Problem 21 - Precalculus : Building Concepts and Connections

Question

Sketch the graph of each function. $$f(x)=2(3)^{-x}$$

Step-by-Step Solution

Answer

The graph of the function \(f(x) = 2(3)^{-x}\) starts at the point (0, 2), approaches the line \(y = 0\) but never crosses it, and decreases rapidly as we move to the left from the y-intercept.

Step 1: Identify the y-intercept

To find the y-intercept, set \(x = 0\) in the function. This gives \(f(0) = 2(3)^{0} = 2\). Therefore, the y-intercept is at the point (0, 2).

Step 2: Identify the horizontal asymptote

For the given exponential decay function, the graph approaches a line called the horizontal asymptote as \(x\) moves towards positive or negative infinity. Here, as \(x\) tends to positive infinity, \(f(x)\) approaches 0. Thus the line \(y = 0\) is the horizontal asymptote.

Step 3: Sketch the graph

Start by drawing the y-intercept at (0, 2). Next, plot the horizontal asymptote, the x-axis (\(y = 0\)). The graph is in the first quadrant and approaches the x-axis as we move to the right. The line never crosses the x-axis. It also rapidly decreases as we move to the left from the y-intercept.

Key Concepts

Understanding the Y-Intercept

The y-intercept of a graph is a crucial concept in understanding the overall behavior of a function. Simply put, the y-intercept is the point where the graph of the function crosses the y-axis. In order to determine the y-intercept for the function \(f(x)=2(3)^{-x}\), you set \(x = 0\) since the y-intercept occurs at this point. Evaluating the function at \(x = 0\) gives us \(f(0) = 2(3)^{0} = 2\). Thus, the y-intercept is at the point \((0, 2)\). This means that at \(x = 0\), the function has a value of 2, and this point will be a starting point when sketching the graph of the function.

Exploring the Horizontal Asymptote

A horizontal asymptote provides insight into the end behavior of a function and represents a horizontal line that the graph approaches but never touches. For exponential decay functions like \(f(x)=2(3)^{-x}\), as \(x\) moves towards positive infinity, the function tends to get closer and closer to a particular value. In our given function, as \(x\) increases, \(f(x)\) approaches 0 but never actually reaches it. Hence, the line \(y=0\) acts as a horizontal asymptote. This means that as you sketch the graph, the curve will get infinitely closer to the x-axis (which is \(y=0\)) as it extends to the right, reflecting the behavior of the function trend.

Getting Started with Graph Sketching

When sketching the graph of an exponential decay function like \(f(x)=2(3)^{-x}\), it's essential to use both the y-intercept and horizontal asymptote as guides. Begin by plotting the y-intercept at the point \((0, 2)\). This is your initial anchor point on the graph.Next, draw the horizontal asymptote, which we've identified as the line \(y = 0\), or the x-axis. The curve of the graph should pass through the y-intercept and gradually get closer to the horizontal asymptote as \(x\) moves right. Make sure to note that the graph will not cross the x-axis. Additionally, as \(x\) moves left from the y-intercept, the graph sharply decreases. This sketch will reveal the characteristic downward trend consistent with exponential decay.