Problem 94 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptote of the graph. $$f(x)=e^{3 x}$$

Step-by-Step Solution

Answer

For the function \(f(x)=e^{3x}\), the graph steeply rises for positive \(x\) values and approaches zero as \(x\) becomes negative, meaning \(y = 0\) is the horizontal asymptote.

Step 1: Construct a Table of Values

Use a graphing utility to generate a table of values for the function by plugging several values of \(x\) into the function \(f(x)=e^{3x}\). The range of \(x\) values should be broad enough to offer a comprehensive view of the graph, for example, from -2 to 2.

Step 2: Sketch the Graph

On a coordinate plane, plot the points from the table of values and draw a smooth curve that connects these points. The curve should show the general shape of the function \(f(x)=e^{3x}\). It should rise steeply for positive \(x\) values and approach zero as \(x\) goes to negative.

Step 3: Identify Asymptote

The asymptote is the line to which the curve approaches but never reaches. For the function \(f(x)=e^{3x}\), the curve approaches the x-axis as \(x\) goes to negative, and so the x-axis, or \(y = 0\), is the horizontal asymptote.

Key Concepts

Graphing Utility

To begin graphing an exponential function like \(f(x) = e^{3x}\), a graphing utility is a great tool. It helps visualize complex mathematical concepts by allowing you to input functions and instantly see their graphs. A graphing utility can be a standalone calculator, a computer program, or even an online tool. These utilities provide features that are especially useful for understanding exponential graphs.

When using a graphing utility:

• Enter the equation \(f(x) = e^{3x}\) into the input area.
• Explore settings to adjust the viewing window so it best captures the behavior of the function across a range of \(x\) values.
• Observe how the graph behaves for both positive and negative \(x\).

Employing a graphing utility makes it easier to interpret how the function grows and to identify key characteristics like asymptotes and intercepts.

Table of Values

Creating a table of values is fundamental in understanding and plotting the function accurately, especially if you're manually sketching the graph. With the function \(f(x) = e^{3x}\), you select several \(x\) values, plug them into the function, and compute the corresponding \(f(x)\) values.

For example, choose a range of \(x\) values like -2, -1, 0, 1, and 2:

• For \(x = -2\), \(f(-2) = e^{-6} \approx 0.0025\)
• For \(x = -1\), \(f(-1) = e^{-3} \approx 0.05\)
• For \(x = 0\), \(f(0) = e^{0} = 1\)
• For \(x = 1\), \(f(1) = e^{3} \approx 20.09\)
• For \(x = 2\), \(f(2) = e^{6} \approx 403.43\)

These points can be plotted on a coordinate grid to form the shape of the exponential curve, which shows rapid growth for positive \(x\) and approaching zero for negative \(x\).

Horizontal Asymptote

A key feature of exponential functions is the horizontal asymptote. This is a line that the curve gets infinitely close to but never touches. In the case of \(f(x) = e^{3x}\), as \(x\) approaches negative infinity, the values of \(f(x)\) become small and approach zero.

This indicates that the x-axis, or \(y = 0\), serves as the horizontal asymptote for this particular function. It helps to visualize that while the curve will rise steeply in the positive \(x\) direction, it will never cross or precisely meet the x-axis as it stretches to the left. Identifying the horizontal asymptote is crucial because it provides insight into the long-term behavior of the function.

Exponential Growth

Exponential growth describes a process that increases rapidly over time, and this is evident in the graph of \(f(x) = e^{3x}\). As you plot the function or observe it on a graphing utility, you'll notice:

• For positive \(x\) values, there's a sharp upward curve indicating rapid increase.
• The rate of growth becomes faster as \(x\) moves further to the right.
• The graph never touches the x-axis, maintaining a distance that diminishes exponentially as \(x\) becomes more negative.

Understanding exponential growth is essential in recognizing patterns in fields like biology, finance, and technology. In these fields, exponential functions model phenomena such as population dynamics, compound interest, and even data processing advancements.