Transformations Of Exponential Functions G(x) = 3(2)^(-x) + 2 Explained

Understanding transformations of functions is a cornerstone of mathematics, particularly when dealing with exponential functions. The question at hand asks us to dissect the transformation of the function g(x) = 3(2)^(-x) + 2 from its parent function, f(x) = 2^x. To accurately describe this transformation, we must carefully consider the impact of each component within the equation of g(x). This exploration will not only provide the answer but also deepen our understanding of how transformations work in general.

Parent Function: f(x) = 2^x

Before diving into the transformations, it's crucial to understand the parent function, f(x) = 2^x. This is a basic exponential function with a base of 2. Its graph starts near the x-axis on the left side (as x approaches negative infinity) and increases rapidly as x increases. The function passes through the point (0, 1) because any number raised to the power of 0 is 1. The exponential growth is a key characteristic of this function, making it a fundamental building block for understanding more complex exponential functions.

The graph of f(x) = 2^x serves as our baseline. It's a smooth, increasing curve that never actually touches the x-axis (it has a horizontal asymptote at y = 0). This parent function will be transformed by several operations in the given function g(x), and recognizing its initial shape helps us visualize the effects of each transformation. Understanding the behavior of the parent function is paramount to deciphering the transformations applied to it. We will use this understanding to analyze the changes step by step as we move from f(x) to g(x).

Analyzing the Transformations in g(x) = 3(2)^(-x) + 2

To unravel the transformations applied to the parent function, let's break down the equation g(x) = 3(2)^(-x) + 2 step-by-step. We have three key elements to consider: the negative sign in the exponent (-x), the multiplication by 3, and the addition of 2. Each of these elements corresponds to a specific type of transformation that alters the graph of the function in a predictable way. By analyzing them individually, we can piece together the complete transformation from f(x) to g(x). The goal here is not just to identify the transformations, but to comprehend the mechanics behind them, enabling us to apply this knowledge to other functions as well.

1. Reflection across the y-axis: The Impact of -x

The first transformation we encounter is the negative sign in the exponent: 2^(-x). This indicates a reflection across the y-axis. When x is replaced with -x in a function, the graph is mirrored over the y-axis. This is because the y-values for positive x-values in the original function now correspond to the y-values for negative x-values in the transformed function, and vice versa. Think of it as flipping the graph horizontally. This reflection is a fundamental transformation, and recognizing it is crucial for accurately interpreting the function's behavior.

2. Vertical Stretch: The Role of the Multiplier 3

Next, we have the multiplication by 3: 3(2)^(-x). This represents a vertical stretch by a factor of 3. Multiplying a function by a constant greater than 1 stretches the graph vertically away from the x-axis. In this case, every y-value of the reflected function 2^(-x) is multiplied by 3, making the graph taller. A vertical stretch affects the overall shape and amplitude of the function. It's important to distinguish this from a horizontal stretch or compression, which would involve multiplying the x-value inside the function.

The vertical stretch by a factor of 3 significantly alters the function's range. While the original function f(x) = 2^x and the reflected function 2^(-x) have a range of (0, ∞), multiplying by 3 stretches this range, but it remains unbounded above and still bounded by 0 on the lower end. Visualizing this stretch is key to understanding the transformation's effect on the function's values. Every point on the graph is effectively moved three times further away from the x-axis, creating a taller, more elongated curve.

3. Vertical Shift: The Effect of Adding 2

Finally, we have the addition of 2: 3(2)^(-x) + 2. This signifies a vertical shift of 2 units upwards. Adding a constant to a function shifts the entire graph vertically. In this case, every point on the graph of 3(2)^(-x) is moved 2 units up. This shift affects the function's range and its horizontal asymptote. The horizontal asymptote, which was originally at y = 0, is now shifted to y = 2. Vertical shifts are straightforward transformations, but they can have a significant impact on the function's key features.

The vertical shift of 2 units changes the asymptotic behavior of the function. The original asymptote at y = 0 is lifted to y = 2, meaning the function will now approach 2 as x approaches infinity (or negative infinity, depending on the other transformations). This shift also affects the range of the function, changing it from (0, ∞) after the stretch to (2, ∞). Understanding how vertical shifts impact both the asymptote and the range is vital for sketching the transformed graph and predicting its behavior.

Putting it All Together: The Complete Transformation

Now, let's synthesize our analysis to describe the complete transformation of g(x) = 3(2)^(-x) + 2 from the parent function f(x) = 2^x. We've identified three key transformations:

1. Reflection across the y-axis: This is due to the -x in the exponent.
2. Vertical stretch by a factor of 3: This is caused by the multiplication by 3.
3. Vertical shift 2 units up: This results from the addition of 2.

Therefore, the function g(x) is obtained by reflecting f(x) across the y-axis, then stretching the graph vertically by a factor of 3, and finally shifting it 2 units upwards. This sequence of transformations completely defines the relationship between the parent function and the transformed function. Visualizing each step helps solidify our understanding of how these transformations work together to create the final graph.

The order in which we apply these transformations is crucial. Generally, reflections and stretches/compressions should be performed before shifts. This is because shifts affect the final position of the transformed graph, and performing them earlier might lead to an incorrect result. In this case, reflecting across the y-axis, then stretching vertically, and finally shifting upwards gives us the correct transformed function. Understanding this order of operations is essential for accurately transforming functions.

Conclusion: Describing the Transformation

In conclusion, the transformation of g(x) = 3(2)^(-x) + 2 from the parent function f(x) = 2^x is best described as a reflection across the y-axis, a vertical stretch by a factor of 3, and a shift of 2 units upwards. This detailed analysis highlights the importance of understanding each component of a function's equation and how it contributes to the overall transformation. By breaking down the equation step-by-step, we can accurately describe the changes in the graph and gain a deeper appreciation for the behavior of exponential functions. This understanding is not only useful for solving specific problems but also for building a solid foundation in function transformations.

Mastering transformations is a key skill in mathematics. It allows us to predict and understand the behavior of functions by analyzing their equations. By understanding the effects of reflections, stretches, and shifts, we can visualize the transformed graph and gain insights into its properties. This detailed explanation of the transformation of g(x) = 3(2)^(-x) + 2 serves as a valuable example of how to approach similar problems and develop a strong understanding of function transformations.