Finding The Y-Intercept Of G(x) = 2f(x) + 1 Where F(x) = E^x

At the heart of our exploration lies the exponential function f(x) = e^x, a cornerstone of mathematical analysis. This function, characterized by its rapid growth, forms the foundation for understanding various phenomena in science, engineering, and finance. The graph of f(x) = e^x is a classic exponential curve that starts near the x-axis on the left and rises sharply as x increases. A key feature of this function is its y-intercept, the point where the graph crosses the y-axis. To find this point, we set x = 0 and evaluate f(0). This gives us f(0) = e^0 = 1, meaning the graph of f(x) = e^x intersects the y-axis at the point (0, 1). This intercept serves as the starting point for the exponential growth, as the function value increases exponentially from this point forward.

Delving Deeper into the Exponential Function:

To fully grasp the behavior of f(x) = e^x, it's essential to consider its properties beyond the y-intercept. The function is always positive, never touching or crossing the x-axis. This is because e raised to any power, whether positive, negative, or zero, will always yield a positive result. As x approaches negative infinity, f(x) approaches zero, indicating a horizontal asymptote along the negative x-axis. Conversely, as x approaches positive infinity, f(x) grows without bound, demonstrating the characteristic exponential growth. The base e, known as Euler's number, is approximately equal to 2.71828 and is a fundamental constant in mathematics. It arises naturally in various contexts, including compound interest, population growth, and radioactive decay. The exponential function with base e, denoted as e^x, is often referred to as the natural exponential function due to its widespread applications and mathematical properties.

The derivative of f(x) = e^x is itself, e^x, a unique property that makes it particularly important in calculus and differential equations. This means that the rate of change of the function at any point is equal to its value at that point. This property is crucial in modeling phenomena where the rate of change is proportional to the current value, such as population growth and radioactive decay. The integral of e^x is also e^x, plus a constant of integration, further highlighting its special status in calculus. Understanding these properties of f(x) = e^x is crucial for analyzing the behavior of the transformed function g(x), which is the focus of our problem. By manipulating f(x) through multiplication and addition, we can explore how these transformations affect the key features of the graph, including the y-intercept.

Now, let's shift our focus to the transformed function g(x) = 2f(x) + 1. This transformation involves two key operations: a vertical stretch and a vertical shift. The multiplication by 2, represented by 2f(x), stretches the graph of f(x) vertically by a factor of 2. This means that every y-coordinate on the graph of f(x) is doubled, making the graph steeper. The addition of 1, represented by + 1, shifts the entire graph upward by 1 unit. This means that every point on the graph of 2f(x) is moved up by 1 unit, changing the position of the graph relative to the axes. These transformations significantly alter the appearance of the original exponential function, affecting its y-intercept and overall behavior.

Analyzing the Impact of Transformations:

To understand how these transformations affect the y-intercept, we need to consider their individual contributions. The vertical stretch by a factor of 2 doubles the y-coordinate of every point on the graph of f(x). Since the y-intercept of f(x) is (0, 1), the vertical stretch transforms this point to (0, 2). This intermediate graph, represented by 2f(x), has a y-intercept of 2. The subsequent vertical shift by 1 unit then moves this point upward, changing the y-coordinate from 2 to 3. Therefore, the y-intercept of the transformed function g(x) is (0, 3). This illustrates how vertical stretches and shifts can be used to manipulate the graph of a function and control its key features, such as the y-intercept. Understanding these transformations is crucial for analyzing and interpreting the behavior of functions in various mathematical and real-world contexts.

Furthermore, it's important to note that these transformations also affect other aspects of the graph. The vertical stretch changes the rate of growth of the exponential function, making it grow faster than the original function. The vertical shift changes the horizontal asymptote of the function. While f(x) = e^x has a horizontal asymptote at y = 0, the transformed function g(x) = 2f(x) + 1 has a horizontal asymptote at y = 1. This is because the addition of 1 shifts the entire graph upward, including the asymptote. By analyzing the effects of these transformations, we gain a deeper understanding of the relationship between the original function and its transformed version.

The $y$-intercept of a function is the point where the graph intersects the $y$-axis. This occurs when $x = 0$. To find the $y$-intercept of $g(x)$, we need to evaluate $g(0)$. Given that $g(x) = 2f(x) + 1$ and $f(x) = e^x$, we can substitute $x = 0$ into both functions.

Calculating the Y-Intercept:

First, we evaluate $f(0)$: f(0) = e^0 = 1. Next, we substitute this value into $g(x)$: g(0) = 2f(0) + 1 = 2(1) + 1 = 2 + 1 = 3. Therefore, the $y$-intercept of the function $g(x)$ is 3. This means that the graph of $g(x)$ intersects the $y$-axis at the point (0, 3). This confirms our previous analysis of the transformations, where we determined that the vertical stretch and shift would result in a y-intercept of 3. The y-intercept is a crucial characteristic of a function, as it provides a starting point for understanding its behavior and how it relates to other functions. In this case, the y-intercept of $g(x)$ indicates the value of the function when $x = 0$, which is a key piece of information for graphing and analyzing the function.

Visualizing the Y-Intercept:

To further understand the significance of the y-intercept, it's helpful to visualize the graph of $g(x)$. The graph is an exponential curve that starts at the point (0, 3) and grows rapidly as $x$ increases. The y-intercept serves as the initial value of the function, and the exponential growth builds upon this starting point. By knowing the y-intercept and the general shape of the exponential curve, we can sketch a rough graph of the function and gain insights into its behavior. The y-intercept is also useful for comparing different functions and understanding how they relate to each other. For example, if we were to compare $g(x)$ to another exponential function with a different y-intercept, we could see how the starting values affect the overall behavior of the functions.

In conclusion, by understanding the properties of the exponential function $f(x) = e^x$ and the transformations applied to create $g(x) = 2f(x) + 1$, we have successfully determined that the $y$-intercept of $g(x)$ is 3. This point, (0, 3), marks where the graph of $g(x)$ crosses the $y$-axis and provides a crucial reference point for analyzing the function's behavior. This exercise highlights the importance of understanding function transformations and their effects on key features like the $y$-intercept. The combination of vertical stretching and shifting alters the original function's graph, leading to a new $y$-intercept that reflects these changes. By systematically evaluating the function at $x = 0$, we can precisely determine this intercept and gain valuable insights into the function's characteristics. This understanding is essential for further analysis, graphing, and application of exponential functions in various mathematical and real-world scenarios.

By mastering the concepts of function transformations and $y$-intercepts, we equip ourselves with powerful tools for analyzing and interpreting mathematical functions. This knowledge extends beyond the specific example of exponential functions, providing a foundation for understanding a wide range of mathematical relationships and their applications in diverse fields.