Graphing Exponential Functions: A Step-by-Step Guide

Understanding Exponential Functions

Welcome! Today, we're diving into the fascinating world of exponential functions. These functions are all about rapid growth or decay, and understanding how to graph them is a fundamental skill in mathematics. Our focus will be on the function $f(x) = 20 \left(\frac{1}{4}\right)^x$, and we'll walk through the very first step Chelsea took: plotting the initial value. This initial value, often referred to as the y-intercept, is crucial because it establishes a starting point for our graph. It's the value of the function when $x=0$. Think of it as the moment you begin observing a process – whether it's the population of a bacteria colony, the value of an investment, or the amount of a radioactive substance decaying over time. Without this starting point, visualizing the subsequent behavior of the function would be significantly more challenging. The general form of an exponential function is $f(x) = a \cdot b^x$, where 'a' is the initial value (the y-intercept) and 'b' is the base, which determines whether the function grows or decays. In Chelsea's function, $f(x) = 20 \left(\frac{1}{4}\right)^x$, the 'a' value is 20, and the base 'b' is $\frac{1}{4}$. The base $\frac{1}{4}$ is between 0 and 1, which tells us this function will exhibit exponential decay. This means as 'x' increases, the value of $f(x)$ will decrease, getting closer and closer to zero but never quite reaching it. Conversely, if the base were greater than 1, we would see exponential growth. So, the initial value of 20 is incredibly significant; it's the anchor of our graph, the point from which all other values will be calculated and plotted. It's where the curve of the exponential function begins its journey, either ascending into infinity or descending towards the x-axis.

Plotting the Initial Value: Chelsea's First Step

Chelsea's first step in graphing the function $f(x) = 20 \left(\frac{1}{4}\right)^x$ is precisely what we've been discussing: plotting the initial value. To find the initial value, we substitute $x=0$ into the function. This is because the initial value always occurs at $x=0$, representing the starting point of our observation or measurement. So, we calculate $f(0) = 20 \left(\frac{1}{4}\right)^0$. A key rule in exponents is that any non-zero number raised to the power of 0 is equal to 1. Therefore, $\left(\frac{1}{4}\right)^0 = 1$. Substituting this back into our equation, we get $f(0) = 20 \cdot 1$. This simplifies to $f(0) = 20$. This means the initial value of the function is 20. On a coordinate plane, this corresponds to the point $(0, 20)$. This point is the y-intercept of the graph. It's the exact spot where the curve crosses the y-axis. When Chelsea plots this point, she is establishing the starting point for her graph. This single point, $(0, 20)$, provides the foundational information needed to sketch the rest of the curve. It's the anchor that dictates the scale and position of the entire graph. Without this initial point, visualizing the exponential decay dictated by the base $\frac{1}{4}$ would be purely speculative. The value 20 tells us how high the graph starts on the y-axis. As we move to the right (increasing x), the y-values will decrease from this starting height of 20. As we move to the left (decreasing x), the y-values would theoretically increase exponentially, but for practical graphing, we often focus on the behavior for $x \ge 0$. Therefore, Chelsea's first and most critical step is accurately placing the point $(0, 20)$ on her graph. This point is the cornerstone upon which the entire exponential curve will be built, visually representing the starting magnitude of the phenomenon described by the function. The precision of this first plotted point is paramount, as it sets the stage for all subsequent points and the overall shape of the exponential curve.

