Parent Function Transformations

Oct 13, 2025 by talynt.org 107 views

Introduction

The parent function $y=\frac{1}{x}$ is a fundamental function in mathematics, and it has numerous applications in various fields, including calculus, algebra, and analysis. However, when we encounter a function like $y=\frac{2}{x+1}-3$ , it is not immediately clear how it relates to the parent function. In this article, we will explore the transformations that have occurred to the parent function $y=\frac{1}{x}$ to obtain the given function.

Horizontal Shift

The first transformation that we notice is a horizontal shift. The parent function $y=\frac{1}{x}$ has a vertical asymptote at $x=0$ , but the given function $y=\frac{2}{x+1}-3$ has a vertical asymptote at $x=-1$ . This indicates that the function has been shifted horizontally to the left by 1 unit.

To understand this transformation, let's consider the equation of the given function:

y=\frac{2}{x+1}-3

We can rewrite this equation as:

y=\frac{2}{(x+1)-1}-3

This shows that the function has been shifted horizontally to the left by 1 unit, resulting in a new vertical asymptote at $x=-1$ .

Vertical Stretch

In addition to the horizontal shift, the given function also undergoes a vertical stretch. The parent function $y=\frac{1}{x}$ has a vertical asymptote at $y=0$ , but the given function has a vertical asymptote at $y=-3$ . This indicates that the function has been stretched vertically by a factor of 3.

To understand this transformation, let's consider the equation of the given function:

y=\frac{2}{x+1}-3

We can rewrite this equation as:

y=\frac{2}{x+1}-3(0)

This shows that the function has been stretched vertically by a factor of 3, resulting in a new vertical asymptote at $y=-3$ .

Vertical Translation

The given function also undergoes a vertical translation. The parent function $y=\frac{1}{x}$ has a vertical asymptote at $y=0$ , but the given function has a vertical asymptote at $y=-3$ . This indicates that the function has been translated vertically by 3 units downward.

To understand this transformation, let's consider the equation of the given function:

y=\frac{2}{x+1}-3

We can rewrite this equation as:

y=\frac{2}{x+1}-3(1)

This shows that the function has been translated vertically by 3 units downward, resulting in a new vertical asymptote at $y=-3$ .

Conclusion

In conclusion, the given function $y=\frac{2}{x+1}-3$ has undergone three transformations to obtain the parent function $y=\frac{1}{x}$ . These transformations include a horizontal shift to the left by 1 unit, a vertical stretch by a factor of 3, and a vertical translation downward by 3 units. Understanding these transformations is essential in mathematics, as it allows us to analyze and manipulate functions in a more efficient and effective manner.

Parent Function Transformations

Horizontal Shift

A horizontal shift is a transformation that moves the function to the left or right. In the case of the given function, the horizontal shift is to the left by 1 unit.

Vertical Stretch

A vertical stretch is a transformation that stretches the function vertically. In the case of the given function, the vertical stretch is by a factor of 3.

Vertical Translation

A vertical translation is a transformation that moves the function up or down. In the case of the given function, the vertical translation is downward by 3 units.

Real-World Applications

Understanding the transformations of the parent function $y=\frac{1}{x}$ has numerous real-world applications. For example, in physics, the function $y=\frac{1}{x}$ is used to model the behavior of electric circuits. In economics, the function $y=\frac{1}{x}$ is used to model the behavior of supply and demand curves.

Conclusion

References

[1] "Calculus" by Michael Spivak
[2] "Algebra" by Michael Artin
[3] "Analysis" by Walter Rudin

Glossary

Horizontal Shift: A transformation that moves the function to the left or right.
Vertical Stretch: A transformation that stretches the function vertically.
Vertical Translation: A transformation that moves the function up or down.
Parent Function: The original function that undergoes transformations.
Transformations: Changes made to the parent function to obtain a new function.
Q&A: Transformations of the Parent Function $y=\frac{1}{x}$ ===========================================================

Introduction

In our previous article, we explored the transformations that have occurred to the parent function $y=\frac{1}{x}$ to obtain the given function $y=\frac{2}{x+1}-3$ . In this article, we will answer some frequently asked questions about the transformations of the parent function.

Q: What is the parent function?

A: The parent function is the original function that undergoes transformations. In this case, the parent function is $y=\frac{1}{x}$ .

Q: What are the three transformations that occur to the parent function?

A: The three transformations that occur to the parent function are:

Horizontal Shift: The function is shifted horizontally to the left by 1 unit.
Vertical Stretch: The function is stretched vertically by a factor of 3.
Vertical Translation: The function is translated vertically downward by 3 units.

Q: Why is the horizontal shift to the left by 1 unit?

A: The horizontal shift to the left by 1 unit is due to the presence of the term $(x+1)$ in the denominator of the given function. This term shifts the function to the left by 1 unit.

Q: Why is the vertical stretch by a factor of 3?

A: The vertical stretch by a factor of 3 is due to the presence of the term $2$ in the numerator of the given function. This term stretches the function vertically by a factor of 3.

Q: Why is the vertical translation downward by 3 units?

A: The vertical translation downward by 3 units is due to the presence of the term $-3$ in the given function. This term translates the function downward by 3 units.

Q: How do the transformations affect the graph of the function?

A: The transformations affect the graph of the function in the following ways:

The horizontal shift to the left by 1 unit moves the graph to the left by 1 unit.
The vertical stretch by a factor of 3 stretches the graph vertically by a factor of 3.
The vertical translation downward by 3 units moves the graph downward by 3 units.

Q: What are some real-world applications of the transformations of the parent function?

A: The transformations of the parent function have numerous real-world applications, including:

Modeling the behavior of electric circuits in physics
Modeling the behavior of supply and demand curves in economics
Analyzing the behavior of complex systems in engineering

Q: How can I apply the transformations of the parent function to other functions?

A: To apply the transformations of the parent function to other functions, you can follow these steps:

Identify the parent function and the given function.
Determine the transformations that have occurred to the parent function.
Apply the transformations to the parent function to obtain the given function.

Conclusion

In conclusion, the transformations of the parent function $y=\frac{1}{x}$ are essential in mathematics and have numerous real-world applications. By understanding these transformations, you can analyze and manipulate functions in a more efficient and effective manner.

References

[1] "Calculus" by Michael Spivak
[2] "Algebra" by Michael Artin
[3] "Analysis" by Walter Rudin

Glossary

Parent Function: The original function that undergoes transformations.
Transformations: Changes made to the parent function to obtain a new function.
Horizontal Shift: A transformation that moves the function to the left or right.
Vertical Stretch: A transformation that stretches the function vertically.
Vertical Translation: A transformation that moves the function up or down.