Understanding Vertical Stretching in Functions

Oct 13, 2025 by miodeya.com 27 views

Which of the Following Tables Shows That $$y=k f(x)$$ Results in the Parent Function $$y=|x+4|$$ Being Vertically Stretched by a $$k$$ Factor of 3?

Understanding Vertical Stretching in Functions

In mathematics, functions are used to describe the relationship between variables. When we talk about vertical stretching in functions, we are referring to the process of changing the scale of the function's output. This is typically achieved by multiplying the function by a constant factor, known as the vertical stretch factor. In this article, we will explore how to identify vertical stretching in functions and apply this concept to a specific problem.

What is Vertical Stretching?

Vertical stretching is a transformation that changes the scale of a function's output. When a function is vertically stretched, its output values are multiplied by a constant factor, resulting in a new function with a different scale. This transformation can be represented algebraically by multiplying the original function by a constant factor, denoted as $$k$$.

The Parent Function $$y=|x+4|$$

The parent function $$y=|x+4|$$ is a quadratic function that represents a V-shaped graph. This function has a vertex at $$( -4, 0 )$$ and opens upwards. The absolute value function is used to ensure that the graph is symmetric about the y-axis.

The Effect of Vertical Stretching on the Parent Function

When the parent function $$y=|x+4|$$ is vertically stretched by a factor of $$k$$, the resulting function is $$y=k f(x)$$. This means that the output values of the original function are multiplied by $$k$$, resulting in a new function with a different scale.

Analyzing the Tables

To determine which table shows that $$y=k f(x)$$ results in the parent function $$y=|x+4|$$ being vertically stretched by a $$k$$ factor of 3, we need to examine each table carefully.

Table 1

$x$	$y=k f(x)$
0	12
-4	0
4	12

Table 2

$x$	$y=k f(x)$
0	3
-4	0
4	3

Table 3

$x$	$y=k f(x)$
0	6
-4	0
4	6

Table 4

$x$	$y=k f(x)$
0	9
-4	0
4	9

Which Table Shows the Correct Vertical Stretching?

To determine which table shows the correct vertical stretching, we need to examine the output values of each table. The correct table should have output values that are 3 times the output values of the parent function.

Table 1 Analysis

In Table 1, the output value for $$x=0$$ is 12, which is not 3 times the output value of the parent function. Therefore, Table 1 does not show the correct vertical stretching.

Table 2 Analysis

In Table 2, the output value for $$x=0$$ is 3, which is 3 times the output value of the parent function. This suggests that Table 2 shows the correct vertical stretching.

Table 3 Analysis

In Table 3, the output value for $$x=0$$ is 6, which is not 3 times the output value of the parent function. Therefore, Table 3 does not show the correct vertical stretching.

Table 4 Analysis

In Table 4, the output value for $$x=0$$ is 9, which is not 3 times the output value of the parent function. Therefore, Table 4 does not show the correct vertical stretching.

Conclusion

Based on the analysis of the tables, Table 2 shows that $$y=k f(x)$$ results in the parent function $$y=|x+4|$$ being vertically stretched by a $$k$$ factor of 3. This is because the output values of Table 2 are 3 times the output values of the parent function, indicating a vertical stretch factor of 3.

Final Thoughts

In conclusion, vertical stretching is an important concept in mathematics that can be used to change the scale of a function's output. By understanding how to identify vertical stretching in functions, we can apply this concept to a variety of problems and gain a deeper understanding of mathematical functions. In this article, we analyzed four tables to determine which one shows the correct vertical stretching of the parent function $$y=|x+4|$$ by a $$k$$ factor of 3. The correct table was Table 2, which had output values that were 3 times the output values of the parent function.
Q&A: Understanding Vertical Stretching in Functions

Introduction

In our previous article, we explored the concept of vertical stretching in functions and analyzed four tables to determine which one shows the correct vertical stretching of the parent function $$y=|x+4|$$ by a $$k$$ factor of 3. In this article, we will answer some frequently asked questions about vertical stretching in functions.

Q: What is vertical stretching in functions?

A: Vertical stretching is a transformation that changes the scale of a function's output. When a function is vertically stretched, its output values are multiplied by a constant factor, resulting in a new function with a different scale.

Q: How do I identify vertical stretching in a function?

A: To identify vertical stretching in a function, look for a constant factor, denoted as $$k$$, that is multiplied by the original function. This factor changes the scale of the function's output, resulting in a new function with a different scale.

Q: What is the effect of vertical stretching on the parent function?

A: When the parent function is vertically stretched by a factor of $$k$$, the resulting function is $$y=k f(x)$$. This means that the output values of the original function are multiplied by $$k$$, resulting in a new function with a different scale.

Q: How do I determine the vertical stretch factor?

A: To determine the vertical stretch factor, examine the output values of the function. The vertical stretch factor is the constant factor that is multiplied by the original function to produce the new function.

Q: Can vertical stretching be applied to any function?

A: Yes, vertical stretching can be applied to any function. However, the function must be in the form $$y=f(x)$$, where $$f(x)$$ is the original function.

Q: What are some common applications of vertical stretching in functions?

A: Vertical stretching has many applications in mathematics, science, and engineering. Some common applications include:

Modeling population growth and decline
Analyzing the behavior of physical systems, such as springs and pendulums
Studying the properties of geometric shapes, such as circles and ellipses
Developing mathematical models for real-world problems, such as economics and finance

Q: How do I graph a function that has been vertically stretched?

A: To graph a function that has been vertically stretched, follow these steps:

Graph the original function.
Identify the vertical stretch factor.
Multiply the original function by the vertical stretch factor.
Graph the resulting function.

Q: Can vertical stretching be combined with other transformations?

A: Yes, vertical stretching can be combined with other transformations, such as horizontal stretching, reflection, and rotation. However, the order of the transformations may affect the final result.

Conclusion

Additional Resources

For more information on vertical stretching in functions, we recommend the following resources:

Khan Academy: Vertical Stretching and Shrinking
Mathway: Vertical Stretching and Shrinking
Wolfram Alpha: Vertical Stretching and Shrinking

Final Thoughts

In conclusion, vertical stretching is a powerful tool in mathematics that can be used to change the scale of a function's output. By understanding how to identify vertical stretching in functions, we can apply this concept to a variety of problems and gain a deeper understanding of mathematical functions. We hope that this Q&A article has provided you with a better understanding of vertical stretching in functions.