Problem 36 - Algebra and Trigonometry

Question

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ y=\frac{1}{2}|\boldsymbol{X}| $$

Step-by-Step Solution

Answer

The graph of \( y = \frac{1}{2}|x| \) is a V-shape with a vertex at (0,0) and arms with slope 0.5.

Step 1: Identify the Standard Function

The given function is based on the absolute value function \( |x| \), which is a well-known V-shaped graph with its vertex at the origin (0, 0). The arms of the V symmetrically extend upwards with a slope of 1.

Step 2: Recognize the Vertical Scaling Transformation

The function \( y = \frac{1}{2}|x| \) involves a vertical scaling transformation of the standard absolute value function. Instead of the slope being 1 on each side of the vertex, the slope will be \( \frac{1}{2} \). This transformation compresses the graph vertically by half.

Step 3: Sketch the Transformed Graph

Start at the origin (0, 0), which remains the vertex of the graph. For the right arm (when x > 0), instead of increasing by 1 unit for each unit increase in x, the y-values increase by \( \frac{1}{2} \) times the x-values. For the left arm (when x < 0), the same reduced slope applies, mirroring the right arm.

Key Concepts

Understanding the Absolute Value Function

The absolute value function, represented as \(|x|\), forms the basis of our graph. It is easily recognizable by its V-shape, which is symmetrical about the y-axis. The graph's vertex is at the origin \((0,0)\), which serves as a pivot point where the direction of the graph changes. This function calculates the distance of any number from zero without considering the sign of the number. This means that for any input \(x\), the value \(|x|\) is always non-negative. Key characteristics of the absolute value function include:

• A V-shaped appearance.
• A vertex at the origin.
• Both arms of the V extending upwards with a slope of 1.

Understanding this standard function is crucial before we apply any transformations to it. This forms the foundation, upon which transformations like vertical scaling can be applied.

Essentials of Graph Sketching

Graph sketching is an important skill that combines understanding of standard functions and applying transformations to visualize different functional forms. Instead of plotting individual points, graph sketching starts from a basic, well-known graph—like the absolute value function—and applies transformations systematically. Here's how it works:

• Identify the Base Graph: Begin with a simple graph such as the absolute value function.
• Determine Transformations: Recognize any changes to this base graph, such as translations, reflections, or scalings.
• Visualize the Adjustments: Apply the transformations mentally or on paper to see the new shape of the graph.

By conceptualizing the transformation before drawing, you can effectively sketch graphs without plotting numerous points, making this method more efficient and intuitive.

Vertical Scaling and Its Effects

Vertical scaling is a type of transformation that affects how a graph stretches or compresses in the y-direction. When dealing with the function \(y = \frac{1}{2}|x|\), you are performing a vertical scaling on the standard absolute value function graph.In vertical scaling:

• Scaling Factor Less Than 1: The function is compressed. In our example, the factor is \(\frac{1}{2}\), meaning the graph is squished towards the x-axis by half.
• Scaling Factor Greater Than 1: The graph would stretch away from the x-axis, becoming taller and steeper.
• The vertex remains unchanged, ensuring the graph's focus point stays at the origin \((0,0)\).

Effects are consistent across both sides of the vertex for absolute value functions, leading to a uniform transformation of the graph. Visualizing this transformation helps in correctly sketching the modified graph without detailed calculations, just by altering the slope of the V-branches.