Problem 46 - 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 asymptotes of the graph. $$s(t)=3 e^{-0.2 t}$$

Step-by-Step Solution

Answer

A table of values would show that as t increases, s(t) decreases. The graph of the function starts relatively high and decreases as t increases, getting closer and closer to the t-axis but never quite touching it. The horizontal asymptote of the function is at s(t) = 0.

Step 1: Construct the Table of Values

Use a graphing utility to generate values for function at different points. For example, choose values for t from -1 to 10 and calculate the corresponding s(t). These exact values will vary based on the graphing utility but should result in a table showing that as t increases, s(t) decreases.

Step 2: Sketch the Graph

Based on the table of values, sketch the graph. The graph should start relatively high when t is small and decrease as t increases, getting closer and closer to the t-axis but never quite touching it.

Step 3: Identify the Asymptotes

An asymptote is a line that a graph approaches as the inputs or outputs (t or s(t) in this case) get closer to infinity. For the function \(s(t)=3 e^{-0.2 t}\), as t goes to infinity, \(e^{-0.2 t}\) goes to 0, and so s(t) goes to 0. This suggests a horizontal asymptote at s(t) = 0.

Key Concepts

Table of Values for Exponential Functions

Creating a table of values for an exponential function is an essential first step in graphing it accurately. This table helps to have concrete numerical points to plot on the graph.
To create such a table for the function \( s(t) = 3 e^{-0.2t} \), you can select several different values for \( t \), preferably within the range that interests you (for example, from \( t = -1 \) to \( t = 10 \)).

• Calculate \( s(t) \) for each chosen value of \( t \) using the given exponential equation.
• The result for each \( t \) gives you a pair \((t, s(t))\) which you can plot on a graph.

This process creates a data set that illustrates how the function behaves. You'll notice as \( t \) increases, \( s(t) \) decreases, confirming the function's nature as an exponential decay.

Sketching Graphs of Exponential Functions

Once you have your table of values, you're ready to sketch the graph of the exponential function. Start by plotting each point from your table on a coordinate plane. Connect these points smoothly, as exponential graphs are continuous and don't have sharp angles.
The graph of \( s(t) = 3 e^{-0.2t} \) should start high at times when \( t \) is small, like around \( t = -1 \). As \( t \) grows, the value of \( s(t) \) decreases, demonstrating a downward trend characteristic of exponential decay.

• The curve should get progressively closer to the t-axis.
• Maintain a smooth curve throughout, as exponential functions remain fluent between plotted points.

This sketch provides a visual representation of how the function behaves over the range of \( t \) values you selected.

Understanding Horizontal Asymptotes

Horizontal asymptotes are lines that the graph of a function approaches but never crosses as it extends towards infinity. For the function \( s(t) = 3 e^{-0.2 t} \), as \( t \) increases to infinity, the term \( e^{-0.2 t} \) approaches zero.
Consequently, \( s(t) \) approaches zero, indicating that the horizontal asymptote of this graph is at \( s(t) = 0 \). This means that no matter how large \( t \) becomes, \( s(t) \) will always stay just above the horizontal line at zero, but will never touch or cross this line.

• The function value, \( s(t) \), becomes smaller but never reaches zero.
• This characteristic is crucial in understanding the long-term behavior of exponential decay functions.

Recognizing the horizontal asymptote helps predict the behavior of the function beyond the plotted range, guiding further analysis of the function's trend over time.