Problem 6 - A Graphical Approach to Precalculus with Limits

Question

Write an equation in \(x\) and \(y\) that results in the desired transformation. Do not use a calculator. The absolute value function, vertically shrunk by applying a factor of \(\frac{1}{3}\) and reflected across the \(y\) -axis

Step-by-Step Solution

Answer

The equation is \( y = \frac{1}{3} |x| \).

Step 1: Understanding the Problem

We are tasked with writing an equation that describes the transformation of the absolute value function. Specifically, we need to vertically shrink the graph by a factor of \( \frac{1}{3} \) and reflect it across the \( y \)-axis.

Step 2: Identifying the Base Function

The base function we have is the absolute value function \( y = |x| \). This graph is a V-shaped curve symmetric about the \( y \)-axis.

Step 3: Considering Vertical Shrink

A vertical shrink by a factor of \( \frac{1}{3} \) involves multiplying the function by \( \frac{1}{3} \). Hence, the equation becomes \( y = \frac{1}{3} |x| \).

Step 4: Reflecting Across the y-axis

To reflect the function across the \( y \)-axis, we replace \( x \) with \( -x \). The equation becomes \( y = \frac{1}{3} |-x| \).

Step 5: Final Equation

Since the absolute value property \( |-x| = |x| \) holds, the equation simplifies back to \( y = \frac{1}{3} |x| \). This is because the absolute value negates the effect of reflecting \( x \) over the \( y \)-axis. However, the key effect here is the vertical shrink.

Key Concepts

Absolute Value Function

The absolute value function is a fundamental mathematical concept characterized by its distinct "V" shape when graphed. It's often represented as \( y = |x| \), where the vertical line symmetry along the \( y \)-axis is a key feature. This means for any value of \( x \), whether positive or negative, the output \( y \) is always non-negative, as it measures the distance from zero. The absolute value transforms negative inputs to their positive counterparts.

In practical terms, if \( x = -3 \), then \( |x| = 3 \), and similarly, if \( x = 3 \), then \( |x| = 3 \). This property makes the absolute value function handy when addressing real-world scenarios where only magnitudes matter without regard to direction, like distances or even absolute temperature changes.

• Basic Form: \( y = |x| \)
• Graph Shape: V-shaped
• Symmetric about the \( y \)-axis

Understanding how this function behaves is essential when learning about various transformations like vertical shrinks and reflections.

Vertical Shrink

Vertical shrink is a type of transformation applied to graphs, making them narrower or "squeezed" vertically towards the \( x \)-axis. This occurs when every point on the graph is proportionally lowered by multiplying the function by a factor that is between 0 and 1. Given our task, to perform a vertical shrink on the absolute value function by a factor of \( \frac{1}{3} \), we transform it into \( y = \frac{1}{3} |x| \).

This scaling down maintains the overall "V" shape of the graph but pushes it closer to the \( x \)-axis by each original \( y \)-value being reduced to one-third. The transformation does not affect the \( x \)-coordinates of the graph; it exclusively alters the height of each point.

Consider an initial point \((x, y)\) on the graph \( y = |x| \), which becomes \((x, \frac{1}{3} y)\) on the new graph. Key points like the vertex at \((0, 0)\) remain unchanged because zero multiplied by any fraction is still zero.

• Applied Factor: \( \frac{1}{3} \)
• Effect: Reduces \( y \)-values, squishes vertically
• Vertex remains unchanged at \((0, 0)\)

Recognizing and applying vertical shrink allows for creating more compacted, focused graph representations.

Reflection Across y-axis

Reflection across the \( y \)-axis is another transformation that involves "flipping" the graph over the \( y \)-axis. For absolute functions, this involves changing each \( x \)-coordinate to its negative counterpart, reflecting the function horizontally. Mathematically, it translates into replacing \( x \) with \( -x \) in the equation. However, due to how the absolute value function \( y = |x| \) operates, replacing \( x \) with \( -x \) has no visible effect.

Indeed, though you substitute to obtain \( y = |x| \) or \( y = |-x| \), absolute values confirm that both \( |x| \) and \( |-x| \) result in the same output values, meaning they are identical functions visually. Therefore, the key emphasis in a situation that discusses such reflection is more on understanding the properties of absolute values rather than seeing an alteration in the graph.

• Base Form: \( y = |x| \)
• Transformation: Replace \( x \) with \( -x \)
• Graph Effect: No visible change for the absolute value function

Understanding this reflection transformation highlights the symmetry inherent in the absolute value function and its real-world applications.