Problem 36 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The graph of \( f(x) = -e^x \) is a downward curve, crossing the y-axis at (0, -1) and approaching y = 0 as x increases.

Step 1: Understanding the Function

The function given is an exponential function \( f(x) = -e^x \). This is a standard exponential function \( e^x \) that has been multiplied by -1, reflecting it across the x-axis.

Step 2: Identifying Key Characteristics

The base of the exponential, \( e \), is approximately 2.718. Normally, \( e^x \) increases rapidly, but because of the negative sign, \( -e^x \) will decrease. The function will never touch the x-axis (asymptote at y = 0) and crosses the y-axis at \( (0, -1) \).

Step 3: Sketching the Graph

Begin by plotting the y-intercept at \( (0, -1) \). For other points, choose a few x-values to find corresponding y-values: for example, when \( x = 1, f(x) = -e^1 \approx -2.718 \), and when \( x = -1, f(x) = -e^{-1} \approx -0.368 \). Plot these points and draw a curve that approaches the x-axis as \( x \to \infty \) and extends towards \( -\infty \) for negative x-values.

Step 4: Verifying Axis-Asymptotic Behavior

Ensure the graph approaches the horizontal asymptote (y=0) from below as \( x \to \infty \). The graph should be entirely in the lower half-plane, indicating a reflection across the x-axis compared to \( e^x \).

Key Concepts

Graphing Exponential Functions

Graphing exponential functions may seem challenging at first, but it's a fascinating way to explore how numbers can grow or decrease rapidly. An exponential function often takes the form \( f(x) = a \cdot b^x \), where \( a \) is the initial value, and \( b \) is the base of the exponential. For the specific function \( f(x) = e^x \), \( e \) is approximately 2.718, a special mathematical constant. This function grows quickly as \( x \) increases, showcasing exponential growth.
To graph an exponential function like \( f(x) = e^x \):

• Start by finding the y-intercept by evaluating \( f(x) \) at \( x = 0 \). This gives the point (0,1) for \( e^x \).
• Determine the growth by evaluating other values, such as \( x = 1 \) or \( x = -1 \).
• Draw a smooth curve that passes through these points and reflects the exponential nature.

The graph should stretch upwards steeply as \( x \) becomes positive and flatten out approaching the asymptote as \( x \) becomes more negative. In the case of \( e^x \), this asymptote is the x-axis, which the graph approaches but never crosses. Understanding these steps makes graphing exponential functions clear and straightforward.

Negative Exponential Reflection

A negative exponential reflection occurs when the exponential function includes a negative sign, like \( f(x) = -e^x \). This negative sign in front of \( e^x \) flips the graph over the x-axis. Imagine holding a regular exponential graph and turning it upside down, and you'll have a negative reflection.
When graphing \( f(x) = -e^x \), you'll see:

• The function decreases as \( x \) increases, opposite of the standard exponential growth.
• The x-axis still acts as an asymptote, but this time the curve approaches it from below.
• Each point on the graph of \( e^x \) is mirrored downwards. So, while \( e^x \) goes through (0,1), \( -e^x \) will pass through (0,-1).

This reflection is an essential concept in understanding how variations can drastically change the behavior of an exponential function. Knowing how to manipulate and reflect these functions helps in exploring different types of real-world scenarios that involve decay or decrease instead of growth.

Asymptotes in Graphs

An asymptote is an important feature in graphing, referring to a line that a curve approaches, but never touches. For exponential functions, the horizontal asymptote is typically the x-axis, or \( y = 0 \).
Consider how the graph acts as it extends:

• As \( x \) becomes very large, the value of \( e^x \) grows, making \( y = 0 \) an unreachable, horizontal boundary.
• In the case of \( -e^x \), the graph approaches \( y = 0 \) from below, creeping closer but never actually reaching it.
• Asymptotes help us understand the behavior of functions over an infinite range, predicting outcomes without plotting every possible point.

Asymptotes act as guides. They help establish the end behavior of functions, showing limits and providing a visual understanding of how graphs behave at extremes. Understanding where these invisible boundaries lie is crucial when analyzing and predicting the behavior of exponential and other types of functions.