Problem 4 - Algebra and Trigonometry

Question

1–10 ? 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) Reflect across x-axis; (b) Reflect across y-axis.

Step 1: Understanding the Transformation

For the function transformation, begin by recognizing the impact of the negative sign in front of the function. The given function is modified to form new functions as specified in parts (a) and (b). Each modification changes how the original graph is displayed.

Step 2: Analyze Part (a): Reflection Across the X-axis

For the function in part (a), the transformation is given by \(y = -f(x)\). This represents a reflection of the original graph of \(f(x)\) across the x-axis. Every point (x, y) on the graph of \(f\) is transformed to (x, -y). This means if a point was above the x-axis, it will now be below it, and vice versa.

Step 3: Analyze Part (b): Reflection Across the Y-axis

For part (b), the transformation is given by \(y = f(-x)\). This transformation reflects the graph of \(f(x)\) across the y-axis. Every point (x, y) on the graph of \(f\) is transformed to (-x, y). This means if a point was on the right side of the y-axis, it will now be on the left side, and vice versa.

Key Concepts

Reflection Across the X-axis

A reflection across the x-axis is a type of transformation that flips a graph over this axis. When dealing with the function transformation defined as \( y = -f(x) \), each point on the original graph of \( f(x) \) is reflected over the x-axis. Essentially, the \( y \)-coordinate of every point becomes its negative counterpart.

For example, a point (3, 4) on the graph of \( f(x) \) will be transformed to (3, -4) on the graph of \( -f(x) \). If the point was above the x-axis, it will now be below it, mirroring its original position. This is a crucial concept to understand as it changes the visual presentation of the function, but not its fundamental shape.

• The entire graph flips upside down.
• Points above the x-axis are moved to an equal distance below it.
• Points on the x-axis remain unchanged.

This transformation preserves the x-intercepts of the original graph. That means if the graph of \( f(x) \) intersects the x-axis, these points remain fixed during the reflection.

Reflection Across the Y-axis

A reflection across the y-axis is another fundamental transformation. This transformation involves flipping the graph over the y-axis. For the function described by \( y = f(-x) \), each point on the original graph \( f(x) \) is transformed to the opposite side of the y-axis. Essentially, the \( x \)-coordinate of every point becomes its negative equivalent.

To illustrate, consider a point (2, 5) on the graph of \( f(x) \). In the transformation \( f(-x) \), this point becomes (-2, 5). If a point was originally to the right of the y-axis, it will now be positioned symmetrically to the left.

• The graph is mirrored left-right along the vertical y-axis.
• Points on the y-axis remain unchanged.
• The horizontal symmetry of the graph is preserved.

This transformation keeps the y-intercepts intact. Points where the graph intersects the y-axis do not change position during this reflection.

Function Transformations

Function transformations involve modifying the position or shape of a graph based on specific mathematical rules. These transformations can translate, reflect, stretch, or compress the graph of a function. Understanding these transformations allows us to manipulate functions to meet different criteria or understand their properties more deeply.

Common transformations include:

• Translations: Shifting the graph horizontally or vertically without changing its shape.
• Reflections: Flipping the graph over an axis, such as x-axis or y-axis reflections.
• Stretches and Compressions: Altering the graph's dimensions either by stretching it taller/wider or compressing it shorter/narrower.

Each transformation affects the graph in a predictable manner, allowing us to transform one graph into another. Recognizing and applying these transformations is an essential skill in graphing and understanding functions. By comprehensively understanding transformations, developing insights into the behavior and characteristics of functions becomes significantly easier.