Determine The Vertical And Horizontal Asymptotes Of The Exponential Function Y = 3 ( 2 ) X − 5 Y=3(2)^x-5 Y = 3 ( 2 ) X − 5 .A. V.A.: X = − 5 X=-5 X = − 5 B. V.A.: NoneC. H.A.: Y = − 5 Y=-5 Y = − 5 D. H.A.: None

Introduction

Asymptotes are lines or curves that a function approaches as the input or output values become very large or very small. In this article, we will focus on determining the vertical and horizontal asymptotes of the exponential function $y=3(2)^x-5$. Understanding asymptotes is crucial in mathematics, as they provide valuable information about the behavior of a function.

What are Vertical and Horizontal Asymptotes?

Before we dive into the specifics of the given function, let's briefly discuss what vertical and horizontal asymptotes are.

• Vertical Asymptotes (V.A.): A vertical asymptote is a line that a function approaches as the input value (x) becomes very large or very small. In other words, it's a line that the function gets arbitrarily close to but never touches. Vertical asymptotes occur when the function is undefined at a particular point.
• Horizontal Asymptotes (H.A.): A horizontal asymptote is a line that a function approaches as the output value (y) becomes very large or very small. In other words, it's a line that the function gets arbitrarily close to but never touches. Horizontal asymptotes occur when the function approaches a constant value as the input value becomes very large or very small.

The Exponential Function $y=3(2)^x-5$

Now that we have a basic understanding of vertical and horizontal asymptotes, let's focus on the given exponential function $y=3(2)^x-5$. This function has a base of 2, which is an exponential function, and a coefficient of 3, which is a constant multiplier.

Determining Vertical Asymptotes

To determine the vertical asymptotes of the function, we need to find the values of x that make the function undefined. In the case of the exponential function, the function is undefined when the base (2) raised to the power of x is equal to 0 or infinity.

However, since the base (2) is always positive, the function is never undefined. Therefore, there are no vertical asymptotes for this function.

Determining Horizontal Asymptotes

To determine the horizontal asymptotes of the function, we need to find the values of y that the function approaches as x becomes very large or very small.

As x becomes very large, the term $3(2)^x$ dominates the function, and the constant term -5 becomes negligible. Therefore, as x becomes very large, the function approaches $3(2)^x$.

Similarly, as x becomes very small, the term $3(2)^x$ dominates the function, and the constant term -5 becomes negligible. Therefore, as x becomes very small, the function approaches $3(2)^x$.

Since the function approaches $3(2)^x$ as x becomes very large or very small, the horizontal asymptote is $y=3(2)^x$. However, this is not a constant value, so we need to find the limit of the function as x approaches infinity.

Using the properties of exponents, we can rewrite the function as $y=3(2)^x-5=3(2)^x-5=3(2)^x-5(2)^0$. As x approaches infinity, the term $3(2)^x$ grows much faster than the constant term -5. Therefore, the limit of the function as x approaches infinity is infinity.

However, since the function approaches infinity as x approaches infinity, there is no horizontal asymptote in the classical sense. Instead, we say that the function has a horizontal asymptote at infinity.

Conclusion

In conclusion, the vertical asymptotes of the exponential function $y=3(2)^x-5$ are none, and the horizontal asymptote is none in the classical sense. However, the function has a horizontal asymptote at infinity.

Answer

Based on the analysis above, the correct answer is:

• A. V.A.: none
• B. V.A.: none
• C. H.A.: none
• D. H.A.: none

Introduction

In our previous article, we discussed how to determine the vertical and horizontal asymptotes of the exponential function $y=3(2)^x-5$. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information.

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

A: A vertical asymptote is a line that a function approaches as the input value (x) becomes very large or very small. In other words, it's a line that the function gets arbitrarily close to but never touches. A horizontal asymptote, on the other hand, is a line that a function approaches as the output value (y) becomes very large or very small.

Q: How do you determine the vertical asymptotes of an exponential function?

A: To determine the vertical asymptotes of an exponential function, you need to find the values of x that make the function undefined. In the case of an exponential function, the function is undefined when the base raised to the power of x is equal to 0 or infinity. However, since the base is always positive, the function is never undefined, and there are no vertical asymptotes.

Q: How do you determine the horizontal asymptotes of an exponential function?

A: To determine the horizontal asymptotes of an exponential function, you need to find the values of y that the function approaches as x becomes very large or very small. As x becomes very large, the term with the highest power of x dominates the function, and the constant term becomes negligible. Therefore, as x becomes very large, the function approaches the value of the term with the highest power of x.

Q: What is a horizontal asymptote at infinity?

A: A horizontal asymptote at infinity is a line that a function approaches as x becomes very large or very small. In other words, it's a line that the function gets arbitrarily close to but never touches. However, since the function approaches infinity as x approaches infinity, there is no horizontal asymptote in the classical sense.

Q: Can you provide an example of a function with a horizontal asymptote at infinity?

A: Yes, the function $y=3(2)^x-5$ has a horizontal asymptote at infinity. As x becomes very large, the term $3(2)^x$ grows much faster than the constant term -5, and the function approaches infinity.

Q: How do you determine the horizontal asymptote of a function with a horizontal asymptote at infinity?

A: To determine the horizontal asymptote of a function with a horizontal asymptote at infinity, you need to find the limit of the function as x approaches infinity. In the case of the function $y=3(2)^x-5$, the limit as x approaches infinity is infinity.

Q: What is the significance of determining the vertical and horizontal asymptotes of a function?

A: Determining the vertical and horizontal asymptotes of a function is crucial in mathematics, as it provides valuable information about the behavior of the function. It helps us understand how the function behaves as the input or output values become very large or very small.

Conclusion

In conclusion, determining the vertical and horizontal asymptotes of a function is an essential concept in mathematics. By understanding how to determine these asymptotes, we can gain valuable insights into the behavior of a function and make informed decisions about its applications.

Frequently Asked Questions

• Q: What is the difference between a vertical and horizontal asymptote?
• A: A vertical asymptote is a line that a function approaches as the input value (x) becomes very large or very small. A horizontal asymptote, on the other hand, is a line that a function approaches as the output value (y) becomes very large or very small.
• Q: How do you determine the vertical asymptotes of an exponential function?
• A: To determine the vertical asymptotes of an exponential function, you need to find the values of x that make the function undefined.
• Q: How do you determine the horizontal asymptotes of an exponential function?
• A: To determine the horizontal asymptotes of an exponential function, you need to find the values of y that the function approaches as x becomes very large or very small.

Additional Resources

• For more information on determining vertical and horizontal asymptotes, please refer to our previous article on the topic.
• For additional practice problems and examples, please visit our website or consult a mathematics textbook.

Conclusion

In conclusion, determining the vertical and horizontal asymptotes of a function is an essential concept in mathematics. By understanding how to determine these asymptotes, we can gain valuable insights into the behavior of a function and make informed decisions about its applications.