Identify The Type Of Transformation For The Following Functions, Starting From The Parent Function Y = 1 X Y=\frac{1}{x} Y = X 1 ​ .1. Y = 1 X + 4 − 6 Y=\frac{1}{x+4}-6 Y = X + 4 1 ​ − 6 - Has A Horizontal Translation Of 4 Units To The Left. - Has A Vertical Translation Of 6

Introduction

In mathematics, parent functions are the basic functions from which other functions can be derived by applying various transformations. Understanding these transformations is crucial in algebra and calculus, as it helps us analyze and manipulate functions more effectively. In this article, we will focus on identifying the type of transformation for the given function $y=\frac{1}{x+4}-6$, which is derived from the parent function $y=\frac{1}{x}$.

Parent Function: $y=\frac{1}{x}$

The parent function $y=\frac{1}{x}$ is a reciprocal function, which has a graph that consists of two branches: one in the first quadrant and the other in the third quadrant. The function has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$. The graph of the parent function is shown below:

Graph of Parent Function

  +---------------+
  |               |
  |  y = 1/x     |
  |               |
  +---------------+
  |               |
  |  x = 0       |
  |               |
  +---------------+
  |               |
  |  y = 0       |
  |               |
  +---------------+

Transformation: $y=\frac{1}{x+4}-6$

The given function $y=\frac{1}{x+4}-6$ is derived from the parent function $y=\frac{1}{x}$ by applying two transformations: a horizontal translation and a vertical translation.

Horizontal Translation

The function $y=\frac{1}{x+4}$ has a horizontal translation of 4 units to the left. This means that the graph of the function is shifted 4 units to the left compared to the graph of the parent function. To understand this, let's consider the equation $y=\frac{1}{x+4}$. If we substitute $x+4$ with $x$, we get $y=\frac{1}{x}$. This shows that the function $y=\frac{1}{x+4}$ is equivalent to the parent function $y=\frac{1}{x}$, but with a horizontal translation of 4 units to the left.

Vertical Translation

The function $y=\frac{1}{x+4}-6$ also has a vertical translation of 6 units down. This means that the graph of the function is shifted 6 units down compared to the graph of the function $y=\frac{1}{x+4}$. To understand this, let's consider the equation $y=\frac{1}{x+4}-6$. If we add 6 to both sides of the equation, we get $y=\frac{1}{x+4}$. This shows that the function $y=\frac{1}{x+4}-6$ is equivalent to the function $y=\frac{1}{x+4}$, but with a vertical translation of 6 units down.

Graph of Transformed Function

The graph of the transformed function $y=\frac{1}{x+4}-6$ is shown below:

Graph of Transformed Function

  +---------------+
  |               |
  |  y = 1/(x+4)  |
  |               |
  +---------------+
  |               |
  |  x = -4       |
  |               |
  +---------------+
  |               |
  |  y = 0       |
  |               |
  +---------------+
  |               |
  |  y = -6      |
  |               |
  +---------------+

Conclusion

In conclusion, the function $y=\frac{1}{x+4}-6$ is derived from the parent function $y=\frac{1}{x}$ by applying two transformations: a horizontal translation of 4 units to the left and a vertical translation of 6 units down. Understanding these transformations is crucial in algebra and calculus, as it helps us analyze and manipulate functions more effectively.

Key Takeaways

• The parent function $y=\frac{1}{x}$ is a reciprocal function with a graph that consists of two branches.
• The function $y=\frac{1}{x+4}$ has a horizontal translation of 4 units to the left compared to the parent function.
• The function $y=\frac{1}{x+4}-6$ has a vertical translation of 6 units down compared to the function $y=\frac{1}{x+4}$.
• Understanding transformations is crucial in algebra and calculus, as it helps us analyze and manipulate functions more effectively.

References

• [1] "Algebra and Calculus" by [Author's Name]
• [2] "Mathematics for Dummies" by [Author's Name]

Further Reading

• "Transformations of Parent Functions" by [Author's Name]
• "Algebra and Calculus: A Comprehensive Guide" by [Author's Name]

Introduction

In our previous article, we discussed the transformations of the parent function $y=\frac{1}{x}$ to obtain the function $y=\frac{1}{x+4}-6$. We also explored the concept of horizontal and vertical translations. In this article, we will answer some frequently asked questions related to transformations of parent functions.

Q: What is a parent function?

A: A parent function is a basic function from which other functions can be derived by applying various transformations. Parent functions are used as a reference point to analyze and manipulate functions more effectively.

Q: What are the types of transformations?

A: There are two main types of transformations: horizontal and vertical translations. Horizontal translations involve shifting the graph of a function to the left or right, while vertical translations involve shifting the graph of a function up or down.

Q: How do I determine the type of transformation?

A: To determine the type of transformation, you need to analyze the equation of the function. If the equation has a term in the form of $x+c$ or $x-c$, it indicates a horizontal translation. If the equation has a term in the form of $y+c$ or $y-c$, it indicates a vertical translation.

Q: What is the effect of a horizontal translation on the graph of a function?

A: A horizontal translation shifts the graph of a function to the left or right. If the equation has a term in the form of $x+c$, it shifts the graph $c$ units to the left. If the equation has a term in the form of $x-c$, it shifts the graph $c$ units to the right.

Q: What is the effect of a vertical translation on the graph of a function?

A: A vertical translation shifts the graph of a function up or down. If the equation has a term in the form of $y+c$, it shifts the graph $c$ units up. If the equation has a term in the form of $y-c$, it shifts the graph $c$ units down.

Q: How do I apply transformations to a parent function?

A: To apply transformations to a parent function, you need to follow these steps:

1. Identify the parent function.
2. Determine the type of transformation (horizontal or vertical).
3. Analyze the equation of the function to determine the amount of translation.
4. Apply the transformation to the parent function.

Q: What are some common parent functions?

A: Some common parent functions include:

• Linear functions: $y=mx+b$
• Quadratic functions: $y=ax^2+bx+c$
• Cubic functions: $y=ax^3+bx^2+cx+d$
• Reciprocal functions: $y=\frac{1}{x}$

Q: How do I graph a transformed function?

A: To graph a transformed function, you need to follow these steps:

1. Graph the parent function.
2. Apply the transformation to the parent function.
3. Graph the transformed function.

Conclusion

In conclusion, transformations of parent functions are an essential concept in algebra and calculus. By understanding the types of transformations and how to apply them, you can analyze and manipulate functions more effectively. We hope this Q&A article has provided you with a better understanding of transformations of parent functions.

Key Takeaways

• Parent functions are basic functions from which other functions can be derived by applying various transformations.
• There are two main types of transformations: horizontal and vertical translations.
• Horizontal translations shift the graph of a function to the left or right.
• Vertical translations shift the graph of a function up or down.
• To apply transformations to a parent function, you need to follow the steps outlined above.

References

• [1] "Algebra and Calculus" by [Author's Name]
• [2] "Mathematics for Dummies" by [Author's Name]

Further Reading

• "Transformations of Parent Functions" by [Author's Name]
• "Algebra and Calculus: A Comprehensive Guide" by [Author's Name]

Note: The references and further reading section can be replaced with actual references and further reading materials.