Problem 42 - A Graphical Approach to College Algebra

Question

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$f(x)=\frac{1}{2}|x|$$

Step-by-Step Solution

Answer

Transformed graph is a less steep V-shape with vertex at (0,0).

Step 1: Identify the Parent Function

First, determine the parent function of \(f(x) = \frac{1}{2}|x|\). The absolute value function \(y = |x|\) serves as the parent function, which is a V-shaped graph with its vertex at the origin (0,0) and lines extending at 45-degree angles to the x-axis.

Step 2: Apply Vertical Compression

The function \(f(x) = \frac{1}{2}|x|\) involves a vertical compression by a factor of \(\frac{1}{2}\). This means that each y-value of the parent function is reduced to half its original value. Thus, the vertex remains at (0,0), but the lines of the V-shape are less steep compared to the parent function.

Step 3: Sketch the Transformed Graph

Start by plotting the vertex of the transformed graph at the origin (0,0). Then, for each unit step horizontally from the vertex, go up by \(\frac{1}{2}\) unit (instead of 1 unit in the parent function) to determine the new positions of the lines. Draw the lines creating a less steep V-shape to complete the graph.

Key Concepts

Absolute Value Function

The absolute value function is a foundational concept in algebra, represented mathematically by \( y = |x| \). Its graph is distinctive and easy to recognize. Imagine a giant **V** on the coordinate plane, with its pointed part touching the origin at point (0,0). This shape results from the absolute value operation, which takes any real number and turns it into its non-negative counterpart.

• If \( x \) is positive or zero, \( |x| = x \).
• If \( x \) is negative, \( |x| = -x \).

Whether positive or negative, the absolute value function ensures a non-negative output, reflecting all negative portions of a graph above the x-axis. Knowing this function well will help you understand many transformations applied to it.

Vertical Compression

Vertical compression is a transformation that changes the steepness of a graph without altering its width or horizontal position. In the function \( f(x) = \frac{1}{2}|x| \), we modify the height of the **V** from the parent function \( y = |x| \) by applying a vertical compression factor of \( \frac{1}{2} \).

• This factor essentially means we halve each y-coordinate of the original graph.
• All points on the graph bring half their height on the y-axis, starting from the vertex at the origin.

For example, in the parent function, when \( x = 2 \), \( |x| = 2 \). But in the vertically compressed version, with the same \( x = 2 \), the y-value drops to \( \frac{1}{2} \times 2 = 1 \). This gives the graph a more flattened, spread-out look.

Parent Function

The notion of a parent function is crucial when studying transformations. The parent function is the simplest form of a given type of function and serves as the starting point for other variations that exhibit multiple transformations. In our example, the parent function is \( y = |x| \). It forms the backbone of the graph, with its classic **V** shape originating at (0, 0) with a slope of 1 on each side.

• Identifying the parent function allows us to easily predict and sketch transformations.
• Every transformation (like stretches, compressions, translations) originates from this base structure.

Understanding your parent function well is like having a reliable map—it guides you through any changes or adjustments while maintaining the integrity of the graph's original characteristics.