Graph Changes Of F(x)=10(2)^x When Base B Is Decreased

In mathematics, exponential functions play a crucial role in modeling various real-world phenomena, such as population growth, radioactive decay, and compound interest. An exponential function is generally expressed in the form f(x) = ab^x, where a is the initial value, b is the base, and x is the exponent. The base, b, is a positive real number not equal to 1, and it significantly influences the behavior of the exponential function's graph.

Let's delve into the specific exponential function given: f(x) = 10(2)^x. In this function, the initial value (a) is 10, and the base (b) is 2. This means that the function starts at a value of 10 when x is 0, and the value increases exponentially as x increases. The graph of this function will exhibit a characteristic exponential growth pattern, starting from the point (0, 10) and rising sharply as x moves towards positive infinity.

To fully understand the graph, we can plot a few key points. When x is 0, f(x) is 10. When x is 1, f(x) is 20. When x is 2, f(x) is 40. These points clearly show the exponential growth. Furthermore, as x becomes negative, the function approaches 0 but never actually reaches it, demonstrating the horizontal asymptote at y = 0.

The behavior of the graph is heavily influenced by the base, b. In this case, b = 2 indicates that for every unit increase in x, the function value doubles. This rapid increase is what gives the graph its steep upward curve. Understanding these fundamental aspects of exponential functions and their graphs is essential for analyzing how changes in the parameters, particularly the base, affect the overall behavior of the function.

When analyzing exponential functions of the form f(x) = ab^x, the base b is a critical determinant of the function's growth or decay rate. The given function, f(x) = 10(2)^x, has a base of 2, indicating exponential growth. Let's explore how the graph of this function would change if the value of b is decreased while remaining greater than 1. This scenario is crucial for understanding the nuances of exponential behavior.

Consider decreasing b from 2 to a value such as 1.5. The new function would be f(x) = 10(1.5)^x. The initial value, a, remains at 10, meaning the graph will still pass through the point (0, 10). However, the rate of growth will be different. When b = 2, the function doubles for every unit increase in x. With b = 1.5, the function increases by 50% for every unit increase in x, which is a slower rate of growth compared to the original function.

This change in the growth rate is visually represented in the graph. The new graph with b = 1.5 will still exhibit exponential growth, but it will rise less steeply than the original graph with b = 2. This means that for the same value of x, the function value f(x) will be lower when b is 1.5 compared to when b is 2. Consequently, the graph will appear flatter or more stretched out horizontally.

Another way to think about this is by comparing the function values at specific points. For instance, at x = 2, the original function f(x) = 10(2)^2 yields f(x) = 40, while the modified function f(x) = 10(1.5)^2 gives f(x) = 22.5. This numerical comparison clearly demonstrates that decreasing b reduces the function's value for x > 0, resulting in a less steep graph. Understanding this relationship between the base and the growth rate is essential for interpreting and manipulating exponential functions in various applications.

Given the exponential function f(x) = 10(2)^x, the question at hand is how the graph would change if the b value in the equation is decreased but remains greater than 1. We have established that b represents the base of the exponential function and influences the rate of growth. Decreasing b from 2 to a value between 1 and 2 will have specific effects on the graph's characteristics. Let's explore these changes systematically.

First, consider the initial point of the graph. The y-intercept occurs when x = 0. For the function f(x) = 10(2)^x, when x = 0, f(x) = 10(2)^0 = 10. If we decrease b to, say, 1.5, the function becomes f(x) = 10(1.5)^x. Again, when x = 0, f(x) = 10(1.5)^0 = 10. This demonstrates that the y-intercept remains unchanged at (0, 10). Therefore, the graph will not begin at a lower point on the y-axis; it will still start at the same point.

However, the rate of exponential growth will be affected. The original function doubles its value for every unit increase in x. When b is decreased, the function will still grow exponentially, but at a slower rate. This means that the graph will be less steep. For example, at x = 1, the original function f(x) = 10(2)^x gives f(x) = 20, while the modified function f(x) = 10(1.5)^x gives f(x) = 15. This illustrates that the function values are lower for the same x values when b is smaller.

