Problem 56 - A Graphical Approach to Precalculus with Limits

Question

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. $$f(x)=|-2 x+1|$$

Step-by-Step Solution

Answer

The graph of \( f(x) = |-2x + 1| \) is a horizontally compressed and right-shifted V-shape.

Step 1: Identify the Basic Function

The given function is based on the absolute value function. The basic or parent function here is \( y = |x| \). This is a V-shaped curve where the vertex is at the origin (0, 0).

Step 2: Identify the Transformations

The function \( f(x) = |-2x + 1| \) involves several transformations:1. **Reflection:** The \(-2\) indicates a reflection across the y-axis.2. **Horizontal Shift:** The \(+1\) inside the absolute value suggests a horizontal shift.3. **Horizontal Scaling:** The factor \( -2 \) compresses the graph horizontally.

Step 3: Reflection over y-axis

Reflect the graph of \( |x| \) across the y-axis. This transformation does not appear or affect the quadratic appearance, but affects directionality and other transformations.

Step 4: Horizontal Shift and Interpretations

Rewrite the expression \(|-2x + 1|\) as \(|2(-x + \frac{1}{2})|\). This suggests the graph shifts horizontally to the opposite direction of what \( x - \frac{1}{2} = 0 \) would become. Therefore, the graph shifts to the right by \( \frac{1}{2} \).

Step 5: Horizontal Compression

The factor \(-2\) compresses the graph horizontally by a factor of 2. This makes the slopes of the arms of the \(|x|\) function steeper. Both arms originally had slopes of \(1\) and \(-1\); after compression, they become \(2\) and \(-2\).

Step 6: Graph the Function

Start by plotting the new vertex of the V, which is at \((\frac{1}{2}, 0)\). For the right arm of the V, use the equation \(y = -2(x - \frac{1}{2})\), and for the left arm, use \(y = 2(x - \frac{1}{2})\). This constructs the final graph, which is a V-shaped graph shifted to the right by 0.5 units and its slopes are now more steep.

Key Concepts

Absolute Value Function

The absolute value function, expressed as \( y = |x| \), is a fundamental concept in graph transformations. This function creates a V-shaped graph that is symmetric about the y-axis. At its core, this function captures the distance of a number from zero without regard to its direction on the number line. Therefore, all values of \( y \) in an absolute value function are non-negative.

Key characteristics of an absolute value function include:

• The graph is always V-shaped.
• It has a vertex, typically at the origin (0,0) for the parent function \(y = |x|\).
• The graph exhibits symmetry about the y-axis.

This parent function serves as the building block for more complex functions involving various transformations.

Reflection

A reflection in graph transformations involves flipping the graph over a line, such as the x-axis or y-axis. In the function \( f(x) = |-2x + 1| \), the term \(-2x\) inside the absolute value indicates a reflection.

Specifically, here, the reflection occurs across the y-axis. This transformation modifies how the graph reacts to the x-axis, essentially changing the direction the graph opens. For absolute value graphs like \(y = |x|\), a reflection often does not change the shape significantly but alters how other transformations, such as shifts and scaling, manifest.

In general:

• Reflection across the y-axis replaces \( x \) with \( -x \).
• Reflection across the x-axis happens outside the absolute value, affecting the function as \( -|x| \).

This helps in understanding the initial directionality before applying further transformations.

Horizontal Shift

A horizontal shift involves moving the graph left or right on the coordinate plane. For the function \( f(x) = |-2x + 1| \), rewriting it as \( |-2(x -\frac{1}{2})| \) shows a shift to the right.

Horizontal shifts result from changes inside the function's argument. In this transformation:

• The expression \( x - h \) shifts the graph to the right by \( h \) units.
• The expression \( x + k \) shifts the graph to the left by \( k \) units.

The vertex of the absolute value graph moves from the origin (0,0) to the new position. In this exercise, the graph shifts to the right by \( \frac{1}{2} \) unit, placing the vertex at \( (\frac{1}{2}, 0) \). This rearranges the graph on the plane but keeps its overall shape intact.

Horizontal Scaling

Horizontal scaling, or horizontal compression and expansion, alters the width of the graph without changing its height. In the function \( f(x) = |-2x + 1| \), the coefficient \(-2\) before \(x\) signifies horizontal compression.

Horizontal compression occurs when the absolute value of the coefficient is greater than 1, which makes the graph narrower. Conversely, if the coefficient is between 0 and 1, it results in horizontal expansion. With compression:

• The arms of the V-shape become steeper.
• Slopes increase, as seen from 1 and -1 to 2 and -2.

Horizontal scaling affects how quickly the y-values increase or decrease as you move along the x-axis, emphasizing changes caused by other transformations. This results in a sharper V-shape for the absolute value graph, focusing the viewer's attention on the altered steepness of its arms.