The graph of y = e^x represents an exponential function where the base e (approximately equal to 2.718) is raised to the power of x. This graph is essential in understanding exponential growth, displaying unique characteristics such as an increasing trend, a horizontal asymptote at y = 0, and passing through the point (0, 1). Recognizing these features helps in analyzing real-world applications such as population growth and compound interest.
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The graph of y = e^x always passes through the point (0, 1), which means when x is 0, y equals e^0, resulting in a value of 1.
As x approaches negative infinity, the value of e^x approaches 0 but never actually reaches it, indicating the presence of a horizontal asymptote at y = 0.
The function is continuous and smooth for all real values of x, meaning there are no breaks or gaps in the graph.
The slope of the graph at any point x is equal to the value of e^x at that point, showcasing that the function grows faster as x increases.
In real-world scenarios, this graph is often used to model phenomena like population growth or radioactive decay where quantities grow or shrink at rates proportional to their current value.
Review Questions
How does the graph of y = e^x illustrate the concept of exponential growth?
The graph of y = e^x exemplifies exponential growth by showing a rapid increase in values as x becomes larger. As x increases, the output values rise sharply, demonstrating how quantities can expand significantly over time. The continuous and smooth nature of the graph indicates that growth accelerates, reinforcing the idea that changes compound on themselves with each increment in x.
Discuss the significance of the horizontal asymptote in relation to the graph of y = e^x and its behavior as x approaches negative infinity.
The horizontal asymptote at y = 0 for the graph of y = e^x is significant because it illustrates how the function behaves as x decreases without bound. As x approaches negative infinity, e^x gets closer and closer to 0 but never actually reaches it. This characteristic highlights an essential aspect of exponential functions: they can decrease toward zero but remain positive for all real numbers.
Evaluate how understanding the graph of y = e^x can enhance your ability to solve real-world problems involving growth and decay.
Understanding the graph of y = e^x equips you with essential tools for solving real-world problems related to growth and decay by providing insight into how quantities change over time. For instance, in modeling population growth or investment returns, recognizing how quickly these quantities can rise or fall helps in making predictions and informed decisions. The graphical behavior allows you to visualize trends and derive relevant equations that apply to specific scenarios in areas like finance or biology.