).

Step 3: Apply the horizontal compression

A horizontal compression by a factor of \( \frac{1}{4} \) involves multiplying the \( x \) term by \( 4 \). Thus, we replace \( x \) in the function \( |x| \) with \( 4x \). Therefore, the function becomes \( g(x) = |4x| \).

Step 4: Write the transformed function

After applying the reflection and horizontal compression, the resulting function is \( g(x) = |4x| \). This is because the reflection didn't change the function in terms of the formula (as \(|-x| = |x| \)), and the horizontal compression compressed the input by \( 4 \).

Key Concepts

Absolute Value Function

The absolute value function is one of the essential building blocks in mathematics. It is often denoted by \( f(x) = |x| \). This function has a distinctive, symmetric "V" shape that graphs upward, with the vertex at the origin, \((0, 0)\).
The unique property of the absolute value function is its ability to make all values non-negative. This means whatever the input value of \( x \) is, the output is always positive or zero:

• For \( x > 0 \), \( |x| = x \).
• For \( x < 0 \), \( |x| = -x \).
• For \( x = 0 \), \( |x| = 0 \).

The concept of absolute value is applied across various mathematical phenomena, including distance calculations. In function transformations, it lays the groundwork for many operations like reflections and stretches. Understanding the basic properties of the absolute value function helps in visualizing and performing transformations effectively.

Reflection Over the Y-Axis

A reflection over the y-axis means flipping the graph across this vertical axis, which changes the direction of the graph horizontally. To perform this on any function \( f(x) \), simply replace the variable \( x \) with \( -x \).
However, for the absolute value function, this transformation doesn’t alter its appearance. That's because the absolute value negates the effect of the negative sign:

• The new function becomes \( f(x) = |-x| \).
• Since \( |-x| = |x| \), the graph remains unchanged.

Although the algebraic expression \( |-x| \) looks different initially, its effect on graphing the function is non-existent due to the properties of absolute values. This serves as a reminder that reflections need to be assessed in the context of the function's inherent properties.

Horizontal Compression

Horizontal compression is a transformation that "squeezes" the function towards the y-axis. It modifies how the function's graph stretches horizontally by altering the coefficient of \( x \) within the function.

• For the function \( f(x) \), to achieve a horizontal compression by a factor of \( \frac{1}{4} \), replace \( x \) with \( 4x \).
• Thus, the transformed absolute value function becomes \( g(x) = |4x| \).

What happens here is each point on the graph that was originally at \( x \) now appears at \( x/4 \). This results in the graph being narrower, as each x-value has been effectively reduced by dividing the original input by a factor of 4. Horizontal transformations are crucial in adjusting the spread and width of graphs without altering their vertical positions.