Question

Explain the differences that occur in transforming the graph of the function \(y=f(x)\) to the graph of the function \(y=f(b x)\) as compared to transforming \(y=f(x)\) to \(y=a f(x)\)

Short answer

\(y=f(bx)\) compresses or stretches horizontally, while \(y=af(x)\) compresses or stretches vertically.

Step-by-step solution

Identify the Original Function

The original function provided is \(y=f(x)\). This will serve as the base graph for our transformations.

Understand the Transformation \(y=f(bx)\)

The transformation \(y=f(bx)\) involves a horizontal scaling or compression/stretching. Specifically, if \(|b| > 1\), the graph compresses horizontally, and if \(0 < |b| < 1\), it stretches horizontally. Additionally, if \(b < 0\), there is also a reflection across the y-axis.

Illustrate the Horizontal Transformation \(y=f(bx)\)

To illustrate, consider the point \((x, y)\) on the original graph \(y=f(x)\). On the transformed graph \(y=f(bx)\), this point becomes \(\left(\frac{x}{b}, y\right)\). All x-coordinates are scaled by a factor of \(\frac{1}{b}\).

Understand the Transformation \(y=af(x)\)

The transformation \(y=af(x)\) involves a vertical scaling or compression/stretching. Specifically, if \(|a| > 1\), the graph stretches vertically, and if \(0 < |a| < 1\), it compresses vertically. Additionally, if \(a < 0\), there is a reflection across the x-axis.

Illustrate the Vertical Transformation \(y=af(x)\)

To illustrate, consider the point \((x, y)\) on the original graph \(y=f(x)\). On the transformed graph \(y=af(x)\), this point becomes \((x, ay)\). All y-coordinates are scaled by a factor of \(a\).

Compare the Effects

The primary difference between the two transformations is the direction of the scaling effect. The transformation \(y=f(bx)\) affects the x-coordinates, leading to horizontal changes, whereas the transformation \(y=af(x)\) affects the y-coordinates, resulting in vertical changes.

Key concepts

Understanding Horizontal Scaling

Horizontal scaling happens when we transform the graph of a function by changing the x-values. This usually occurs in the transformation of the form \(y=f(bx)\). The variable \(b\) plays a crucial role in this type of transformation.

If \(|b| > 1\), the graph compresses towards the y-axis, making it narrower. For example, if \(b=2\), every x-coordinate on the graph of \(y=f(x)\) is halved, because \(x\) values are divided by 2. So, a point at \((4, y)\) on the original graph would move to \((2, y)\) on the transformed graph.

If \(0 < |b| < 1\), the graph stretches away from the y-axis, making it wider. For instance, if \(b=0.5\), every x-coordinate doubles, so a point at \((2, y)\) on the original graph moves to \((4, y)\) on the new graph.

If \(b < 0\), there's also a reflection across the y-axis. This flips the graph horizontally.

Exploring Vertical Scaling

Vertical scaling occurs when the y-values of the function are scaled. This kind of transformation is represented by \(y=af(x)\). The variable \(a\) determines how the graph will be scaled vertically.

If \(|a| > 1\), the graph stretches upwards and downwards, making it taller. For instance, with \(a=3\), each y-coordinate is tripled. So, a point at \((x, 1)\) on the original graph moves to \((x, 3)\) on the transformed graph.

If \(0 < |a| < 1\), the graph compresses, making it shorter. For example, if \(a=0.5\), each y-coordinate is halved. Hence, a point at \((x, 2)\) moves to \((x, 1)\).

If \(a < 0\), there's a reflection across the x-axis, meaning the graph flips vertically.

Comprehending Graph Transformations

Graph transformations involve changing the appearance of a graph without altering its basic shape. Horizontal and vertical scaling are just two ways to transform a graph.

Other common transformations include:

• Horizontal shifts: Moving the graph left or right by adding or subtracting a value to \(x\), such as \(y = f(x - h)\)
• Vertical shifts: Moving the graph up or down by adding or subtracting a value to \(y\), such as \(y = f(x) + k\)
• Reflections: Flipping the graph across axes, achieved by multiplying \(x\) or \(y\) by -1

Each transformation affects the graph's position, orientation, or size, making it essential to understand their specific impacts.

Understanding Functions and Their Transformations

Functions are mathematical expressions that relate an input to an output. Transformations change how these functions look on a graph but do not alter the fundamental relationship between inputs and outputs.

By mastering graph transformations, such as horizontal and vertical scaling, shifts, and reflections, one can easily manipulate and understand the behaviors of various functions.

For instance, comparing \(y=f(bx)\) and \(y=af(x)\) helps us see how different transformations impact the graph's dimensions. While one affects the x-coordinates (horizontal scaling), the other changes the y-coordinates (vertical scaling).

Becoming skilled in these transformations allows for deeper insights into how functions work and enables better problem-solving capabilities in mathematics.