Problem 30 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, use a graphing utility to graph the function. $$ s(t)=2^{-t}+3 $$

Step-by-Step Solution

Answer

To plot the function \(s(t)=2^{-t}+3\), use a graphing utility and remember that the function decreases towards zero as t increases due to the base of the exponential function being a fraction. Also, take into account the '+3', which results in a vertical shift upwards by 3 units.

Step 1: Identify the Type of Function

Identify the type of function given. \(s(t)=2^{-t}+3\) is an exponential function because it is in the form \(a^{bt}+c\), where a, b, and c are constants.

Step 2: Identify the Behavior of the Function

Analyze how the function behaves. For \(2^{-t}\), as t increases, the value of the function decreases towards zero, because the base of the exponential function is a fraction. The '+3' is a vertical shift upwards by 3 units, meaning the graph will hover above the t-axis at the line s = 3.

Step 3: Plot the function

Use a graphing utility to plot the function according to the behavior analyzed in the previous step. This should yield an accurate graph of the function \(s(t)=2^{-t}+3\).

Key Concepts

Graphing Utility

A graphing utility is a handy tool that helps visualize mathematical functions by generating their graphical representations. This can include graphing calculators or software like GeoGebra, Desmos, and other similar programs. Using a graphing utility simplifies the process of plotting complex functions, as it can instantly compute values and represent them on a graph.

• Choose the correct type of function from the menu or input the function directly into the software.
• For the function in this exercise, input it as: \( s(t)=2^{-t}+3 \).
• Observe how the graph looks and behaves upon alteration of function parameters.

The main advantage of using a graphing utility is its ability to quickly illustrate the curve, making it easier to analyze properties and behaviors without manually plotting many points.
By visualizing the function \( s(t)=2^{-t}+3 \), students can easily spot its features, such as asymptotes and shifts, which aids in understanding its nature.

Function Behavior

Understanding the behavior of a function is a key aspect of analyzing how it reacts over its domain. For the exponential function \( s(t)=2^{-t}+3 \), we note certain important behavioral traits.

• **Exponential Decay:** Here, the term \( 2^{-t} \) represents exponential decay. As the variable \( t \) increases, the base \( 2^{-t} \) produces ever smaller results since the exponent is negative.
• **Approaching Zero:** As \( t \) moves towards positive infinity, \( 2^{-t} \) approaches zero but never actually reaches it, signifying the graph extends infinitely but never touches the t-axis.
• **Rate of Change:** Exponential decay means the function decreases at a slower rate as time goes on. Initially, this decrease is rapid, but slows considerably as \( t \) increases.

Understanding these functions' nature helps in predicting and rationalizing their long-term behavior, crucial for interpreting real-world data that exhibits similar exponential patterns.

Vertical Shift

A vertical shift in a graph involves moving it up or down along the y-axis. In the function \( s(t)=2^{-t}+3 \), the '+3' impacts this shift.

• **Upward Shift:** The '+3' means the entire function shifts 3 units upward. While the base function \( 2^{-t} \) has a horizontal asymptote at y = 0, \( s(t) \) shifts this to y = 3.
• **Graph hovering:** Instead of leveling out at t-axis, the graph hovers at y = 3, maintaining a consistent minimum distance from this line.
• **Impact on the Range:** The range of the function also shifts. Originally tending towards zero, it now tends towards three, highlighting the fact that this vertical shift impacts the overall output values of the function.

Recognizing vertical shifts makes it easier to analyze how modifications affect real-world models, enabling clearer insights into data interpretation.