Problem 8 - Precalculus : Building Concepts and Connections

Question

Fill in the blank with one of the following: horizontal, vertical. The graph of \(\frac{1}{4} f(x)\) is obtained by a __________ compression of the graph of \(f(x)\) by a factor of \(\frac{1}{4}\).

Step-by-Step Solution

Answer

The correct answer is vertical.

Step 1: Understanding the Transformation

First, recall that the effect of multiplying a function by a constant depends on whether the multiplication is being done to the function itself (inside the function) or to the function's output (outside of the function). If the multiplication is inside the function, like \(f(4x)\), it would be a horizontal transformation. But If it's outside the function, like \(\frac{1}{4}f(x)\), it would be a vertical transformation.

Step 2: Identifying the Type of Transformation

In our exercise, we have \(\frac{1}{4}f(x)\), which means the multiplication \(\frac{1}{4}\) is outside the function. Hence, the transformation is vertical.

Step 3: Identifying the Type of Compression

Since the factor is less than 1, it compresses the graph towards the x-axis. So it's a vertical compression.

Key Concepts

Function Transformations

Function transformations involve altering the basic graph of a function in various ways to produce a new graph. Imagine taking a piece of stretchy elastic with a graph drawn on it and then you pull, push, and twist to manipulate the shape and position of the graph. Mathematically, we apply operations such as translating, scaling, and reflecting to modify the original function graphically.

A key point in understanding function transformations is recognizing whether the transformation will be horizontal or vertical, and this depends on where the constant is applied. If you multiply a function by a constant outside of the function, like \(c \cdot f(x)\), it stretches or compresses the graph vertically. However, if it is within the function, like \(f(c \cdot x)\), then the graph is altered horizontally.

Vertical Compression

Vertical compression is a specific type of transformation where the graph of a function is squeezed towards the x-axis. Think of it as reducing the height of every point on the graph by a certain factor, which is always between 0 and 1. For instance, when we look at the transformation \(\frac{1}{4}f(x)\), every y-value of the graph of \(f(x)\) is multiplied by \(\frac{1}{4}\). This results in a new graph that looks narrower or 'compressed' compared to the original.

Here's an important takeaway: vertical compression doesn't affect the x-values of the graph at all. Only the y-values are changed, which means the graph keeps its original left to right spread while getting pushed closer to the x-axis.

Graphing Functions

Graphing functions is the visual representation of mathematical functions on a coordinate plane. For many students, seeing a function on a graph can make understanding its behavior much easier. When graphing transformations like vertical compression, you need to adjust the original points according to the transformation's rule.

To graph a vertically compressed function, you would start by plotting the original function's points. Next, apply the compression factor to the y-values of these points. For example, a point \( (x, y) \) on the graph of \(f(x)\) would become \( (x, \frac{1}{4}y) \) on the graph of \(\frac{1}{4}f(x)\). This illustrates the same x-coordinate with a y-value that is a quarter of its original value, effectively shrinking the graph towards the x-axis.