The Given Exponential Function

Oct 13, 2025 by talynt.org 14 views

Introduction

In mathematics, the y-intercept and asymptote are two fundamental concepts used to analyze and understand the behavior of functions. The y-intercept is the point where the function intersects the y-axis, while the asymptote is a line that the function approaches as the input values increase or decrease without bound. In this article, we will focus on identifying the y-intercept and asymptote of a given exponential function.

The Given Exponential Function

The given exponential function is:

y = (5)^{x+2} - 3

Calculating the y-intercept

To calculate the y-intercept, we need to set x = 0 in the given function.

y = (5)^{0+2} - 3

y = (5)^{2} - 3

y = 25 - 3

y = 22

Therefore, the y-intercept of the given function is (0, 22).

Identifying the Horizontal Asymptote

To identify the horizontal asymptote, we need to analyze the behavior of the function as x approaches positive or negative infinity.

As x approaches positive infinity, the term $(5)^{x+2}$ grows exponentially, and the function approaches positive infinity.

As x approaches negative infinity, the term $(5)^{x+2}$ approaches 0, and the function approaches -3.

Therefore, the horizontal asymptote of the given function is y = -3.

Discussion

In this article, we have identified the y-intercept and asymptote of a given exponential function. The y-intercept is the point where the function intersects the y-axis, while the asymptote is a line that the function approaches as the input values increase or decrease without bound.

The y-intercept is calculated by setting x = 0 in the given function, while the horizontal asymptote is identified by analyzing the behavior of the function as x approaches positive or negative infinity.

Conclusion

In conclusion, identifying the y-intercept and asymptote of a given function is an essential step in understanding its behavior and analyzing its properties. By following the steps outlined in this article, you can easily identify the y-intercept and asymptote of any given function.

Example Problems

Problem 1

Find the y-intercept of the function:

y = (2)^{x+1} - 2

Solution

To find the y-intercept, we need to set x = 0 in the given function.

y = (2)^{0+1} - 2

y = (2)^{1} - 2

y = 2 - 2

y = 0

Therefore, the y-intercept of the function is (0, 0).

Problem 2

Find the horizontal asymptote of the function:

y = (3)^{x+1} + 1

Solution

To find the horizontal asymptote, we need to analyze the behavior of the function as x approaches positive or negative infinity.

As x approaches positive infinity, the term $(3)^{x+1}$ grows exponentially, and the function approaches positive infinity.

As x approaches negative infinity, the term $(3)^{x+1}$ approaches 0, and the function approaches 1.

Therefore, the horizontal asymptote of the function is y = 1.

Key Takeaways

The y-intercept is the point where the function intersects the y-axis.
The horizontal asymptote is a line that the function approaches as the input values increase or decrease without bound.
To identify the y-intercept, set x = 0 in the given function.
To identify the horizontal asymptote, analyze the behavior of the function as x approaches positive or negative infinity.

Final Thoughts

Introduction

In our previous article, we discussed how to identify the y-intercept and asymptote of a given exponential function. In this article, we will provide a Q&A section to help you better understand the concepts and provide additional examples to reinforce your understanding.

Q1: What is the y-intercept of the function y = (2)^x - 1?

A1: To find the y-intercept, we need to set x = 0 in the given function.

y = (2)^{0} - 1

y = 1 - 1

y = 0

Therefore, the y-intercept of the function is (0, 0).

Q2: What is the horizontal asymptote of the function y = (3)^x + 2?

A2: To find the horizontal asymptote, we need to analyze the behavior of the function as x approaches positive or negative infinity.

As x approaches positive infinity, the term $(3)^{x}$ grows exponentially, and the function approaches positive infinity.

As x approaches negative infinity, the term $(3)^{x}$ approaches 0, and the function approaches 2.

Therefore, the horizontal asymptote of the function is y = 2.

Q3: How do I identify the y-intercept of a function with a negative exponent?

A3: To identify the y-intercept of a function with a negative exponent, we need to set x = 0 in the given function.

For example, consider the function y = (2)^{-x} - 1.

y = (2)^{-0} - 1

y = (2)^{0} - 1

y = 1 - 1

y = 0

Therefore, the y-intercept of the function is (0, 0).

Q4: What is the horizontal asymptote of the function y = (1/2)^x + 1?

A4: To find the horizontal asymptote, we need to analyze the behavior of the function as x approaches positive or negative infinity.

As x approaches positive infinity, the term $(1/2)^{x}$ approaches 0, and the function approaches 1.

As x approaches negative infinity, the term $(1/2)^{x}$ grows exponentially, and the function approaches positive infinity.

Therefore, the horizontal asymptote of the function is y = 1.

Q5: Can I have a function with no y-intercept or asymptote?

A5: Yes, it is possible to have a function with no y-intercept or asymptote. For example, consider the function y = x^2 + 1.

As x approaches positive or negative infinity, the term x^2 grows exponentially, and the function approaches positive infinity.

However, the function does not have a y-intercept because it is not defined at x = 0.

Q6: How do I identify the y-intercept and asymptote of a function with a rational exponent?

A6: To identify the y-intercept and asymptote of a function with a rational exponent, we need to analyze the behavior of the function as x approaches positive or negative infinity.

For example, consider the function y = (x^2 + 1)^x.

As x approaches positive infinity, the term (x^2 + 1)^x grows exponentially, and the function approaches positive infinity.

As x approaches negative infinity, the term (x^2 + 1)^x approaches 0, and the function approaches 1.

Therefore, the horizontal asymptote of the function is y = 1.

Conclusion

In this Q&A article, we have provided additional examples and explanations to help you better understand the concepts of identifying the y-intercept and asymptote of a given function. We hope that this article has been helpful in reinforcing your understanding of these important concepts.