Problem 86 - Algebra and Trigonometry

Question

Sketch graphs of the functions \(f(x)=\|x\|, g(x)=[2 x \|, \text { and } h(x)=\|3 x\| \text { on separate }\) graphs. How are the graphs related? If \(n\) is a positive integer, what does the graph of \(k(x)=\|n x\|\) look like?

Step-by-Step Solution

Answer

The graph of \( k(x) = |nx| \) forms a V-shape with vertex at the origin and slopes \( n \) and \( -n \), making it narrower as \( n \) increases.

Step 1: Understand the Absolute Value Function

The graph of the function \( f(x) = |x| \) is a V-shaped curve. It has a vertex at the origin, \( (0,0) \), and is symmetric about the y-axis. The function is linear for both positive and negative values of \( x \), with slopes of 1 and -1, respectively.

Step 2: Sketch the Graph of \(f(x) = |x|\)

Draw two straight lines: one line starts at the origin and goes up with a slope of 1 for \( x \geq 0 \), while the other line starts at the origin and goes down with a slope of -1 for \( x < 0 \). This creates a V shape centered at the origin.

Step 3: Modify the Absolute Value Function \( g(x) = |2x| \)

The function \( g(x) = |2x| \) is a horizontal compression of \( f(x) = |x| \). Each point on the graph of \( f(x) \) is moved closer to the y-axis, reducing the slope of the lines. The vertex is still at \( (0,0) \), but the slopes become 2 and -2, doubling the rate of increase and decrease.

Step 4: Sketch the Graph of \(g(x) = |2x|\)

Draw two straight lines starting at the origin. The line for \( x \geq 0 \) should have a slope of 2, and the line for \( x < 0 \) should have a slope of -2. It retains the V shape but is narrower than \( f(x) \).

Step 5: Modify the Absolute Value Function \( h(x) = |3x| \)

The function \( h(x) = |3x| \) is a further horizontal compression compared to \( f(x) = |x| \). The slopes become 3 and -3 for \( x \geq 0 \) and \( x < 0 \), respectively, forming an even narrower V shape centered at the origin.

Step 6: Sketch the Graph of \(h(x) = |3x|\)

Draw two straight lines starting at the origin. The line for \( x \geq 0 \) should have a slope of 3, and the line for \( x < 0 \) should have a slope of -3. The V shape is now even narrower than with \( 2x \).

Step 7: Relate the Graphs and Predict \( k(x) = |nx| \)

For any positive integer \( n \), the graph of \( k(x) = |nx| \) will be a V-shaped graph narrower than \( f(x) = |x| \). The vertex remains at the origin \( (0,0) \), and the slopes of the lines are \( n \) and \( -n \). The larger \( n \) is, the narrower the graph becomes.

Key Concepts

Horizontal Compression

Let's talk about horizontal compression, a key concept in graph transformations. A horizontal compression happens when we "squeeze" the graph of a function inward towards the y-axis. This is what occurs with functions like \( g(x) = |2x| \) and \( h(x) = |3x| \).

In simpler terms, each point on the graph moves closer to the y-axis. Imagine taking the original V-shaped graph of \( f(x) = |x| \) and pushing both arms towards the center. The result? A skinnier V shape!

Specifically, for \( g(x) = |2x| \), we compress by a factor of 2. For \( h(x) = |3x| \), the compression factor is 3. In general, \( k(x) = |nx| \) describes a graph compressed by a factor of \( n \), with the slope of the arms increasing as \( n \) increases. The lines of the graph become more steep as the function gets more compressed.

Slope

The slope is a measurement of steepness or incline of a line. In the context of the absolute value function, the slope tells us how much the line rises or falls as we move along the x-axis.

For the basic absolute value function \( f(x) = |x| \), the slopes are 1 and -1. This means:

• From the vertex moving right (x \( \geq 0 \)), the line rises with a slope of 1.
• From the vertex moving left (x \( < 0 \)), the line declines at a slope of -1.

When we look at variations like \( g(x) = |2x| \) or \( h(x) = |3x| \):

• The slopes change to 2 and -2 for \( g(x) \), which means the line is twice as steep in both directions compared to \( f(x) \).
• Similarly, \( h(x) \) has slopes of 3 and -3, making it three times as steep.

Essentially, the steeper the slope, the narrower the V-shaped graph becomes. The slope is directly linked to the compression factor \( n \), as in \( k(x) = |nx| \).

Vertex

The vertex of a function is like the starting point or the "corner" of the graph. For an absolute value function, this is where the direction of the graph changes sharply, forming a V shape.

Let's take \( f(x) = |x| \) as an example; its vertex is at the origin, \((0,0)\). This point is significant because:

• It marks the lowest point on the graph, the point of symmetry.
• All transformations of the graph, such as compressions or reflections, occur around this point.

When we modify the function to \( g(x) = |2x| \) or \( h(x) = |3x| \), the vertex remains unchanged at \((0,0)\).

This consistency means no matter how much we compress or stretch the arms of the graph, the vertex remains fixed, acting as a pivot for these transformations. Understanding the vertex is crucial for sketching and analyzing the behavior of absolute value graphs.

V-shaped graph

A V-shaped graph is very much the hallmark of an absolute value function. This shape comes from the distinct way these functions "take" the absolute value, creating a sharp turn at the vertex.

For \( f(x) = |x| \), the graph naturally forms a V with its point at the origin \((0,0)\).

• For positive x-values, the graph ascends with a constant slope.
• For negative x-values, it declines with a mirrored slope, creating symmetry.

When comparing \( g(x) = |2x| \) and \( h(x) = |3x| \), which are horizontally compressed versions, these V shapes become narrower:

• \( g(x) \) forms a narrower V than \( f(x) \) because of the doubled slope.
• \( h(x) \) presents an even sharper V with a tripled slope.

Overall, the distinctive V shape doesn't change, regardless of the transformations applied. This fundamental shape is what makes absolute value graphs easily recognizable and understandable.