Problem 7 - Algebra and Trigonometry

Question

\(5-14\) . Suppose the graph of \(f\) is given. Describe how the graph of each function can be obtained from the graph of \(f .\) (a) \(y=-f(x) \quad\) (b) \(y=f(-x)\)

Step-by-Step Solution

Answer

(a) Vertical reflection across x-axis; (b) Horizontal reflection across y-axis.

Step 1: Understanding the Problem

The exercise involves transforming the graph of a function \( f(x) \) in specific ways. Two transformations are given: (a) \( y = -f(x) \) and (b) \( y = f(-x) \). We need to describe what happens to the graph of \( f \) under each transformation.

Step 1: Vertical Reflection for \( y = -f(x) \)

For the function \( y = -f(x) \), the graph of \( y = f(x) \) undergoes a vertical reflection across the x-axis. This means that each point \((x, y)\) on the graph of \( f(x) \) is transformed to \((x, -y)\). The x-coordinates stay the same, but the y-values are multiplied by -1.

Step 2: Horizontal Reflection for \( y = f(-x) \)

For the function \( y = f(-x) \), the graph of \( y = f(x) \) undergoes a horizontal reflection across the y-axis. This means that each point \((x, y)\) on the graph of \( f(x) \) is transformed to \((-x, y)\). The y-coordinates remain unchanged, while the x-coordinates are multiplied by -1.

Step 4: Conclusion

In summary, transforming the graph of \( f \) using \( y = -f(x) \) results in a vertical flip, while using \( y = f(-x) \) results in a horizontal flip.

Key Concepts

Vertical Reflection

When dealing with vertical reflection, the graph of a function experiences a flip over the x-axis. Let's break this process down. Taking the function given as

• \( y = f(x) \), a general point on this graph is represented as \((x, y)\).
• In a vertical reflection, represented by \( y = -f(x) \), each point on the graph is transformed to \((x, -y)\).
• The x-coordinates remain unchanged in this transformation, while the y-coordinates are multiplied by -1.

It's like flipping the graph over an invisible line running along the x-axis. Just imagine that you're mirroring the top part of the graph below the x-axis and vice versa.

• As a result, all features of the graph, such as peaks and valleys, are inverted while maintaining their horizontal positions.

Vertical reflection is a straightforward transformation that effectively turns the graph upside down.

Horizontal Reflection

Horizontal reflection involves flipping the graph of a function over the y-axis. This results in a mirror image across the vertical axis. Here's how it works:

• Consider a graph of \( y = f(x) \). Each point on this graph is originally positioned at \((x, y)\).
• For a horizontal reflection, expressed as \( y = f(-x) \), each point becomes \((-x, y)\).
• In this case, y-coordinates remain unchanged, while x-coordinates are multiplied by -1.

Imagine holding the graph in your hand and flipping it to reverse the positions from left to right. The left side of the graph moves to the right, and vice versa.

• As a result, the graph maintains its original height features, but their positions across the y-axis are switched.

This transformation alters the horizontal orientation without changing the vertical attributes of the graph.

Function Transformations

Function transformations are operations that you can apply to the graph of a function to change its shape, position, or orientation. They are powerful tools in mathematics for analyzing, understanding, and visualizing how different algebraic manipulations affect graphs. The main types of transformations include:

• Translations - These shift the graph either vertically or horizontally without altering its shape. Adding or subtracting a constant to \( f(x) \) shifts the graph up/down or left/right.
• Reflections - As discussed, these flip the graph over the x-axis (vertical reflection) or y-axis (horizontal reflection), altering the orientation.
• Scaling - By multiplying the function by a constant, the graph can be stretched or compressed in the y-direction or x-direction.
• Rotations - Although less common in elementary function transformation, they involve rotating the graph around a point or axis.

These transformations enable you to manipulate a graph in numerous ways to explore and analyze the underlying mathematical concepts visually. Understanding them provides deeper insights into how functions behave and allows for versatile problem-solving techniques.