Problem 38 - Intermediate Algebra

Question

For Problems \(33-38\), graph each of the exponential functions. $$ f(x)=e^{-x} $$

Step-by-Step Solution

Answer

Graph resembles exponential decay, decreasing to approach zero as \( x \) grows.

Step 1: Understand the Function

The given function is an exponential function of the form \( f(x) = e^{-x} \). This is an example of exponential decay because the exponent is negative, which means the function decreases as \( x \) increases.

Step 2: Identify Key Points

To graph the function \( f(x) = e^{-x} \), let's find some key points by substituting a few \( x \)-values. Calculating, we find these values: - \( f(0) = e^{0} = 1 \)- \( f(1) = e^{-1} = \frac{1}{e} \approx 0.368 \)- \( f(-1) = e^{1} = e \approx 2.718 \)These values give us specific points: \( (0, 1) \), \( (1, 0.368) \), and \( (-1, 2.718) \).

Step 3: Plot the Points

Plot the points from Step 2 on a coordinate plane: (0, 1), (1, 0.368), (-1, 2.718). These will help us understand the shape and direction of the graph.

Step 4: Draw the Graph

Using the plotted points, draw a smooth curve that starts high on the left (approaching infinity as \( x \) approaches negative infinity) and decreases smoothly to approach the x-axis as \( x \) approaches positive infinity. The curve should never touch the x-axis but get arbitrarily close to it as \( x \) increases.

Step 5: Analyze the Graph

Notice that the function decreases but never reaches zero, illustrating the horizontal asymptote at \( y = 0 \). The graph is a reflection of the function \( f(x) = e^{x} \) across the y-axis, due to the negative exponent.

Key Concepts

Exponential Decay

Exponential decay describes a process where a quantity decreases rapidly at first, and then more and more slowly over time. This concept can be visualized in many natural phenomena, such as radioactive decay or cooling of a hot substance. Mathematically, an exponential decay function has a form similar to \( f(x) = a \cdot e^{-kx} \), where

• \( a \) is the initial amount,
• \( k \) is the decay constant, which must be positive, so that as \( x \) increases, \( f(x) \) decreases.

For the function \( f(x) = e^{-x} \), we see exponential decay because the exponent of \( e \) is negative. This indicates that as \( x \) becomes larger, the output of the function becomes smaller, approaching but never quite reaching zero. Understanding exponential decay is vital as it provides insight into processes where reduction occurs at a rate proportional to the current value, reflecting many real-world scenarios.

Graphing Functions

Graphing functions, especially exponential ones, is essential to understanding their behavior visually. When graphing a function like \( f(x) = e^{-x} \), it’s important to determine key points: specific \( x \)-values and their corresponding \( y \)-values, to help visualize the curve.For \( f(x) = e^{-x} \):

• When \( x = 0 \), \( f(x) = 1 \), indicating that the y-value starts at 1 on the graph.
• As \( x \) increases to 1, \( f(x) = \frac{1}{e} \approx 0.368 \), a decrease showing the declining nature of the function.
• When \( x = -1 \), \( f(x) = e \approx 2.718 \), showing that for negative \( x \), the y-value is greater than 1.

These points are plotted on the coordinate plane to help form the graph. The graph depicts the function rapidly declining initially and then flattening as it approaches the x-axis.

Horizontal Asymptote

A horizontal asymptote represents a value that a function approaches but never quite reaches as \( x \) goes to positive or negative infinity. For the function \( f(x) = e^{-x} \), the horizontal asymptote is the x-axis, or \( y = 0 \).This means, no matter how large \( x \) grows (moving to the right), the value of \( f(x) \) will get closer and closer to zero but never actually equals zero. This behavior is typical of exponential decay, where quantities become infinitely small but persistently positive, maintaining a fraction of their initial value.The horizontal asymptote is a critical feature of the graph of any decaying exponential function, informing us that over time or distance, the value levels off toward a consistent boundary.

Coordinate Plane

The coordinate plane is a fundamental grid system essential to graphing any mathematical function, including exponential ones. Consisting of two perpendicular axes — the x-axis (horizontal) and the y-axis (vertical) — it allows for the systematic plotting of points that represent relationships defined by algebraic equations.When graphing \( f(x) = e^{-x} \), the coordinate plane is used to plot key points calculated from the function:

• Point (0, 1) shows where the function is intersecting the y-axis.
• Point (1, 0.368) helps visualize the decline of the function.
• Point (-1, 2.718) illustrates behavior for negative x-values.

These points are connected smoothly to represent the exponential decay, providing a visual understanding of the function’s properties. The coordinate plane is not just essential for creating graphs; it’s a vital learning tool for seeing the mathematical relationships in action.