Problem 35 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 33-40: Explain how the graph of the function compares to the graph of \(y=f(x)\). For example, for the equation \(y=2 f(x+3)\), the graph of \(f\) is shifted 3 units to the left and stretched vertically by a factor of 2 . $$ y=f(-x)-2 $$

Step-by-Step Solution

Answer

The graph of \(f(x)\) is reflected across the y-axis and shifted down by 2 units.

Step 1: Understand the Function

The given function is \( y = f(-x) - 2 \). This function is derived from the original function \( y = f(x) \). We need to analyze how the transformations affect the graph of \( f(x) \).

Step 2: Horizontal Reflection

The expression \(f(-x)\) represents a horizontal reflection across the y-axis. Any input \(x\) changes to \(-x\), which means that each point on the graph of \(f(x)\) will be mirrored with respect to the y-axis.

Step 3: Vertical Translation

The expression \(-2\) at the end of the function \(y = f(-x) - 2\) indicates a vertical translation downward. This means that every point on the reflected graph is shifted 2 units down.

Key Concepts

Horizontal Reflection

When we talk about a horizontal reflection in the context of graph transformations, we refer to mirroring a graph across the y-axis. This transformation occurs when the function inside modifies as \(f(-x)\).
This means that each point \( (x, y) \) on the original graph of \( y = f(x) \) is transformed to \( (-x, y) \). Essentially, the right side of the graph of the original function will appear on the left side after the reflection, and vice versa.
A good way to visualize it is to imagine flipping the graph over the vertical y-axis like a book closing from left to right. Keep in mind, horizontals reflections affect the x-values of the function while keeping their y-values unchanged.

Vertical Translation

Vertical translation is a way to shift a graph up or down without altering its shape. When the function is written as \( y = f(x) - c \), it indicates that the graph shifts downward by \( c \) units. On the other hand, \( y = f(x) + c \) means it moves upward by \( c \) units.
In our specific function \( y = f(-x) - 2 \), the \(-2\) denotes a vertical shift 2 units downward.
This kind of translation keeps the general shape and orientation of the graph but simply changes its position on the y-axis. You'll notice that every point drops 2 units, making it easy to implement this transformation by simply subtracting 2 from each y-coordinate.

Graph Transformations

Graph transformations involve several operations that alter the position, size, or orientation of a graph. They include translations, reflections, stretches, and compressions. By understanding each transformation's effect, we can predict how a graph will look after being modified.
Here's a simple rundown of common transformations:

• Translation: Shifts the graph horizontally or vertically.
• Reflection: Flips the graph over the x-axis or y-axis.
• Stretch/Compression: Alters the graph's width or height, changing how steep or wide it appears.

For instance, in the function \( y = f(-x) - 2 \), we see a combination of a horizontal reflection and a vertical translation.
Understanding the order of transformations is also crucial; typically, reflections and stretches come before translations to achieve the correct final graph. With practice, these concepts become intuitive, enabling you to manipulate and visualize complex functions smoothly.