Problem 26 - Precalculus : Building Concepts and Connections

Question

Identify the underlying basic function, and use transformations of the basic function to sketch the graph of the given function. $$f(x)=\left|\frac{x}{2}\right|$$

Step-by-Step Solution

Answer

The given function \(f(x)=\left|\frac{x}{2}\right|\) is a vertical stretching by a factor of 2 of the basic function \(y=|x|\). The graph of \(f(x)\) will accordingly look like a 'V' that is more stretched along the x-axis compared to the graph of \(y=|x|\).

Step 1: Identifying the Basic Function

The underlying basic function here is \(y=|x|\). The graph of this function is a 'V' shape along the x-axis. The function is positive for all real numbers, and it reflects any input on the negative x-axis onto the positive y-axis.

Step 2: Identifying the Transformation

The given function \(f(x)=\left|\frac{x}{2}\right|\) can be seen as a vertical stretching of the basic function \(y=|x|\) by a factor of 2. Every y-coordinate on the basic graph will be halved in the graph of the given function.

Step 3: Sketching the Graph

Now that we know our basic function and the transformation applied to it, we can sketch the graph of our given function. Begin by graphing the basic function \(y=|x|\). Then apply the transformation by dividing every y-coordinate by 2. The result is the graph of the function \(f(x)=\left|\frac{x}{2}\right|\), which will look like a 'V' that is more stretched along the x-axis compared to the graph of the basic function.

Key Concepts

Basic Function

To start with graphing transformations, one needs to understand what a basic function is. A basic function is the simplest form of a function that acts as the starting block for transforming into more complex functions. For instance, the function y = |x| represents the absolute value function, which is a fundamental example of a basic function. It has characteristic traits like being non-negative for all real numbers, and its graph is a recognizable 'V' shape centered on the origin. Understanding a basic function is crucial because it serves as the reference point from which we make alterations or transformations to obtain a new graph that represents the altered function.

Absolute Value Function

The absolute value function, represented as f(x) = |x|, is a cornerstone in understanding graph transformations. By definition, this function takes any real number x and returns its non-negative value. Mathematically, this translates to f(x) = x if x >= 0 and f(x) = -x if x < 0. Its graph produces the classic 'V' shape with the vertex at the origin of the coordinate plane.

The significance of the absolute value function is its ability to reflect negative inputs to the positive y-axis. This unique reflection property is essential to comprehend as it underlies many transformations involving vertical flips or reflections.

Vertical Stretching

When it comes to transforming functions, vertical stretching is a vital concept to grasp. It involves altering a function in a way that expands or compresses its graph vertically. Think of it as pulling or pushing the graph of a function away from or towards the x-axis without altering its width along the x-axis.

Vertical stretching is achieved by multiplying the original y-values by a constant factor. If we take the absolute value function as an example, f(x) = |x/2| is a vertical stretch of the basic function by a factor of 2. This means all the y-values of the basic function are halved. In general, a multiplier greater than 1 stretches the graph away from the x-axis, while a multiplier between 0 and 1 compresses it towards the x-axis.

Function Graph Sketching

Graphing or sketching a function is a fundamental skill in understanding how mathematical functions behave visually. The process involves plotting points, which represent the function's output for specific inputs, and connecting these points to reveal the shape of the graph. The graph offers visual insights into the function's properties, like symmetry, intercepts, and intervals of increase or decrease.

For graph sketching, knowing the basic function's shape is crucial. Once this is established, various transformations can be applied systematically. In the context of the absolute value function f(x) = |x/2|, we start by graphing the basic 'V' shape of y = |x| and then apply the predetermined transformations. A proper sketch will show the vertical stretching and provide a true representation of the function's behavior across all its domain.