1-2 Additional Practice Transformations Of Functions

1-2 Additional Practice Transformations of Functions: Mastering Function Manipulation

Transformations of functions are a cornerstone of algebra and pre-calculus, providing a powerful tool to manipulate and understand the behavior of various functions. This article delves into two crucial function transformations – horizontal stretching/compression and vertical stretching/compression – providing detailed explanations, examples, and practice problems to solidify your understanding. We will explore how these transformations affect the graph of a function and how to apply them systematically.

Understanding Function Transformations: A Quick Recap

Before we dive into horizontal and vertical stretching/compression, let's briefly review the fundamental transformations:

• Vertical Shifts: Adding a constant 'k' to a function, f(x) + k, shifts the graph vertically upwards by 'k' units if k > 0, and downwards by |k| units if k < 0.

• Horizontal Shifts: Adding a constant 'h' inside the function, f(x+h), shifts the graph horizontally to the left by 'h' units if h > 0, and to the right by |h| units if h < 0.

• Reflections: -f(x) reflects the graph across the x-axis, while f(-x) reflects the graph across the y-axis.

Horizontal Stretching and Compression: Expanding and Shrinking Horizontally

Horizontal stretching and compression involves manipulating the input values of a function. This transformation affects the graph's width. The general form is:

f(bx), where 'b' is a constant.

• Compression (0 < |b| < 1): When |b| is between 0 and 1 (e.g., 1/2, 1/3), the graph is compressed horizontally. The graph appears narrower, as the x-values are scaled down. Think of it as squeezing the graph towards the y-axis.

• Stretching (|b| > 1): When |b| is greater than 1 (e.g., 2, 3), the graph is stretched horizontally. The graph appears wider, as the x-values are scaled up. Think of it as pulling the graph away from the y-axis.

Important Note: The effect on the x-coordinates is the inverse of what you might initially expect. A value of b > 1 widens the graph, while 0 < b < 1 narrows it.

Example 1: Horizontal Compression

Let's consider the function f(x) = x². Now let's apply a horizontal compression with b = 1/2: g(x) = f(2x) = (2x)² = 4x².

The graph of g(x) = 4x² is narrower than the graph of f(x) = x². Every x-coordinate is halved, resulting in a compressed graph.

Example 2: Horizontal Stretching

Consider the function f(x) = √x. Let's apply a horizontal stretch with b = 1/3: g(x) = f(x/3) = √(x/3).

The graph of g(x) = √(x/3) is wider than the graph of f(x) = √x. Every x-coordinate is tripled, resulting in a stretched graph.

Practice Problems: Horizontal Transformations

1. Problem: Given f(x) = |x|, sketch the graph of g(x) = f(3x). Describe the transformation.

2. Problem: Given f(x) = x³, sketch the graph of h(x) = f(x/2). Describe the transformation.

3. Problem: The graph of y = f(x) passes through the points (1, 2), (2, 4), and (3, 6). What are the coordinates of the points on the graph of y = f(2x)?

4. Problem: If the graph of y = g(x) is horizontally compressed by a factor of 1/4, what is the new function?

Vertical Stretching and Compression: Altering the Height

Vertical stretching and compression involves multiplying the entire function by a constant. This transformation affects the graph's height. The general form is:

af(x), where 'a' is a constant.

• Compression (0 < |a| < 1): When |a| is between 0 and 1, the graph is compressed vertically. The graph appears shorter, as the y-values are scaled down. Think of it as squeezing the graph towards the x-axis.

• Stretching (|a| > 1): When |a| is greater than 1, the graph is stretched vertically. The graph appears taller, as the y-values are scaled up. Think of it as pulling the graph away from the x-axis.

Example 3: Vertical Compression

Consider the function f(x) = x³. Let's apply a vertical compression with a = 1/2: g(x) = (1/2)f(x) = (1/2)x³.

The graph of g(x) = (1/2)x³ is shorter than the graph of f(x) = x³. Every y-coordinate is halved.

Example 4: Vertical Stretching

Consider the function f(x) = sin(x). Let's apply a vertical stretch with a = 3: g(x) = 3f(x) = 3sin(x).

The graph of g(x) = 3sin(x) is taller than the graph of f(x) = sin(x). The amplitude is tripled.

Practice Problems: Vertical Transformations

1. Problem: Given f(x) = √x, sketch the graph of g(x) = 4f(x). Describe the transformation.

2. Problem: Given f(x) = eˣ, sketch the graph of h(x) = (1/3)f(x). Describe the transformation.

3. Problem: The graph of y = f(x) passes through the points (1, 1), (2, 4), and (3, 9). What are the coordinates of the points on the graph of y = 2f(x)?

4. Problem: If the graph of y = g(x) is vertically stretched by a factor of 5, what is the new function?

Combining Transformations: A Symphony of Changes

The true power of function transformations lies in combining multiple transformations. You can sequentially apply vertical shifts, horizontal shifts, reflections, stretching, and compression to create complex transformations. The order of operations matters! Generally, transformations within the function (horizontal shifts, stretches/compressions) are applied before transformations outside the function (vertical shifts, stretches/compressions).

Example 5: Combining Transformations

Let's transform f(x) = x² as follows: g(x) = -2f(x+1) + 3.

1. Horizontal Shift: f(x+1) shifts the graph one unit to the left.
2. Vertical Stretch: 2f(x+1) stretches the graph vertically by a factor of 2.
3. Reflection: -2f(x+1) reflects the graph across the x-axis.
4. Vertical Shift: -2f(x+1) + 3 shifts the graph three units upwards.

This results in a parabola that opens downwards, is narrower than the original, shifted one unit to the left, and three units upwards.

Practice Problems: Combined Transformations

1. Problem: Describe the transformations applied to f(x) = |x| to obtain g(x) = -|x-2| + 5. Sketch the graph of g(x).

2. Problem: Given f(x) = x³, find the equation of the function that results from shifting f(x) two units to the right, stretching it vertically by a factor of 3, and then reflecting it across the x-axis.

3. Problem: A function g(x) is obtained by compressing f(x) = √x horizontally by a factor of 1/2, shifting it 1 unit to the left, stretching it vertically by 2, and then shifting it 3 units upwards. Find the equation for g(x).

Conclusion: Mastering Function Transformations

Understanding and applying function transformations is crucial for success in mathematics and related fields. By mastering horizontal and vertical stretching/compression, along with the other basic transformations, you gain the ability to manipulate and analyze functions effectively. Practice is key; work through the provided problems and explore other examples to build your intuition and skill. The ability to visualize and predict the effect of transformations on a function’s graph is a highly valuable mathematical skill. Remember to pay close attention to the order of operations when combining multiple transformations. With consistent practice and a clear understanding of the principles, you will become proficient in this powerful tool for analyzing and manipulating functions.