4.4.4 Practice Modeling Stretching And Compressing Functions

4.4.4 Practice: Modeling Stretching and Compressing Functions

This comprehensive guide delves into the intricacies of stretching and compressing functions, a crucial concept in mathematics, particularly within the realm of transformations and function manipulation. We'll explore the underlying principles, provide detailed examples, and offer practical exercises to solidify your understanding. This isn't just about memorizing formulas; it's about developing an intuitive grasp of how these transformations affect the graph of a function.

Understanding Transformations: Stretching and Compressing

Transformations of functions involve altering their graphs by shifting, stretching, compressing, or reflecting them. Stretching and compressing, specifically, modify the scale of the graph along the x-axis (horizontal) or the y-axis (vertical).

Key Concepts:

• Vertical Stretch/Compression: This affects the y-values of the function. Multiplying the function by a constant 'a' (where |a| > 1) results in a vertical stretch, while multiplying by 'a' (where 0 < |a| < 1) results in a vertical compression.

• Horizontal Stretch/Compression: This affects the x-values of the function. Replacing 'x' with 'x/b' (where |b| > 1) results in a horizontal stretch, while replacing 'x' with 'bx' (where 0 < |b| < 1) results in a horizontal compression.

• Absolute Value: The absolute value of 'a' and 'b' determines the magnitude of the stretch or compression. The sign of 'a' determines whether there's a reflection across the x-axis (negative 'a'), and the sign of 'b' determines whether there's a reflection across the y-axis (negative 'b').

Vertical Stretching and Compressing

Let's consider a parent function, f(x). A vertical stretch or compression is represented by:

g(x) = a * f(x)

• If |a| > 1: The graph of f(x) is stretched vertically by a factor of 'a'. Each y-coordinate is multiplied by 'a', making the graph appear taller and narrower.

• If 0 < |a| < 1: The graph of f(x) is compressed vertically by a factor of 'a'. Each y-coordinate is multiplied by 'a', making the graph appear shorter and wider.

• If a < 0: In addition to the stretch or compression, the graph is reflected across the x-axis.

Example:

Let f(x) = x².

1. g(x) = 2f(x) = 2x²: This represents a vertical stretch by a factor of 2. The parabola becomes narrower.

2. g(x) = (1/2)f(x) = (1/2)x²: This represents a vertical compression by a factor of 1/2. The parabola becomes wider.

3. g(x) = -f(x) = -x²: This represents a reflection across the x-axis. The parabola opens downwards.

Horizontal Stretching and Compressing

Horizontal stretching and compressing are represented by:

g(x) = f(x/b)

• If |b| > 1: The graph of f(x) is stretched horizontally by a factor of 'b'. Each x-coordinate is multiplied by 'b', making the graph appear wider and shorter.

• If 0 < |b| < 1: The graph of f(x) is compressed horizontally by a factor of 'b'. Each x-coordinate is multiplied by 'b', making the graph appear narrower and taller.

• If b < 0: In addition to the stretch or compression, the graph is reflected across the y-axis.

Example:

Again, let f(x) = x².

1. g(x) = f(x/2) = (x/2)²: This represents a horizontal stretch by a factor of 2. The parabola becomes wider.

2. g(x) = f(2x) = (2x)²: This represents a horizontal compression by a factor of 1/2. The parabola becomes narrower.

3. g(x) = f(-x) = (-x)²: This represents a reflection across the y-axis (although in this specific case, it doesn't visually change the parabola because it's an even function).

Combining Transformations

Often, you'll encounter functions involving multiple transformations simultaneously. The order of operations matters: transformations are applied from the inside out.

General Form:

g(x) = a * f( (x-h)/b ) + k

Where:

• 'a' affects vertical stretch/compression and reflection across the x-axis.
• 'b' affects horizontal stretch/compression and reflection across the y-axis.
• 'h' affects horizontal shift (translation).
• 'k' affects vertical shift (translation).

Example:

Consider f(x) = √x. Let's analyze g(x) = -2√(x+1) -3.

1. (x+1): Horizontal shift 1 unit to the left.
2. √(x+1): The square root function is applied.
3. -2√(x+1): Vertical stretch by a factor of 2 and reflection across the x-axis.
4. -2√(x+1) -3: Vertical shift 3 units down.

Practice Problems

Let's solidify our understanding with some practice problems. For each problem, identify the parent function and describe the transformations applied. Then, sketch the graph.

Problem 1: g(x) = 3(x-2)² + 1

Solution:

• Parent Function: f(x) = x²
• Transformations:
  • Horizontal shift 2 units to the right.
  • Vertical stretch by a factor of 3.
  • Vertical shift 1 unit up.

Problem 2: g(x) = -1/2|x+3| -2

Solution:

• Parent Function: f(x) = |x|
• Transformations:
  • Horizontal shift 3 units to the left.
  • Vertical compression by a factor of 1/2.
  • Reflection across the x-axis.
  • Vertical shift 2 units down.

Problem 3: g(x) = 2sin(x/3)

Solution:

• Parent Function: f(x) = sin(x)
• Transformations:
  • Vertical stretch by a factor of 2.
  • Horizontal stretch by a factor of 3.

Problem 4: g(x) = -√(4x) +5

Solution:

• Parent Function: f(x) = √x
• Transformations:
  • Horizontal compression by a factor of 1/4.
  • Reflection across the x-axis.
  • Vertical shift 5 units up.

Problem 5: g(x) = 1/3(x+2)³ -4

Solution:

• Parent Function: f(x) = x³
• Transformations:
  • Horizontal shift 2 units to the left.
  • Vertical compression by a factor of 1/3.
  • Vertical shift 4 units down.

Advanced Considerations

Understanding the interplay between horizontal and vertical transformations is critical for accurately predicting the behavior of transformed functions. Pay close attention to the order of operations and practice sketching graphs to build a strong intuition. Remember, practice is key to mastering this essential mathematical skill.

This detailed guide provides a solid foundation for comprehending and applying stretching and compressing functions. As you work through more complex examples, you'll further hone your ability to accurately predict and visualize these transformations. By understanding the interplay between vertical and horizontal changes, you'll develop a deeper understanding of function manipulation and its applications in various mathematical contexts. Remember to always consider the parent function, the order of transformations, and the impact of each transformation on the graph to achieve a comprehensive understanding of this topic. Consistent practice is the key to mastery.