Function Transformation And Evaluation Finding G(0)

In this article, we delve into the fascinating world of function transformations and explore how they impact the graph and values of a function. We'll start with the function f(x) = (x - 8)(x - 5)(x - 3), a cubic polynomial that serves as the foundation for our exploration. Understanding the characteristics of this function, such as its roots and overall shape, is crucial for grasping the subsequent transformation. This particular function, f(x), is a cubic polynomial expressed in factored form. The factored form immediately reveals the roots of the function, which are the x-values where the function equals zero. These roots are x = 8, x = 5, and x = 3, as these are the values that make each factor equal to zero. When graphed on the xy-plane, these roots correspond to the points where the curve intersects the x-axis. Beyond the roots, the factored form provides insight into the function's behavior. As x approaches positive or negative infinity, the leading term (which would be x³ if the polynomial were expanded) dominates the function's behavior. This means that as x becomes very large, f(x) also becomes very large, and as x becomes very negative, f(x) becomes very negative. The shape of the cubic curve will have a general “S” shape, rising from the bottom left, crossing the x-axis at each root, and continuing to the top right. The factored form also allows us to analyze the sign of the function in different intervals. For example, between the roots 3 and 5, the factors (x - 8) and (x - 5) are negative, while (x - 3) is positive, resulting in a positive function value. Analyzing these intervals helps to sketch the curve accurately. Understanding the roots, end behavior, and sign changes is crucial for visualizing the graph of f(x) and for understanding how transformations will affect it. This initial analysis sets the stage for exploring the transformation that will be applied to f(x) to create the new function g(x). By fully understanding f(x), we can more easily predict and interpret the effects of the vertical translation.

Now, let's introduce the concept of vertical translations. A vertical translation involves shifting the graph of a function up or down along the y-axis. In our case, the graph of y = g(x) is obtained by translating the graph of y = f(x) upwards by 3 units. This transformation has a specific impact on the function's equation. When a function's graph is shifted vertically, it directly affects the y-values of the function. Translating a graph upwards means adding a constant value to the original function's output. In this case, since the graph of f(x) is translated up by 3 units to obtain g(x), the equation for g(x) can be expressed as: g(x) = f(x) + 3. This simple addition encapsulates the entire vertical shift. Every point on the graph of f(x) is moved upwards by 3 units to create the graph of g(x). This means that if f(a) = b, then g(a) = b + 3. Vertical translations are one of the fundamental transformations in function analysis. They preserve the shape of the original graph but shift its position on the coordinate plane. Unlike horizontal transformations, which can sometimes be counterintuitive, vertical translations behave exactly as one might expect – adding a positive constant shifts the graph upwards, and subtracting a constant shifts it downwards. The simplicity of vertical translations makes them a useful tool in manipulating functions and understanding their behavior. In this context, understanding that g(x) is simply f(x) shifted upwards by 3 units is key to solving the problem. We can now move on to evaluating g(0), which will require us to find f(0) first and then apply the translation.

The core of our problem lies in determining the value of g(0). To achieve this, we'll first need to find the value of f(0), which serves as the foundation for calculating g(0). Recall that f(x) = (x - 8)(x - 5)(x - 3). To find f(0), we substitute x = 0 into the equation:

f(0) = (0 - 8)(0 - 5)(0 - 3) = (-8)(-5)(-3) = -120.

Thus, the value of the original function at x = 0 is -120. This means that the point (0, -120) lies on the graph of y = f(x). Now that we have found f(0), we can use the relationship between f(x) and g(x) to find g(0). We know that g(x) = f(x) + 3. Therefore, to find g(0), we substitute x = 0 into the equation for g(x):

g(0) = f(0) + 3.

We already found that f(0) = -120, so we substitute this value into the equation:

g(0) = -120 + 3 = -117.

Therefore, the value of g(0) is -117. This means that the point (0, -117) lies on the graph of y = g(x). The vertical translation has shifted the point (0, -120) on the graph of f(x) upwards by 3 units to the point (0, -117) on the graph of g(x). The process of evaluating g(0) highlights the importance of understanding function transformations and their impact on specific function values. By first evaluating the original function f(x) at x = 0 and then applying the vertical translation, we were able to efficiently find the value of the transformed function g(x) at the same point.

In conclusion, we have successfully determined the value of g(0) by understanding and applying the concept of vertical translations. We started with the function f(x) = (x - 8)(x - 5)(x - 3) and recognized that the function g(x) is a vertical translation of f(x) upwards by 3 units. This led us to the key relationship: g(x) = f(x) + 3. To find g(0), we first evaluated f(0), which gave us f(0) = -120. We then used the relationship between f(x) and g(x) to find g(0) = f(0) + 3 = -120 + 3 = -117. Therefore, the value of g(0) is -117. This solution demonstrates the power of function transformations in manipulating and understanding mathematical relationships. Vertical translations, in particular, provide a straightforward way to shift the graph of a function up or down, affecting the function's values in a predictable manner. This concept is fundamental in calculus and other advanced mathematical fields. Moreover, this problem highlights the importance of breaking down complex problems into smaller, manageable steps. By first understanding the original function, then recognizing the transformation, and finally evaluating the functions at the specific point x = 0, we were able to arrive at the solution efficiently. This approach is applicable to a wide range of mathematical problems. Understanding transformations of functions is crucial not only in mathematics but also in various scientific and engineering disciplines where mathematical models are used to represent real-world phenomena. The ability to shift, stretch, and reflect functions allows for the creation of models that accurately capture the behavior of complex systems. The key takeaway from this analysis is the connection between the equation of a function and its graphical representation. A vertical translation in the graph corresponds to a simple addition in the equation, and understanding this relationship is essential for solving problems involving function transformations. This problem serves as a valuable exercise in applying these concepts and solidifying one's understanding of function transformations.