The Parent Exponential Function

Oct 13, 2025 by windailys.com 37 views

Introduction

In mathematics, transformations of functions are essential concepts that help us understand how functions can be manipulated to create new functions. In this article, we will focus on the transformation of the parent exponential function, ${$ f(x)=2^x $}$, to create a new function, ${$ h(x)=-3 \cdot 2^x $}$. We will analyze the given function and determine which statement is true.

The Parent Exponential Function

The parent exponential function is defined as ${$ f(x)=2^x $}$. This function has a base of 2 and an exponent of x. The graph of this function is a curve that increases exponentially as x increases.

The Transformed Function

The transformed function, ${$ h(x)=-3 \cdot 2^x $}$, is a transformation of the parent exponential function. The negative sign in front of the 3 indicates a reflection across the x-axis, while the 3 is a vertical stretch of the function. The ${$ 2^x $}$ term is the same as the parent function, but the negative sign and the 3 change the shape and position of the graph.

Reflection and Vertical Stretch

A reflection across the x-axis is a transformation that flips the graph of a function upside down. In the case of the transformed function, ${$ h(x)=-3 \cdot 2^x $}$, the negative sign in front of the 3 causes the graph to be reflected across the x-axis.

A vertical stretch is a transformation that changes the scale of a function. In the case of the transformed function, the 3 is a vertical stretch of the parent function. This means that the graph of the transformed function is 3 times as tall as the graph of the parent function.

Which Statement is True?

Now that we have analyzed the transformed function, we can determine which statement is true.

A. Function ${$ h $}$ is a reflection and vertical stretch of the parent function.

B. Function ${$ h $}$ is a reflection of the parent function.

C. Function ${$ h $}$ is a vertical stretch of the parent function.

D. Function ${$ h $}$ is neither a reflection nor a vertical stretch of the parent function.

Based on our analysis, we can conclude that statement A is true. The transformed function, ${$ h(x)=-3 \cdot 2^x $}$, is a reflection and vertical stretch of the parent exponential function, ${$ f(x)=2^x $}$.

Conclusion

In conclusion, transformations of functions are essential concepts in mathematics. By understanding how functions can be manipulated to create new functions, we can gain a deeper understanding of mathematical concepts. In this article, we analyzed the transformed function, ${$ h(x)=-3 \cdot 2^x $}$, and determined that it is a reflection and vertical stretch of the parent exponential function, ${$ f(x)=2^x $}$. We hope that this article has provided valuable insights into the world of mathematics.

References

[1] "Exponential Functions" by Math Open Reference
[2] "Transformations of Functions" by Khan Academy
[3] "Reflection and Vertical Stretch" by IXL Math

Glossary

Exponential function: A function of the form ${$ f(x)=a^x $}$, where a is a positive constant.
Transformation: A change in the position, scale, or orientation of a function.
Reflection: A transformation that flips a function upside down.
Vertical stretch: A transformation that changes the scale of a function.
Q&A: Understanding Transformations of Exponential Functions ===========================================================

Introduction

In our previous article, we explored the transformation of the parent exponential function, ${$ f(x)=2^x $}$, to create a new function, ${$ h(x)=-3 \cdot 2^x $}$. We analyzed the transformed function and determined that it is a reflection and vertical stretch of the parent exponential function. In this article, we will answer some frequently asked questions about transformations of exponential functions.

Q: What is a transformation of a function?

A: A transformation of a function is a change in the position, scale, or orientation of the function. This can include reflections, vertical stretches, horizontal shifts, and more.

Q: What is a reflection of a function?

A: A reflection of a function is a transformation that flips the graph of the function upside down. This can be a reflection across the x-axis, y-axis, or a line other than the x-axis or y-axis.

Q: What is a vertical stretch of a function?

A: A vertical stretch of a function is a transformation that changes the scale of the function. This can make the graph of the function taller or shorter.

Q: How do I determine if a function is a reflection or vertical stretch?

A: To determine if a function is a reflection or vertical stretch, look for the following:

A negative sign in front of the function: This indicates a reflection.
A coefficient (such as 3) in front of the function: This indicates a vertical stretch.

Q: Can a function be both a reflection and vertical stretch?

A: Yes, a function can be both a reflection and vertical stretch. For example, the function ${$ h(x)=-3 \cdot 2^x $}$ is both a reflection and vertical stretch of the parent exponential function.

Q: How do I graph a transformed function?

A: To graph a transformed function, follow these steps:

Graph the parent function.
Apply the transformations (such as reflections and vertical stretches) to the parent function.
Graph the resulting function.

Q: What are some common transformations of exponential functions?

A: Some common transformations of exponential functions include:

Reflections across the x-axis or y-axis
Vertical stretches or compressions
Horizontal shifts
Reflections across a line other than the x-axis or y-axis

Q: How do I determine the domain and range of a transformed function?

A: To determine the domain and range of a transformed function, follow these steps:

Determine the domain and range of the parent function.
Apply the transformations to the parent function.
Determine the new domain and range of the transformed function.

Conclusion

In conclusion, transformations of exponential functions are essential concepts in mathematics. By understanding how functions can be manipulated to create new functions, we can gain a deeper understanding of mathematical concepts. We hope that this article has provided valuable insights into the world of mathematics.

References

[1] "Exponential Functions" by Math Open Reference
[2] "Transformations of Functions" by Khan Academy
[3] "Reflection and Vertical Stretch" by IXL Math

Glossary

Exponential function: A function of the form ${$ f(x)=a^x $}$, where a is a positive constant.
Transformation: A change in the position, scale, or orientation of a function.
Reflection: A transformation that flips a function upside down.
Vertical stretch: A transformation that changes the scale of a function.