Shifting the Graph Down by 2 Units

Oct 13, 2025 by windailys.com 39 views

Introduction

Exponential functions are a fundamental concept in mathematics, and they have numerous applications in various fields, including science, engineering, and economics. The graph of the exponential function $y = e^x$ is a fundamental tool for understanding the behavior of exponential functions. In this article, we will use the graph of $y = e^x$ and transformations to sketch the exponential function $f(x) = e^x - 2$ . We will also determine the domain and range of the function, the $y$ -intercept, and the equation of the horizontal asymptote.

The Graph of $y = e^x$

The graph of the exponential function $y = e^x$ is a continuous, increasing curve that passes through the point $(0, 1)$ . The graph has a horizontal asymptote at $y = 0$ and a vertical asymptote at $x = -\infty$ . The graph is also symmetric about the $y$ -axis.

Transformations of the Graph of $y = e^x$

To sketch the graph of $f(x) = e^x - 2$ , we can use the graph of $y = e^x$ and apply transformations. The function $f(x) = e^x - 2$ is obtained by shifting the graph of $y = e^x$ down by 2 units.

Shifting the Graph Down by 2 Units

When we shift the graph of $y = e^x$ down by 2 units, the new graph will have the same shape as the original graph, but it will be shifted down by 2 units. This means that the new graph will have the same $x$ -intercepts as the original graph, but the $y$ -intercept will be shifted down by 2 units.

Domain and Range of the Function

The domain of the function $f(x) = e^x - 2$ is the set of all real numbers, since the exponential function $e^x$ is defined for all real numbers. The range of the function is the set of all real numbers greater than or equal to -2, since the minimum value of the function is -2.

$y$ -Intercept

The $y$ -intercept of the function $f(x) = e^x - 2$ is the point where the graph intersects the $y$ -axis. Since the graph is shifted down by 2 units, the $y$ -intercept will be at the point $(0, -2)$ .

Equation of the Horizontal Asymptote

The horizontal asymptote of the function $f(x) = e^x - 2$ is the horizontal line that the graph approaches as $x$ approaches infinity. Since the graph is shifted down by 2 units, the horizontal asymptote will be at the line $y = -2$ .

Conclusion

In this article, we used the graph of $y = e^x$ and transformations to sketch the exponential function $f(x) = e^x - 2$ . We determined the domain and range of the function, the $y$ -intercept, and the equation of the horizontal asymptote. The graph of $f(x) = e^x - 2$ is a continuous, increasing curve that passes through the point $(0, -2)$ . The domain of the function is the set of all real numbers, and the range is the set of all real numbers greater than or equal to -2. The $y$ -intercept is at the point $(0, -2)$ , and the horizontal asymptote is at the line $y = -2$ .

References

[1] Larson, R. E., & Edwards, B. I. (2019). Calculus: Early Transcendentals. Cengage Learning.
[2] Rogawski, J., & Adams, C. (2015). Calculus: Early Transcendentals. W.H. Freeman and Company.
[3] Stewart, J. (2019). Calculus: Early Transcendentals. Cengage Learning.

Introduction

In our previous article, we used the graph of $y = e^x$ and transformations to sketch the exponential function $f(x) = e^x - 2$ . We determined the domain and range of the function, the $y$ -intercept, and the equation of the horizontal asymptote. In this article, we will answer some frequently asked questions about the graph of $f(x) = e^x - 2$ .

Q: What is the domain of the function $f(x) = e^x - 2$ ?

A: The domain of the function $f(x) = e^x - 2$ is the set of all real numbers, since the exponential function $e^x$ is defined for all real numbers.

Q: What is the range of the function $f(x) = e^x - 2$ ?

A: The range of the function $f(x) = e^x - 2$ is the set of all real numbers greater than or equal to -2, since the minimum value of the function is -2.

Q: What is the $y$ -intercept of the function $f(x) = e^x - 2$ ?

A: The $y$ -intercept of the function $f(x) = e^x - 2$ is the point where the graph intersects the $y$ -axis. Since the graph is shifted down by 2 units, the $y$ -intercept will be at the point $(0, -2)$ .

Q: What is the equation of the horizontal asymptote of the function $f(x) = e^x - 2$ ?

A: The horizontal asymptote of the function $f(x) = e^x - 2$ is the horizontal line that the graph approaches as $x$ approaches infinity. Since the graph is shifted down by 2 units, the horizontal asymptote will be at the line $y = -2$ .

Q: How do I sketch the graph of $f(x) = e^x - 2$ ?

A: To sketch the graph of $f(x) = e^x - 2$ , you can use the graph of $y = e^x$ and apply transformations. Shift the graph of $y = e^x$ down by 2 units to obtain the graph of $f(x) = e^x - 2$ .

Q: What are some real-world applications of the exponential function $f(x) = e^x - 2$ ?

A: The exponential function $f(x) = e^x - 2$ has many real-world applications, including modeling population growth, chemical reactions, and financial transactions.

Q: How do I determine the domain and range of the function $f(x) = e^x - 2$ ?

A: To determine the domain and range of the function $f(x) = e^x - 2$ , you can use the properties of the exponential function $e^x$ . Since the exponential function $e^x$ is defined for all real numbers, the domain of the function $f(x) = e^x - 2$ is the set of all real numbers. The range of the function is the set of all real numbers greater than or equal to -2.

Q: What is the relationship between the graph of $y = e^x$ and the graph of $f(x) = e^x - 2$ ?

A: The graph of $f(x) = e^x - 2$ is obtained by shifting the graph of $y = e^x$ down by 2 units. This means that the graph of $f(x) = e^x - 2$ has the same shape as the graph of $y = e^x$ , but it is shifted down by 2 units.

Conclusion

In this article, we answered some frequently asked questions about the graph of $f(x) = e^x - 2$ . We determined the domain and range of the function, the $y$ -intercept, and the equation of the horizontal asymptote. We also discussed some real-world applications of the exponential function $f(x) = e^x - 2$ and provided tips on how to sketch the graph of the function.

References

[1] Larson, R. E., & Edwards, B. I. (2019). Calculus: Early Transcendentals. Cengage Learning.
[2] Rogawski, J., & Adams, C. (2015). Calculus: Early Transcendentals. W.H. Freeman and Company.
[3] Stewart, J. (2019). Calculus: Early Transcendentals. Cengage Learning.