Problem 72 - College Algebra with Modeling and Visualization

Question

For the given representation of a function \(f,\) graph the reflection across the \(x\)-axis and graph the reflection across the \(y\)-axis. \(f(x)=\frac{1}{2}|x-2|+2\)

Step-by-Step Solution

Answer

The x-axis reflection is \( g(x) = -\frac{1}{2}|x-2| - 2 \) and the y-axis reflection is \( h(x) = \frac{1}{2}|x+2| + 2 \).

Step 1: Identify the Original Graph

The original function is given as \( f(x) = \frac{1}{2}|x-2| + 2 \). This is an absolute value function, \( g(x) = |x-2| \), vertically compressed by \(\frac{1}{2}\) and shifted up by 2 units. Additionally, it is shifted to the right by 2 units. The vertex of the original graph is at the point (2, 2).

Step 2: Reflect Original Graph Across the x-axis

Reflecting a graph across the \(x\)-axis involves changing the sign of the \(y\)-values of the function. Therefore, for the reflection of \(f(x) = \frac{1}{2}|x-2| + 2\) across the \(x\)-axis, we get \( g(x) = -\left(\frac{1}{2}|x-2| + 2\right) = -\frac{1}{2}|x-2| - 2 \). This reflection will have its vertex at the point (2, -2).

Step 3: Reflect Original Graph Across the y-axis

Reflecting a graph across the \(y\)-axis involves changing \(x\) to \(-x\) in the function. For \( f(x) = \frac{1}{2}|x-2| + 2 \), \( x \) becomes \(-x\), resulting in \( h(x) = \frac{1}{2}|-x-2| + 2 \), which simplifies to \( h(x) = \frac{1}{2}|x+2| + 2 \). This reflection has a vertex at (-2, 2).

Step 4: Sketch the Graphs

Graph the original function \( f(x) \) with its vertex at (2, 2). For the reflection across the \(x\)-axis, sketch \( g(x) = -\frac{1}{2}|x-2| - 2 \) with a vertex at (2, -2). For the reflection across the \(y\)-axis, sketch \( h(x) = \frac{1}{2}|x+2| + 2 \) with a vertex at (-2, 2). Make sure to maintain the shape of the function as a V-shape with different locations for each graph.

Key Concepts

Absolute Value Functions

An absolute value function is a type of piecewise function defined using the absolute value of a variable. The absolute value is always non-negative, reflecting the function's graphical representation as a "V" shape. In this context, the function is represented as \( f(x) = \frac{1}{2}|x-2| + 2 \). To understand how this function behaves, consider the following steps:

• Identify that \(|x-2|\) shifts the basic absolute value function \(|x|\) two units to the right on the x-axis.
• The coefficient \(\frac{1}{2}\) compresses the graph vertically, making the "V" less steep compared to the standard \(|x|\) function.
• Adding 2 to the entire function shifts the graph upward by two units.

The vertex form here provides an easy way to see these transformations, beginning with a horizontally shifted and vertically compressed V-shape whose apex is now located at the point (2, 2) on the graph.

Reflections Across Axes

Reflections across axes involve flipping a graph over a specified axis, thus changing the orientation of the graph.
When reflecting across the x-axis:

• Change every \( y \) value to \(-y\) resulting in the equation \( g(x) = -\left(\frac{1}{2}|x-2| + 2\right) = -\frac{1}{2}|x-2| - 2 \).
• This flips the graph upside down, with the vertex now located at (2, -2).

Reflecting across the y-axis on the other hand requires applying the operation directly to the variable \( x\):

• Substitute \(x\) with \(-x\) thus transforming the equation into \( h(x) = \frac{1}{2}|-x-2| + 2 \), which simplifies to \( h(x) = \frac{1}{2}|x+2| + 2 \).
• This moves the vertex of the function to (-2, 2), effectively mirroring the entire graph horizontally about the y-axis.

These reflections create new functions that are transformed versions of the original "V" shaped graph.

Vertex Form

The vertex form of a function makes it easy to identify key characteristics of a graph, particularly the vertex itself, which is crucial for graphing purposes.
For an absolute value function like \( f(x) = \frac{1}{2}|x-2| + 2 \):

• This form is clearly structured as \( y = a|x-h| + k \), where \((h, k)\) is the vertex.
• In our equation, \( (h, k) = (2, 2) \), showing the point where the graph changes direction.
• The coefficient \( a = \frac{1}{2} \) affects the steepness of the graph's arms.

Having the function in vertex form provides a simple way to apply transformations, as any shifts, reflections, or scaling can easily be identified and implemented directly, making understanding and drawing these graphs considerably straightforward.