Graph $f(x)=3 imes (0.5)^x$ And Match The Graph

Understanding Exponential Functions

When we talk about graphing exponential functions, we're diving into a fascinating area of mathematics where quantities grow or decay at a rate proportional to their current value. The function $f(x) = 3 imes (0.5)^x$ is a classic example of an exponential function. At its core, an exponential function takes the form $f(x) = a imes b^x$, where 'a' is the initial value (the y-intercept when $x=0$), and 'b' is the base, which determines whether the function is increasing or decreasing. In our specific case, $f(x) = 3 imes (0.5)^x$, the initial value 'a' is 3, and the base 'b' is 0.5. The base being less than 1 (but greater than 0) is a key indicator that this function will represent exponential decay. This means as 'x' gets larger, the value of $f(x)$ gets smaller, approaching zero. Conversely, as 'x' becomes more negative, $f(x)$ will increase rapidly. To truly grasp how to graph this, let's break down the process step-by-step, focusing on identifying key points and understanding the overall behavior of the curve. This understanding is crucial not just for solving problems but for appreciating the dynamic nature of exponential relationships in the real world, from population dynamics to financial investments.

Plotting Key Points for $f(x)=3 imes (0.5)^x$

To accurately graph the function $f(x) = 3 imes (0.5)^x$, the most effective approach is to calculate a few key points. These points will serve as our anchors on the coordinate plane, allowing us to sketch the curve with confidence. Let's start by evaluating the function at some simple integer values for 'x'. The easiest point to find is always when $x=0$. Plugging in 0 for 'x', we get $f(0) = 3 imes (0.5)^0$. Remember that any non-zero number raised to the power of 0 is 1. So, $f(0) = 3 imes 1 = 3$. This tells us that the graph passes through the point (0, 3). This is our y-intercept, and it aligns with the 'a' value in the general form of the exponential function. Now, let's try a positive value for 'x', say $x=1$. We calculate $f(1) = 3 imes (0.5)^1$. Since $(0.5)^1$ is simply 0.5, we have $f(1) = 3 imes 0.5 = 1.5$. So, another point on our graph is (1, 1.5). As expected, the y-value decreased from 3 to 1.5 as x increased from 0 to 1, confirming the decay nature. Let's try $x=2$: $f(2) = 3 imes (0.5)^2$. First, we calculate $(0.5)^2$, which is $0.5 imes 0.5 = 0.25$. Then, $f(2) = 3 imes 0.25 = 0.75$. This gives us the point (2, 0.75). The y-value continues to decrease, getting closer and closer to zero. Now, let's explore what happens for negative values of 'x', as this is where exponential growth becomes apparent. Let's try $x=-1$: $f(-1) = 3 imes (0.5)^{-1}$. A negative exponent means we take the reciprocal of the base. So, $(0.5)^{-1} = (1/2)^{-1} = 2/1 = 2$. Therefore, $f(-1) = 3 imes 2 = 6$. Our point here is (-1, 6). Notice how the y-value increased significantly as 'x' became more negative. Let's try $x=-2$: $f(-2) = 3 imes (0.5)^{-2}$. First, $(0.5)^{-2} = (1/2)^{-2} = (2/1)^2 = 4$. Then, $f(-2) = 3 imes 4 = 12$. This gives us the point (-2, 12). We can see a rapid increase in the y-value as 'x' goes further into the negative numbers. By calculating these points – (0, 3), (1, 1.5), (2, 0.75), (-1, 6), and (-2, 12) – we have a solid foundation for sketching the graph of $f(x) = 3 imes (0.5)^x$.

