Question

Use words and mapping notation to describe how the graph of each function can be found from the graph of the function \(y=f(x)\) a) \(y=4 f(x)\) b) \(y=f(3 x)\) c) \(y=-f(x)\) d) \(y=f(-x)\)

Short answer

a) Vertical stretch by 4: \( (x, y) \rightarrow (x, 4y) \) b) Horizontal compression by 3: \( (x, y) \rightarrow \left(\frac{x}{3}, y\right) \) c) Reflect over x-axis: \( (x, y) \rightarrow (x, -y) \) d) Reflect over y-axis: \( (x, y) \rightarrow (-x, y) \)

Step-by-step solution

Understanding Function Transformations

Each transformation of the graph of the function can be described using vertical or horizontal scaling, and reflections. Let's analyze each function one by one.

Vertical Stretch by Factor of 4

For the function \(y=4 f(x)\), the graph of \(y=f(x)\) is stretched vertically by a factor of 4. This means that every y-coordinate of the original graph is multiplied by 4.

Horizontal Compression by Factor of 3

For the function \(y=f(3x)\), the graph of \(y=f(x)\) is compressed horizontally by a factor of 3. This means that every x-coordinate of the original graph is divided by 3.

Reflection Over the x-axis

For the function \(y=-f(x)\), the graph of \(y=f(x)\) is reflected over the x-axis. Every y-coordinate of the original graph is multiplied by -1.

Reflection Over the y-axis

For the function \(y=f(-x)\), the graph of \(y=f(x)\) is reflected over the y-axis. Every x-coordinate of the original graph is multiplied by -1.

Mapping Notation

To summarize using mapping notation: a) \(y=4 f(x)\) maps \((x, y) \rightarrow (x, 4y)\) b) \(y=f(3 x)\) maps \((x, y) \rightarrow \left(\frac{x}{3}, y\right)\) c) \(y=-f(x)\) maps \((x, y) \rightarrow (x, -y)\) d) \(y=f(-x)\) maps \((x, y) \rightarrow (-x, y)\)

Key concepts

Vertical Scaling

Vertical scaling changes the y-values of a function's graph. When we vertically scale a graph by a factor, we multiply all the y-coordinates by that factor. For example, in the function \(y=4f(x)\), the graph of \(y=f(x)\) is stretched vertically by a factor of 4. Every point on the graph of \(y=f(x)\) is transformed by multiplying the y-coordinate by 4. So a point \((x, y)\) on \(y=f(x)\) will map to \((x, 4y)\) on \(y=4f(x)\). This makes the graph taller, but the x-coordinates remain the same.

This transformation is very useful in applications that require adjusting the amplitude of a function, like in audio signal processing.

Horizontal Scaling

Horizontal scaling alters the x-values of a function's graph. Instead of stretching or compressing the graph vertically, horizontal scaling stretches or compresses it horizontally. For example, with the function \(y=f(3x)\), the graph of \(y=f(x)\) is compressed horizontally by a factor of 3. This transformation divides the x-coordinates by 3. So, a point \((x, y)\) on \(y=f(x)\) will transform to \((\frac{x}{3}, y)\) on \(y=f(3x)\). As a result, the graph appears to be squished horizontally but retains the same y-values.

This kind of scaling can be crucial when considering time-dependant processes where you need to speed up or slow down an event.

Reflection Transformations

Reflection transformations flip the graph over a specific axis. There are two main types of reflections:

1. **Reflection over the x-axis**: In the function \(y=-f(x)\), the graph of \(y=f(x)\) is flipped over the x-axis. This means every y-coordinate is multiplied by -1, transforming each point \((x, y)\) to \((x, -y)\). It looks like the graph is mirrored upside down.

2. **Reflection over the y-axis**: For the function \(y=f(-x)\), the graph of \(y=f(x)\) is flipped over the y-axis. Here, we multiply the x-coordinates by -1. As a result, a point \((x, y)\) on \(y=f(x)\) is transformed to \((-x, y)\). This mirrors the graph to the other side of the y-axis.

Reflections are common in physics and engineering to analyze symmetrical properties and behaviors.

Graph Mapping Notation

Graph mapping notation helps to formally describe how points on a graph transform under various function changes. This notation uses the form \((x, y) \rightarrow (new x, new y)\) to define transformations.

For the given exercises, using mapping notation clearly shows:

• In \(y=4f(x)\): \((x, y) \rightarrow (x, 4y)\)
• In \(y=f(3x)\): \((x, y) \rightarrow (\frac{x}{3}, y)\)
• In \(y=-f(x)\): \((x, y) \rightarrow (x, -y)\)
• In \(y=f(-x)\): \((x, y) \rightarrow (-x, y)\)

Mapping notation is essential for clearly and concisely describing how transformations affect the coordinates of graph points, making it easier to understand the overall movement and deformation of the graph.