Problem 74 - College Algebra

Question

Describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation. $$n(x)=\frac{1}{3}|x-2|$$

Step-by-Step Solution

Answer

Shift \( |x| \) right by 2 units and compress vertically by \( \frac{1}{3} \).

Step 1: Identify the Toolkit Function

The given function is \( n(x) = \frac{1}{3}|x - 2| \). The toolkit function for this is the absolute value function \( f(x) = |x| \), which forms the basis of the transformation.

Step 2: Determine Horizontal Shift

The expression \( |x - 2| \) indicates a horizontal shift of the graph of \( |x| \) to the right by 2 units. Typically, \( f(x) = |x - h| \) shifts the graph \( h \) units to the right if \( h > 0 \).

Step 3: Determine Vertical Compression

The coefficient \( \frac{1}{3} \) in \( \frac{1}{3}|x - 2| \) indicates a vertical compression (shrink) of the graph by a factor of \( \frac{1}{3} \). This means that all the y-values in the graph are multiplied by \( \frac{1}{3} \).

Step 4: Sketch the Transformed Graph

Start by sketching the basic \( |x| \) graph. Then, shift this graph 2 units to the right. Next, apply the vertical compression by reducing the height of each point on the graph by a factor of \( \frac{1}{3} \), resulting in the graph of \( n(x) = \frac{1}{3}|x - 2| \).

Key Concepts

Absolute Value Function

The absolute value function, denoted as \( f(x) = |x| \), is a pivotal function in the world of mathematics. It is defined as the non-negative value of \( x \), meaning it represents the distance of \( x \) from zero on the number line. This function creates a distinctive "V" shape when graphed.
The function is characterized by:

• A vertex at the origin \( (0,0) \).
• Symmetry about the y-axis, making it an even function.
• Two linear pieces: one with a slope of 1 to the right of the vertex and one with a slope of -1 to the left.

If you visualize the graph, it slopes upwards equally on both sides, giving that tidy V shape. The basic absolute value graph is the backdrop upon which various transformations can be carried out.

Horizontal Shift

A horizontal shift of a graph involves moving it left or right along the x-axis. In our function \( |x - 2| \), the graph of the absolute value function moves to the right by 2 units.
This shift occurs because the variable \( x \) has been adjusted by subtracting 2, expressed generally as \( f(x - h) \). Here, \( h = 2 \), indicating a shift to the right by 2 units:

• Positive values of \( h \) in \( |x - h| \) translate the graph to the right.
• Negative values of \( h \) translate it to the left.

Understanding these shifts is crucial for plotting the transformed function accurately. This simple horizontal move doesn't affect the "V" shape of the absolute value graph, just its position relative to the y-axis.

Vertical Compression

Vertical compression affects the steepness of a graph's slopes. In \( n(x) = \frac{1}{3}|x - 2| \), the graph undergoes a vertical compression by a factor of \( \frac{1}{3} \).
This transformation multiplies all the y-values by \( \frac{1}{3} \), effectively making the graph "shorter," or closer to the x-axis:

• All y-values get scaled by the factor, reducing the y-coordinates.
• The slopes of the "V" become less steep as compared to the original graph.

Vertical compression maintains the overall shape and symmetry of the graph but affects its height. Visualizing this transformation helps see that every point on the graph is brought closer by a consistent ratio, giving it a compressed appearance. This transformation is pivotal in function graphing, providing a clear understanding of how factors affect graph scale.