Vertical Stretching And Compressing Functions Homework Answers

Vertical Stretching and Compressing Functions: Homework Answers and Comprehensive Guide

Understanding vertical stretching and compressing of functions is crucial in grasping transformations in algebra and pre-calculus. This comprehensive guide will not only provide answers to common homework problems but also delve into the underlying concepts, offering a robust understanding of these transformations. We'll explore the effects of multiplying a function by a constant, both greater and less than 1, and how this impacts its graph. Furthermore, we’ll address common misconceptions and provide strategies for tackling various problem types.

Understanding Vertical Transformations

Before diving into specific problems, let's solidify our understanding of vertical stretching and compression. These transformations are a type of vertical scaling applied to the graph of a function. They change the y-coordinates of every point on the graph, effectively stretching or compressing the graph vertically.

The general form of a vertical transformation is:

g(x) = af(x)

Where:

• f(x) is the original function.
• a is the constant that determines the vertical scaling.

Vertical Stretching (|a| > 1)

When the absolute value of a is greater than 1 (|a| > 1), the graph of f(x) is stretched vertically. Each y-coordinate is multiplied by a, resulting in a taller, narrower graph. The larger the value of a, the greater the stretch.

Example: If f(x) = x² and g(x) = 3f(x) = 3x², the graph of g(x) is a vertically stretched version of f(x). Every y-coordinate is tripled.

Vertical Compression (0 < |a| < 1)

When the absolute value of a is between 0 and 1 (0 < |a| < 1), the graph of f(x) is compressed vertically. Each y-coordinate is multiplied by a, resulting in a shorter, wider graph. The closer a is to 0, the greater the compression.

Example: If f(x) = x² and g(x) = (1/2)f(x) = (1/2)x², the graph of g(x) is a vertically compressed version of f(x). Every y-coordinate is halved.

Vertical Reflection (a < 0)

If a is negative, the graph is reflected across the x-axis in addition to being stretched or compressed. This is because multiplying the y-coordinates by a negative number reverses their sign.

Working Through Homework Problems

Let's tackle some common homework problems involving vertical stretching and compression. Remember to always analyze the value of 'a' to determine the type of transformation.

Problem 1:

Given f(x) = x³, describe the transformation applied to obtain g(x) = 2f(x). Sketch both graphs.

Answer:

The transformation applied is a vertical stretch by a factor of 2. Each y-coordinate of f(x) is multiplied by 2 to obtain g(x) = 2x³. The graph of g(x) will be taller and narrower than the graph of f(x). (Sketching the graphs requires graphing software or manual plotting of key points. Remember to show the key features of cubic functions, such as the inflection point and general shape.)

Problem 2:

Given f(x) = √x, what is the equation of the function g(x) obtained by compressing f(x) vertically by a factor of 1/3?

Answer:

The equation of g(x) is g(x) = (1/3)√x. This represents a vertical compression of f(x) where every y-coordinate is multiplied by 1/3, resulting in a shorter, wider graph.

Problem 3:

Describe the transformation applied to f(x) = |x| to obtain g(x) = -4|x|. Sketch the graphs.

Answer:

The transformation applied involves two parts:

1. Vertical Stretch: The graph is stretched vertically by a factor of 4.
2. Reflection: The graph is reflected across the x-axis due to the negative sign.

Therefore, g(x) = -4|x| represents a vertical stretch by a factor of 4 followed by a reflection about the x-axis. (Again, sketching requires either graphing tools or manual plotting, highlighting the V-shape of the absolute value function and its reflection.)

Problem 4:

If f(x) = sin(x), how does the graph of g(x) = 0.5sin(x) differ from the graph of f(x)?

Answer:

The graph of g(x) = 0.5sin(x) is a vertical compression of f(x) = sin(x) by a factor of 0.5. The amplitude of the sine wave is halved, making the oscillations smaller. The period and other key features of the sine function remain unchanged.

Problem 5: (More challenging)

Given f(x) = x² - 2x + 1, find the equation of the function g(x) that results from vertically stretching f(x) by a factor of 5 and then shifting it 3 units upward.

Answer:

First, vertically stretch f(x) by a factor of 5: 5(x² - 2x + 1) = 5x² - 10x + 5. Then shift the result 3 units upward: 5x² - 10x + 5 + 3 = 5x² - 10x + 8. Therefore, g(x) = 5x² - 10x + 8. This demonstrates the combination of vertical stretching and vertical shifting.

Common Mistakes and How to Avoid Them

• Confusing stretching and compression: Remember that |a| > 1 signifies stretching, while 0 < |a| < 1 signifies compression.
• Ignoring the negative sign: A negative value of a indicates both stretching/compressing and reflection across the x-axis. Don’t overlook this crucial aspect.
• Misinterpreting the order of operations: When multiple transformations are involved (e.g., stretching and shifting), perform the transformations in the correct order, as specified in the problem.
• Not checking your work: After applying the transformation, visually inspect the graph (or key points) to confirm it aligns with the expected result.

Advanced Applications and Further Exploration

The concepts of vertical stretching and compressing are fundamental to understanding more complex transformations and applications in various fields. Here are some areas for further exploration:

• Horizontal transformations: Explore how horizontal stretching and compression affect the graph of a function.
• Combining transformations: Practice problems involving multiple transformations (vertical and horizontal stretching/compressing, reflections, and shifts).
• Applications in calculus: Vertical stretching and compression play a significant role in understanding derivatives and integrals.
• Real-world applications: Explore how these transformations are used to model real-world phenomena, such as population growth or decay, wave functions, and many other applications across STEM fields.

By mastering the principles of vertical stretching and compression, you'll develop a strong foundation in function transformations, setting you up for success in more advanced mathematical concepts and their numerous applications. Remember to practice regularly, paying close attention to the value of 'a' and the combined effects of multiple transformations. Good luck with your homework!