Complete The Tables To Show That $y=k F(x$\] Results In The Parent Function Being Vertically Compressed When $k=\frac{1}{2}$.Table 1: $y=x^4$- $x = 0$, $y = 0$- $x = 1$, $y = 1$- $x =

Introduction

In mathematics, functions are a fundamental concept that helps us describe the relationship between variables. One of the essential properties of functions is their behavior under different transformations, such as vertical compression. In this article, we will explore the concept of vertical compression and how it affects the parent function $y = x^4$. We will also complete the tables to demonstrate the effect of vertical compression on the parent function when $k = \frac{1}{2}$.

What is Vertical Compression?

Vertical compression is a type of transformation that involves scaling the function vertically. It is represented by the equation $y = k f(x)$, where $k$ is a constant that determines the degree of compression. When $k$ is greater than 1, the function is stretched vertically, and when $k$ is less than 1, the function is compressed vertically.

The Parent Function: $y = x^4$

The parent function $y = x^4$ is a quartic function that has a minimum value of 0 at $x = 0$. It has a positive leading coefficient, which means that it opens upwards. The function has a single turning point at $x = 0$, and its graph is symmetric about the y-axis.

Table 1: $y = x^4$

Vertical Compression: $y = \frac{1}{2} x^4$

When we apply vertical compression to the parent function $y = x^4$ by multiplying it by $\frac{1}{2}$, we get the function $y = \frac{1}{2} x^4$. This function has the same shape as the parent function, but it is compressed vertically by a factor of $\frac{1}{2}$.

Table 2: $y = \frac{1}{2} x^4$

Comparing the Tables

By comparing the two tables, we can see that the function $y = \frac{1}{2} x^4$ has the same shape as the parent function $y = x^4$, but it is compressed vertically by a factor of $\frac{1}{2}$. The values of $y$ are halved for each corresponding value of $x$.

Conclusion

In conclusion, vertical compression is a type of transformation that involves scaling the function vertically. By applying vertical compression to the parent function $y = x^4$, we get the function $y = \frac{1}{2} x^4$, which has the same shape as the parent function but is compressed vertically by a factor of $\frac{1}{2}$. The tables demonstrate the effect of vertical compression on the parent function, showing that the values of $y$ are halved for each corresponding value of $x$.

Further Exploration

To further explore the concept of vertical compression, we can apply it to other functions and observe the effects on their graphs. We can also investigate how vertical compression affects the properties of functions, such as their domain, range, and symmetry.

References

• [1] "Functions" by Khan Academy
• [2] "Vertical Compression" by Math Open Reference
• [3] "Quartic Functions" by Wolfram MathWorld

Glossary

• Vertical Compression: A type of transformation that involves scaling the function vertically.
• Parent Function: The original function that is being transformed.
• Quartic Function: A function of the form $y = ax^4 + bx^3 + cx^2 + dx + e$.
• Symmetry: The property of a function that describes its reflection about a line or axis.
  Vertical Compression of Functions: A Comprehensive Q&A ===========================================================

Introduction

In our previous article, we explored the concept of vertical compression and its effect on the parent function $y = x^4$. We also completed the tables to demonstrate the effect of vertical compression on the parent function when $k = \frac{1}{2}$. In this article, we will answer some frequently asked questions about vertical compression and provide additional insights into this important concept.

Q&A

Q: What is vertical compression?

A: Vertical compression is a type of transformation that involves scaling the function vertically. It is represented by the equation $y = k f(x)$, where $k$ is a constant that determines the degree of compression.

Q: How does vertical compression affect the graph of a function?

A: Vertical compression affects the graph of a function by scaling it vertically. When $k$ is greater than 1, the function is stretched vertically, and when $k$ is less than 1, the function is compressed vertically.

Q: What is the effect of vertical compression on the parent function $y = x^4$?

A: When we apply vertical compression to the parent function $y = x^4$ by multiplying it by $\frac{1}{2}$, we get the function $y = \frac{1}{2} x^4$. This function has the same shape as the parent function, but it is compressed vertically by a factor of $\frac{1}{2}$.

Q: How do we determine the degree of vertical compression?

A: The degree of vertical compression is determined by the value of $k$. When $k$ is greater than 1, the function is stretched vertically, and when $k$ is less than 1, the function is compressed vertically.

Q: Can vertical compression be applied to any function?

A: Yes, vertical compression can be applied to any function. However, the effect of vertical compression will depend on the specific function and the value of $k$.

Q: How does vertical compression affect the properties of a function?

A: Vertical compression affects the properties of a function, such as its domain, range, and symmetry. For example, when a function is compressed vertically, its range will be reduced.

Q: Can vertical compression be combined with other transformations?

A: Yes, vertical compression can be combined with other transformations, such as horizontal compression, reflection, and rotation.

Q: How do we graph a function that has been vertically compressed?

A: To graph a function that has been vertically compressed, we can use the same techniques as graphing the original function. However, we need to take into account the degree of vertical compression.

Q: What are some real-world applications of vertical compression?

A: Vertical compression has many real-world applications, such as in physics, engineering, and computer science. For example, it can be used to model the behavior of objects under different forces or to analyze the performance of systems.

Conclusion

In conclusion, vertical compression is an important concept in mathematics that has many real-world applications. By understanding how vertical compression affects the graph of a function, we can better analyze and model complex systems. We hope that this Q&A article has provided you with a deeper understanding of vertical compression and its many uses.

Further Exploration

To further explore the concept of vertical compression, we recommend the following resources:

• [1] "Functions" by Khan Academy
• [2] "Vertical Compression" by Math Open Reference
• [3] "Quartic Functions" by Wolfram MathWorld

Glossary

• Vertical Compression: A type of transformation that involves scaling the function vertically.
• Parent Function: The original function that is being transformed.
• Quartic Function: A function of the form $y = ax^4 + bx^3 + cx^2 + dx + e$.
• Symmetry: The property of a function that describes its reflection about a line or axis.