Sketching the Graph on Paper

With our key points in hand, we can now proceed to sketch the graph of $f(x)=3 imes (0.5)^x$ on paper. Begin by drawing a standard Cartesian coordinate system, with a horizontal x-axis and a vertical y-axis. Mark the origin (0,0). Now, plot the points we calculated: (0, 3), (1, 1.5), (2, 0.75), (-1, 6), and (-2, 12). Remember to label your axes and choose an appropriate scale. For the y-axis, since our values range from close to zero to 12 (and can go higher for more negative x), a scale that accommodates this range is necessary. After plotting these points, observe their positions. You'll notice that as you move from left to right (increasing x), the points descend. As you move from right to left (decreasing x), the points ascend rapidly. Connect these plotted points with a smooth, continuous curve. Crucially, remember that this curve should not touch or cross the x-axis. The x-axis acts as a horizontal asymptote for this function. This means that as x approaches positive infinity, the value of $f(x)$ approaches 0, but it never actually reaches 0. This is characteristic of exponential decay. For negative values of x, the graph rises steeply. Ensure your curve reflects this rapid ascent. The shape should be graceful, not jagged. The curve will be concave up. This means that if you were to draw tangent lines at various points along the curve, they would lie below the curve. The point (0, 3) is your y-intercept. Points to the right of the y-axis (positive x) will be below the y-intercept and get progressively closer to the x-axis. Points to the left of the y-axis (negative x) will be above the y-intercept and rise increasingly sharply. When you have finished sketching, examine the overall shape. Does it have the characteristic 'S' curve of exponential decay, starting high on the left and dropping low on the right, flattening out as it approaches the x-axis? Does it pass through your calculated points? Is the y-intercept at (0, 3)? If your sketch exhibits these features, you have successfully graphed $f(x) = 3 imes (0.5)^x$. This visual representation is key to understanding the behavior of exponential functions and how different parameters affect their shape and position.

Identifying the Correct Graph Choice

Once you have your hand-drawn graph of $f(x) = 3 imes (0.5)^x$, the next step is to determine which answer choice matches the graph you drew. This requires careful comparison of your sketch with the provided options. Look for the fundamental characteristics of the graph you created. First, verify the y-intercept. Your graph must pass through the point (0, 3). Any graph that does not cross the y-axis at 3 can be immediately eliminated. Next, consider the behavior of the function as 'x' increases (moving to the right on the graph). Since the base is 0.5 (which is between 0 and 1), the function should exhibit exponential decay. This means the graph should be decreasing as you move from left to right. If you see a graph that is increasing as x increases, it's not your function. Also, check if the graph is flattening out as it moves towards the x-axis on the right side. This indicates the horizontal asymptote at $y=0$. The graph should get closer and closer to the x-axis but never touch or cross it. Now, examine the behavior as 'x' decreases (moving to the left on the graph). For exponential decay with a base between 0 and 1, as 'x' becomes more negative, the function's value should increase rapidly. This means the graph should be rising steeply as you move to the left. Ensure the graph shows this steep ascent for negative x-values. Finally, consider the concavity. The graph of $f(x) = 3 imes (0.5)^x$ should be concave up. This means it curves upwards, like a bowl right-side up. If a graph appears concave down, it's likely incorrect. By systematically checking these features – the y-intercept, the decreasing trend for positive x, the horizontal asymptote, the steep increase for negative x, and the concavity – you can confidently identify the correct graph among the choices. It's like being a detective, matching the evidence from your calculations and sketch to the visual representations provided.

Conclusion: Mastering Exponential Function Graphs

In summary, graphing exponential functions like $f(x) = 3 imes (0.5)^x$ involves understanding the role of the initial value ('a') and the base ('b'). We identified that 'a=3' means the y-intercept is at (0, 3), and 'b=0.5' signifies exponential decay, meaning the graph decreases as 'x' increases and approaches the x-axis as a horizontal asymptote. By calculating key points such as (-1, 6), (0, 3), (1, 1.5), and (2, 0.75), we were able to plot these on a coordinate plane and connect them with a smooth, continuous curve that accurately represents the function's behavior. This process of calculating points, sketching the curve, and then matching it to given options is a fundamental skill in mathematics. It reinforces the visual understanding of algebraic concepts. Being able to accurately graph and identify exponential functions is not just about solving textbook problems; it's about developing a mathematical intuition that can be applied to real-world scenarios involving growth and decay. Whether you're analyzing population changes, financial investments, or radioactive decay, understanding these graphical representations provides invaluable insights. The ability to translate an equation into a visual form, and vice-versa, is a powerful tool in your mathematical arsenal.

For further exploration and practice on graphing and understanding exponential functions, you can visit reputable educational websites like Khan Academy or Wolfram MathWorld.