Problem 31 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Sketch the graphs of the given functions on the same axes. \(y=0.5 e^{-x}, y=e^{-x}\), and \(y=2 e^{-x}\)

Step-by-Step Solution

Answer

The \(y\)-intercepts for the given functions are \((0, 0.5)\) for \(y=0.5e^{-x}\), \((0, 1)\) for \(y=e^{-x}\), and \((0, 2)\) for \(y=2e^{-x}\). To sketch the graphs, plot the \(y\)-intercepts for each function and draw a decreasing curve from the y-intercept towards the x-axis without intersecting it. The function with a greater value of \(a\) will have a steeper decay.

Step 1: Identify the properties of the exponential functions

All given functions are exponential functions of the form \(y=ae^{-x}\), with different values of \(a\). Exponential functions have the following properties: - Positive values: The function values are always positive for any value of x. - Intercept: The functions have a y-intercept at \((0, a)\). - Decay: All given functions are decreasing for the entire domain since the exponential function's rate is negative. These properties will help us explore the behavior of the given exponential functions.

Step 2: Determine y-intercepts for each function

Recall that the y-intercept for each function will be located at \((0, a)\). For \(y=0.5e^{-x}\): The y-intercept is at \((0, 0.5)\). For \(y=e^{-x}\): The y-intercept is at \((0, 1)\). For \(y=2e^{-x}\): The y-intercept is at \((0, 2)\).

Step 3: Sketch the graphs of the given functions

Using the properties of exponential functions and the y-intercepts determined in Step 2, we will now sketch the graphs of the given functions. 1. For \(y=0.5e^{-x}\): - Plot the y-intercept point at \((0, 0.5)\). - Since the function is always positive and decaying, draw a decreasing curve from the y-intercept towards the x-axis without intersecting it. 2. For \(y=e^{-x}\): - Plot the y-intercept point at \((0, 1)\). - Similar to the previous case, draw a decreasing curve from the y-intercept towards the x-axis without intersecting it. The graph's decay is steeper than the previous case since \(a=1\). 3. For \(y=2e^{-x}\): - Plot the y-intercept point at \((0, 2)\). - As before, draw a decreasing curve from the y-intercept towards the x-axis without intersecting it. The graph's decay is steeper than the above two cases since \(a=2\). After following these steps, the graphs of the given functions should be successfully drawn on the same axes, displaying their behavior and differences.

Key Concepts

Graphing Exponential Functions

Graphing exponential functions involves understanding the form and characteristics of these equations. Exponential functions are typically expressed in the form \(y = ae^{-x}\), where \(a\) is a constant that influences the graph's shape.
A few key properties to keep in mind when graphing:

• The curve is always above the x-axis because the outputs are positive values.
• The graph shows a consistent rate of decay as \(x\) increases, meaning the function values decrease.
• It heads closely towards the x-axis without ever actually touching or crossing it. This line that the graph approaches but never meets is called the asymptote.

For the given functions, plotting starts by marking the y-intercept, which also serves as a pivotal point for sketching the decaying curve. Ensure your graph displays the steepness representing the value of \(a\). A larger \(a\) makes a steeper graph.

Y-Intercept of Exponential Functions

The y-intercept of an exponential function is important because it indicates where the function will intersect the y-axis. This is a helpful starting point when sketching a graph.
For any function of the form \(y = ae^{-x}\), the y-intercept is at \((0, a)\). This means:

• For \(y=0.5e^{-x}\), the y-intercept is \((0, 0.5)\).
• For \(y=e^{-x}\), the y-intercept is \((0, 1)\).
• For \(y=2e^{-x}\), the y-intercept is \((0, 2)\).

Plotting these points on the graph provides a guide to draw the rest of the curve accurately. The y-intercept here represents the initial value from which the exponential decay begins, setting each function apart despite them following the same general form.

Exponential Decay

Exponential decay is a process where quantities reduce at a decreasing rate relative to their values. In the functions given, \(e^{-x}\) denotes that they are decaying exponential functions.
Key characteristics include:

• The values continuously approach zero but never actually reach it.
• The size of \(a\) determines the rate of decay. A larger \(a\) results in faster decay, evident by a steeper curve.
• Each curve is smooth and decreases consistently from left to right.

Understanding decay helps explain why the graphs tend to flatten as \(x\) becomes larger. While the initial portion of the curves may be steep due to rapid decay, they eventually smoothen as they approach the asymptote. This reflects how the quantity's reduction slows down over time.