Interpreting the Graph and Its Behavior

Once the initial value is plotted, the next logical step in understanding the graph of $f(x) = 20 \left(\frac{1}{4}\right)^x$ is to interpret its behavior. We've already identified that the base, $\frac{1}{4}$, is between 0 and 1, indicating exponential decay. This means that as the input value, $x$, increases, the output value, $f(x)$, decreases. Let's explore a few more points to visualize this decay. We already know $f(0) = 20$. Now, let's calculate $f(1)$: $f(1) = 20 \left(\frac{1}{4}\right)^1 = 20 \cdot \frac{1}{4} = 5$. So, at $x=1$, the value of the function is 5. This means the point $(1, 5)$ will be on our graph. Notice how the y-value has dropped from 20 to 5 in just one unit increase in $x$. Now let's calculate $f(2)$: $f(2) = 20 \left(\frac{1}{4}\right)^2 = 20 \cdot \left(\frac{1}{16}\right) = \frac{20}{16} = \frac{5}{4} = 1.25$. So, at $x=2$, the function's value is 1.25. The point $(2, 1.25)$ is also on the graph. The rate of decrease slows down as $x$ gets larger, but the y-values continue to approach zero. If we were to calculate for negative values of $x$, say $x=-1$: $f(-1) = 20 \left(\frac{1}{4}\right)^{-1} = 20 \cdot 4 = 80$. This shows that as $x$ decreases, the function's value increases rapidly. The graph will rise steeply as we move to the left of the y-axis. The asymptote for this type of exponential function is the x-axis, which is the line $y=0$. The graph will get infinitely close to the x-axis as $x$ approaches positive infinity, but it will never touch or cross it. This is a defining characteristic of exponential decay. The initial value of 20 sets the starting height, and the base of $\frac{1}{4}$ dictates the rate at which the function approaches the x-axis. When sketching the graph, after plotting $(0, 20)$, Chelsea would then plot $(1, 5)$ and $(2, 1.25)$, and perhaps $(0.5, 10)$ if she wanted more detail. Connecting these points with a smooth, curved line that gets progressively flatter as $x$ increases and approaches the x-axis visually represents the exponential decay. The graph will be a curve that starts at $(0, 20)$, goes down through $(1, 5)$, and continues to get closer and closer to the x-axis as $x$ increases. The steepness of the curve is determined by the base; a base closer to 0 would result in a faster decay, while a base closer to 1 would result in a slower decay. The initial value determines the starting height, impacting the overall vertical positioning of the decay curve. Understanding these elements – the initial value, the base, and the concept of an asymptote – allows for a complete interpretation of the exponential function's graphical representation and its real-world implications.

Choosing the Correct Graph

Now, let's consider how to identify the correct graph representing Chelsea's first step. The question asks which graph represents her first step, which is plotting the initial value. As we've thoroughly established, the initial value is found by setting $x=0$ in the function $f(x) = 20 \left(\frac{1}{4}\right)^x$. Calculating this gives us $f(0) = 20 \left(\frac{1}{4}\right)^0 = 20 \cdot 1 = 20$. Therefore, the initial value is 20, which corresponds to the point $(0, 20)$ on the coordinate plane. This point is the y-intercept. When looking at potential graphs, you should be searching for a graph that clearly shows a point plotted at $(0, 20)$. This means the graph should intersect the y-axis at the value 20. Any graph that doesn't have a point marked or does not show the y-intercept at 20 is incorrect for representing Chelsea's first step. The first step is purely about pinpointing this initial value. Subsequent steps would involve calculating and plotting other points, like $(1, 5)$ or $(2, 1.25)$, and then sketching the curve. However, the question specifically isolates the very beginning of the process. So, among the given options (which would typically be visual representations of graphs), you need to locate the one that accurately depicts the point $(0, 20)$. Some distractors might show graphs that exhibit exponential decay but don't highlight or start with this specific point. Others might show a point on the y-axis but at an incorrect value, such as 0, 1, or some other number not equal to 20. You must ensure the plotted point is precisely at the intersection of the y-axis ($x=0$) and the value 20. If the graphs are presented with multiple points plotted, you should look for the point that is the y-intercept and verify its value. If only one point is plotted, it must be $(0, 20)$ for it to represent Chelsea's first step. Remember, the initial value is the foundation. A solid understanding of how to find and plot this initial value is key to mastering the graphing of exponential functions. It's the anchor that grounds the entire visualization of the function's behavior, ensuring accuracy from the very outset of the graphing process.

Conclusion

In summary, graphing an exponential function like $f(x) = 20 \left(\frac{1}{4}\right)^x$ begins with a critical first step: plotting the initial value. This is achieved by evaluating the function at $x=0$, which yields the y-intercept. For Chelsea's function, this point is $(0, 20)$. This initial point serves as the anchor for the entire graph, from which the curve of exponential decay unfolds. Understanding this foundational step is paramount for accurately visualizing and analyzing the behavior of exponential functions, whether they represent growth or decay.

For further exploration into the concepts of exponential functions and their graphs, you can visit reliable resources like Khan Academy or Wolfram MathWorld.