Which Function Represents A Vertical Stretch Of An Exponential Function?A. $f(x) = 3\left(\frac{1}{2}\right)^x$B. $\tilde{n}(x) = \frac{1}{2}(3)^x$C. $f(x) = (3)^{2x}$D. $f(x) = 3^{\left(\frac{1}{2}x\right)}$

Introduction

Exponential functions are a fundamental concept in mathematics, and understanding their transformations is crucial for solving various mathematical problems. One of the essential transformations of an exponential function is the vertical stretch, which involves multiplying the function by a constant factor. In this article, we will explore which function represents a vertical stretch of an exponential function.

What is a Vertical Stretch?

A vertical stretch is a transformation that involves multiplying the function by a constant factor. This transformation affects the amplitude or the height of the function, making it taller or shorter. In the case of an exponential function, a vertical stretch can be represented by multiplying the function by a constant factor.

Exponential Functions

An exponential function is a function of the form $f(x) = a^x$, where $a$ is a positive constant and $x$ is the variable. The base $a$ determines the rate at which the function grows or decays. For example, if $a = 2$, the function $f(x) = 2^x$ represents an exponential growth, while if $a = 1/2$, the function $f(x) = (1/2)^x$ represents an exponential decay.

Vertical Stretch of an Exponential Function

To represent a vertical stretch of an exponential function, we need to multiply the function by a constant factor. Let's consider the function $f(x) = a^x$. A vertical stretch of this function can be represented by multiplying it by a constant factor $k$, where $k$ is a positive constant. The resulting function is $f(x) = ka^x$.

Analyzing the Options

Now, let's analyze the options given to determine which one represents a vertical stretch of an exponential function.

Option A: $f(x) = 3\left(\frac{1}{2}\right)^x$

This function represents a vertical stretch of the exponential function $f(x) = \left(\frac{1}{2}\right)^x$ by a factor of 3. The base of the exponential function is $\frac{1}{2}$, and the function is multiplied by 3, which represents a vertical stretch.

Option B: $\tilde{n}(x) = \frac{1}{2}(3)^x$

This function represents a horizontal stretch of the exponential function $f(x) = (3)^x$ by a factor of 2, not a vertical stretch. The base of the exponential function is 3, and the function is multiplied by $\frac{1}{2}$, which represents a horizontal stretch.

Option C: $f(x) = (3)^{2x}$

This function represents a horizontal compression of the exponential function $f(x) = (3)^x$ by a factor of 2, not a vertical stretch. The base of the exponential function is 3, and the exponent is $2x$, which represents a horizontal compression.

Option D: $f(x) = 3^{\left(\frac{1}{2}x\right)}$

This function represents a horizontal stretch of the exponential function $f(x) = 3^x$ by a factor of 2, not a vertical stretch. The base of the exponential function is 3, and the exponent is $\frac{1}{2}x$, which represents a horizontal stretch.

Conclusion

Based on the analysis of the options, we can conclude that the function that represents a vertical stretch of an exponential function is:

Option A: $f(x) = 3\left(\frac{1}{2}\right)^x$

This function represents a vertical stretch of the exponential function $f(x) = \left(\frac{1}{2}\right)^x$ by a factor of 3.

Final Answer

Introduction

In our previous article, we discussed the concept of vertical stretch in exponential functions and identified the correct function that represents a vertical stretch of an exponential function. In this article, we will provide a Q&A section to further clarify any doubts and provide additional information on the topic.

Q: What is a vertical stretch in exponential functions?

A: A vertical stretch is a transformation that involves multiplying the function by a constant factor. This transformation affects the amplitude or the height of the function, making it taller or shorter.

Q: How do you represent a vertical stretch of an exponential function?

A: To represent a vertical stretch of an exponential function, you need to multiply the function by a constant factor. Let's consider the function $f(x) = a^x$. A vertical stretch of this function can be represented by multiplying it by a constant factor $k$, where $k$ is a positive constant. The resulting function is $f(x) = ka^x$.

Q: What is the difference between a vertical stretch and a horizontal stretch?

A: A vertical stretch involves multiplying the function by a constant factor, while a horizontal stretch involves changing the exponent of the function. In the case of an exponential function, a vertical stretch affects the amplitude or the height of the function, while a horizontal stretch affects the rate at which the function grows or decays.

Q: Can you provide an example of a vertical stretch of an exponential function?

A: Yes, consider the function $f(x) = 2^x$. A vertical stretch of this function can be represented by multiplying it by a constant factor of 3, resulting in the function $f(x) = 3(2)^x$.

Q: How do you determine if a function represents a vertical stretch or a horizontal stretch?

A: To determine if a function represents a vertical stretch or a horizontal stretch, you need to examine the transformation applied to the function. If the function is multiplied by a constant factor, it represents a vertical stretch. If the exponent of the function is changed, it represents a horizontal stretch.

Q: Can you provide a summary of the key points discussed in this article?

A: Yes, here is a summary of the key points discussed in this article:

• A vertical stretch is a transformation that involves multiplying the function by a constant factor.
• To represent a vertical stretch of an exponential function, you need to multiply the function by a constant factor.
• A vertical stretch affects the amplitude or the height of the function, making it taller or shorter.
• A horizontal stretch involves changing the exponent of the function, affecting the rate at which the function grows or decays.

Conclusion

In conclusion, a vertical stretch is an essential transformation in exponential functions that affects the amplitude or the height of the function. By understanding how to represent a vertical stretch of an exponential function, you can better analyze and solve mathematical problems involving exponential functions.

Final Answer

The final answer is: A vertical stretch is a transformation that involves multiplying the function by a constant factor.