Describing The Graph Of F(x)=4(1.5)^x An Exponential Function

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When we delve into the world of functions, understanding their graphical representations is crucial. The function f(x) = 4(1.5)^x is a prime example of an exponential function, and deciphering its graph provides valuable insights into its behavior. To truly grasp the essence of this function's graph, let's meticulously analyze its key features and characteristics. We will explore how the base (1.5) and the coefficient (4) influence the curve, and we will address the question: Which best describes the graph of the function f(x)=4(1.5)^x?

Understanding Exponential Functions

Before we dive into the specifics of our function, let's establish a firm understanding of exponential functions in general. An exponential function typically takes the form f(x) = a * b^x, where 'a' represents the initial value or y-intercept, 'b' is the base that determines the rate of growth or decay, and 'x' is the independent variable. When the base 'b' is greater than 1, as in our case with 1.5, the function represents exponential growth. This means that as the x-values increase, the y-values increase at an accelerating rate. Conversely, if 'b' were between 0 and 1, we would observe exponential decay. Understanding these fundamental concepts is essential for interpreting the behavior of any exponential function's graph. For the function f(x) = 4(1.5)^x, the initial value 'a' is 4, which tells us where the graph intersects the y-axis. The base 'b' is 1.5, indicating that for every unit increase in x, the y-value is multiplied by 1.5. This multiplicative growth is what characterizes exponential functions and distinguishes them from linear or polynomial functions. Furthermore, the exponential nature of this function means that it will never actually reach zero; it will only get increasingly closer to the x-axis as x becomes more and more negative, illustrating a horizontal asymptote at y=0. Exploring these properties forms a solid foundation for analyzing and describing the graph of our specific exponential function.

Key Features of f(x) = 4(1.5)^x

The function f(x) = 4(1.5)^x possesses several key features that dictate its graphical representation. Let's dissect these elements to paint a clearer picture of the curve. Firstly, the coefficient 4 acts as the y-intercept. This means the graph intersects the y-axis at the point (0, 4). This is because when x is 0, 1.5 raised to the power of 0 is 1, and 4 multiplied by 1 equals 4. So, we can definitively say that the graph passes through the point (0, 4). Secondly, the base 1.5 determines the exponential growth rate. This means that for every increase of 1 in the x-values, the y-values are multiplied by 1.5. This can be expressed as: for each increase of 1 in the x-values, the y-values increase by a factor of 1.5. It's essential to note that the y-values do not increase by a constant amount of 1.5; instead, they increase by a factor of 1.5, signifying exponential growth. For example, when x is 1, f(x) is 4 * 1.5 = 6, and when x is 2, f(x) is 4 * (1.5)^2 = 9. The difference between 6 and 9 is not 1.5, but rather, 6 multiplied by 1.5 gives us 9, demonstrating the multiplicative nature of exponential growth. Thirdly, the function has a horizontal asymptote at y = 0. This means that as x approaches negative infinity, the graph gets closer and closer to the x-axis but never actually touches it. This is a characteristic of exponential functions where the base is a positive number. Understanding these key features – the y-intercept, the growth factor, and the horizontal asymptote – is crucial for accurately describing the graph of f(x) = 4(1.5)^x.

Analyzing the Given Descriptions

Now, let's turn our attention to the provided descriptions and evaluate which one best represents the graph of f(x) = 4(1.5)^x. The descriptions highlight two key aspects: the point the graph passes through and the behavior of y-values as x-values increase. The first part of the description states that the graph passes through the point (0, 4). As we established earlier, this is indeed true. When x is 0, f(x) is 4 * (1.5)^0 = 4 * 1 = 4. Therefore, the graph definitely intersects the y-axis at (0, 4). The second part of the description presents two alternatives. One states that for each increase of 1 in the x-values, the y-values increase by 1.5. This statement is slightly misleading. While the y-values do increase as x-values increase, they don't increase by a constant amount of 1.5. Instead, they increase by a factor of 1.5, indicating exponential growth. This means the y-value is multiplied by 1.5 for every unit increase in x, not added to by 1.5. The other alternative states that for each increase of 1 in the x-values, the y-values increase by a factor of 1.5. This is the accurate description. As we discussed in the previous section, the base 1.5 acts as the growth factor. For every unit increase in x, the y-value is multiplied by 1.5, reflecting the exponential nature of the function. Therefore, to accurately describe the graph, it's crucial to emphasize the multiplicative increase rather than an additive one. This nuanced understanding distinguishes exponential growth from linear growth, where values increase by a constant amount.

The Correct Description

Based on our analysis, the description that best characterizes the graph of f(x) = 4(1.5)^x is: The graph passes through the point (0, 4), and for each increase of 1 in the x-values, the y-values increase by a factor of 1.5. This description encapsulates the two fundamental aspects of the function's graph: its y-intercept and its exponential growth. The y-intercept (0, 4) provides a starting point, indicating where the graph begins on the y-axis. The phrase