Graphing Exponential Functions Finding The Next Point On F(x)=25(3/5)^x
Introduction to Exponential Functions and Graphing
In this article, we'll explore an engaging problem involving exponential functions and graphing. Our focus will be on understanding how to plot points on the graph of an exponential function, specifically the function $f(x)=25(\frac{3}{5})^x$. We'll follow Tanisha's journey as she graphs this function and determine the next point she should plot after already plotting the point $(1,15)$. This exercise is a fantastic way to reinforce our understanding of exponential functions, their properties, and how they are visually represented on a graph. Exponential functions are a cornerstone of mathematics and have widespread applications in various fields, from finance and population growth to radioactive decay and compound interest. Mastering the art of graphing these functions is crucial for developing a strong foundation in mathematical concepts. We will delve into the characteristics of exponential functions, such as their rapid growth or decay, and how the base of the exponent influences the shape of the graph. By working through this problem, we'll not only identify the correct next point but also gain valuable insights into the behavior of exponential functions and how to effectively plot them. This exploration will enhance your ability to analyze and interpret exponential functions in various contexts. The process of graphing exponential functions involves understanding the relationship between the input (x) and the output (f(x)), and how this relationship manifests as a curve on a coordinate plane. Each point on the graph represents a specific (x, f(x)) pair, and by plotting multiple points, we can accurately trace the curve of the function. This skill is essential for visualizing and understanding the behavior of exponential functions, which are used to model a wide range of real-world phenomena.
Understanding the Function $f(x)=25(\frac{3}{5})^x$
Let's begin by deeply understanding the exponential function at hand: $f(x)=25(\frac{3}{5})^x$. This function is a classic example of an exponential decay function. The general form of an exponential function is $f(x) = a imes b^x$, where 'a' is the initial value and 'b' is the base. In our case, $a = 25$ and $b = \frac{3}{5}$. The base 'b' plays a crucial role in determining whether the function represents growth or decay. When 'b' is greater than 1, the function represents exponential growth, meaning the function's value increases rapidly as 'x' increases. Conversely, when 'b' is between 0 and 1, as in our case, the function represents exponential decay, meaning the function's value decreases as 'x' increases. The initial value 'a' represents the function's value when x = 0. In our case, when x = 0, $f(0) = 25 imes (\frac{3}{5})^0 = 25 imes 1 = 25$. This means the graph of the function will intersect the y-axis at the point (0, 25). Now, let's consider the impact of the base $(\frac{3}{5})$ on the function's behavior. As 'x' increases, $(\frac{3}{5})^x$ will decrease, causing the overall value of $f(x)$ to decrease. This decay is characteristic of exponential functions with a base between 0 and 1. The rate of decay is determined by the value of the base; a smaller base will result in a faster decay. Understanding these fundamental aspects of the function is crucial for accurately plotting its graph. We know that the graph will start at (0, 25) and decrease as 'x' increases. The rate of decrease will be determined by the base $(\frac{3}{5})$. By carefully analyzing the function's components, we can predict its behavior and identify points that lie on its graph. This understanding will help us verify the options provided and determine the correct next point Tanisha should plot.
Verifying the Given Point (1, 15)
Tanisha begins by plotting the point $(1,15)$. To verify this point, we need to substitute $x = 1$ into the function $f(x)=25(\frac3}{5})^x$ and check if the result is indeed 15. Let's perform the calculation{5})^1 = 25 \times \frac{3}{5} = \frac{25 \times 3}{5} = \frac{75}{5} = 15$. As we can see, when $x = 1$, $f(x) = 15$. This confirms that the point $(1,15)$ lies on the graph of the function. This verification step is crucial because it ensures that we are starting with a correct reference point. If the initial point were incorrect, any subsequent calculations and point plotting would be based on a flawed foundation, leading to an inaccurate graph. Now that we've verified the initial point, we can proceed with confidence to determine the next point Tanisha should plot. The fact that (1, 15) lies on the graph gives us a starting point for understanding the function's behavior. We know that the function is decreasing, so the next point should have a y-coordinate less than 15. This understanding will help us narrow down the options and identify the correct next point. By carefully substituting values of 'x' into the function and calculating the corresponding 'f(x)' values, we can accurately plot points and trace the curve of the exponential function. This process of verification and calculation is fundamental to graphing functions and understanding their behavior. It allows us to translate the mathematical expression of a function into a visual representation, making it easier to analyze and interpret.
Evaluating the Potential Next Points
Now, let's evaluate the potential next points provided in the options. We have three candidates: $(2,9)$, $(2,-10)$, and $(2,14.4)$. To determine which of these points could be the next point Tanisha plots on the graph, we need to substitute $x = 2$ into the function $f(x)=25(\frac3}{5})^x$ and compare the result with the y-coordinates of the given points. Let's calculate $f(2)${5})^2 = 25 \times \frac{9}{25} = \frac{25 \times 9}{25} = 9$. The calculation shows that when $x = 2$, $f(x) = 9$. This means the point $(2,9)$ lies on the graph of the function. Now, let's compare this result with the other options. The point $(2,-10)$ has a y-coordinate of -10, which is significantly different from 9. Since exponential functions of this form (with a positive initial value and a base between 0 and 1) will always have positive y-values, this point cannot lie on the graph. The point $(2,14.4)$ also has a y-coordinate that does not match our calculated value of 9. Therefore, it cannot be the next point on the graph. By systematically evaluating each option, we have confirmed that only the point $(2,9)$ satisfies the function's equation when $x = 2$. This process of substitution and comparison is a fundamental technique for verifying points on a graph. It allows us to directly connect the algebraic representation of a function with its graphical representation. Understanding how to perform these evaluations is crucial for accurately plotting graphs and analyzing functions. In this case, by calculating $f(2)$, we were able to definitively identify the correct next point Tanisha should plot.
Conclusion: The Next Point on the Graph
In conclusion, by carefully evaluating the given options and calculating the value of the function $f(x)=25(\frac{3}{5})^x$ at $x = 2$, we have determined that the next point Tanisha could plot on the graph is $(2,9)$. This point satisfies the function's equation and aligns with the expected behavior of an exponential decay function. This exercise has reinforced our understanding of how to graph exponential functions and how to verify points on a graph. We began by understanding the function's components, including the initial value and the base, and how these components influence the function's behavior. We then verified the initial point $(1,15)$ to ensure a solid foundation for our analysis. Next, we systematically evaluated the potential next points by substituting $x = 2$ into the function and comparing the results with the given y-coordinates. This process allowed us to definitively identify $(2,9)$ as the correct next point. This problem-solving approach is applicable to a wide range of mathematical problems involving functions and graphs. By breaking down the problem into smaller, manageable steps, we can effectively analyze the given information and arrive at the correct solution. Understanding the properties of exponential functions and how to graph them is a valuable skill that extends beyond the classroom and into various real-world applications. Whether it's modeling population growth, calculating compound interest, or understanding radioactive decay, exponential functions play a crucial role in describing and predicting these phenomena. By mastering the concepts and techniques discussed in this article, you'll be well-equipped to tackle more complex problems involving exponential functions and their applications. The key takeaway is the importance of understanding the relationship between a function's equation and its graph, and how to use this relationship to solve problems and gain insights.