Consequently, the graph will appear flatter or stretched horizontally compared to the original graph. The exponential curve will still exist, but it will not rise as rapidly. This understanding is crucial for visually interpreting how changes in the base b affect the shape of the exponential graph. In summary, decreasing b while keeping it greater than 1 will not change the starting point on the y-axis, but it will reduce the steepness of the exponential growth, making the graph rise more gradually.

To fully address how the graph of f(x) = 10(2)^x changes when the base b is decreased but remains greater than 1, it's essential to dissect the transformations that occur. Decreasing the base affects the growth rate, and this change is visually reflected in the graph's shape. Let's break down the specific impacts on the graph.

First, let's reiterate that the y-intercept remains constant. The y-intercept is the point where the graph intersects the y-axis, which occurs when x = 0. In the original function, f(0) = 10(2)^0 = 10. When we decrease b, for example, to 1.5, the new function becomes f(x) = 10(1.5)^x, and f(0) = 10(1.5)^0 = 10. Thus, the y-intercept remains at (0, 10), indicating that the graph does not begin at a lower point on the y-axis.

However, the key change lies in the steepness of the graph. The base b governs the rate of exponential growth. A larger b value means a faster rate of growth, and a smaller b value (greater than 1) means a slower rate of growth. Consider the original function f(x) = 10(2)^x. For every unit increase in x, the function value doubles. Now, let's compare this to f(x) = 10(1.5)^x. For every unit increase in x, the function value increases by 50%, which is less than doubling.

Visually, this translates to the graph becoming less steep. The curve will still rise exponentially, but it will do so more gradually. If you were to plot both functions on the same coordinate plane, the graph of f(x) = 10(1.5)^x would appear flatter or more stretched horizontally compared to the graph of f(x) = 10(2)^x. This reduced steepness is the direct result of the slower growth rate caused by decreasing the base b.

Another way to think about this is to compare the function values at a specific x value greater than 0. For instance, at x = 2, the original function gives f(2) = 10(2)^2 = 40, while the modified function gives f(2) = 10(1.5)^2 = 22.5. This significant difference in function values further illustrates that the graph of the function with the smaller base will be lower for x > 0, resulting in a less steep curve. Understanding this interplay between the base and the graph's shape is crucial for analyzing and interpreting exponential functions.

In conclusion, when considering the exponential function f(x) = 10(2)^x and evaluating the impact of decreasing the base b while maintaining it above 1, several key changes occur in the graph. The most significant change is the reduction in the rate of exponential growth, which directly affects the steepness of the graph. Let's recap the major points to provide a comprehensive understanding.

Firstly, it's essential to emphasize that the y-intercept remains constant. The initial value, represented by the coefficient a in the general form f(x) = ab^x, determines where the graph intersects the y-axis. In this case, a = 10, so the graph will always pass through the point (0, 10) regardless of the value of b. Therefore, decreasing b does not cause the graph to begin at a lower point on the y-axis.

However, the rate at which the function grows is fundamentally altered by the base b. A larger b value corresponds to a faster growth rate, while a smaller b value (greater than 1) corresponds to a slower growth rate. In the original function, f(x) = 10(2)^x, the base b = 2 results in the function value doubling for every unit increase in x. When b is decreased, such as to 1.5, the function f(x) = 10(1.5)^x grows at a slower pace, increasing by only 50% for each unit increase in x.

This difference in growth rate manifests visually as a change in the graph's steepness. The graph of the function with a smaller b value will be less steep, appearing flatter or more stretched horizontally compared to the original graph. The exponential curve is still present, but it rises more gradually. Comparing function values at specific points, such as x = 2, further illustrates this difference. The original function yields f(2) = 40, while the modified function yields f(2) = 22.5, clearly demonstrating the reduced growth.

Understanding these transformations is crucial for interpreting and manipulating exponential functions in various contexts. By decreasing b, we effectively moderate the exponential growth, resulting in a less aggressive upward curve. This knowledge is invaluable for applications ranging from modeling population dynamics to analyzing financial investments. Therefore, the key takeaway is that decreasing b while keeping it greater than 1 primarily affects the growth rate, leading to a less steep graph, while the y-intercept remains unchanged.

Answer:

Based on the analysis, the correct answer is:

The graph will be less steep.