Problem 40 - Algebra for College Students

Question

Graph each of the exponential functions. See answer section. $$ f(x)=e^{x}-2 $$

Step-by-Step Solution

Answer

The graph is an upward sloping curve starting below -2 with a horizontal asymptote at \( y = -2 \).

Step 1: Understand the Basic Form of the Function

The given function is \( f(x) = e^x - 2 \). This is an exponential function where the base is \( e \), the natural logarithm base, approximately equal to 2.718. The \(-2\) indicates a vertical shift downwards by 2 units.

Step 2: Identify Key Features of the Graph

The function \( f(x) = e^x \) has a horizontal asymptote at \( y = 0 \). Since our function is \( f(x) = e^x - 2 \), this asymptote shifts down to \( y = -2 \). The graph passes through the point \( (0, e^0 - 2) = (0, -1) \), since \( e^0 = 1 \).

Step 3: Choose Additional Points for Plotting

Choose additional \( x \) values to find corresponding \( y \) values. For example, when \( x = 1 \), \( y = e^1 - 2 \approx 0.718 \). When \( x = -1 \), \( y = e^{-1} - 2 \approx -1.632 \). This helps in drawing a more accurate graph.

Step 4: Draw the Graph

Using the points calculated, plot them on the coordinate plane. Draw a smooth curve that approaches the asymptote \( y = -2 \) as \( x \to -\infty \) and rises without bound as \( x \to \infty \). Ensure the curve passes through the calculated points.

Key Concepts

Graphing Exponential Functions

When graphing exponential functions like \( f(x) = e^x - 2 \), it's important to understand the curve's overall behavior. These functions are characterized by rapid growth or decay depending on the base. In this case, the base \( e \) indicates natural growth. The function starts off with lower values on the left and increases as it moves right.

To effectively graph such functions, follow these steps:

• Determine the y-intercept: Here, when \( x = 0 \), \( f(0) = e^0 - 2 = -1 \), giving a starting point for the graph at (0, -1).
• Calculate additional points for accuracy: Use stepwise values like \( x = 1 \) and \( x = -1 \). For \( x = 1 \), \( f(1) \approx 0.718 \) and for \( x = -1 \), \( f(-1) \approx -1.632 \).
• Sketch the curve: Begin at the y-intercept and smoothly draw a curve through the additional points, rising gradually as it extends to the right and approaching the asymptote on the left.

Remember that for exponential growth models, the function will rise faster as \( x \) increases, creating a characteristic upward sweeping graph.

Asymptotes in Graphs

Asymptotes are lines that a graph approaches but never actually touches. For the function \( f(x) = e^x - 2 \), the horizontal asymptote is crucial for understanding its long-term behavior.

Exponential functions with the form \( e^x \) naturally have a horizontal asymptote at \( y = 0 \). This reflects the lowest y-value the function can reach, as it approaches zero without ever actually reaching it. However, the \(-2\) in \( e^x - 2 \) shifts this asymptote down, meaning the function now flattens out closer to \( y = -2 \) instead.

Understanding this horizontal shift helps:

• Predict the function's behavior as \( x \) approaches negative infinity. It will get closer and closer to \( y = -2 \), but never quite reach it.
• Aid in graphing: Ensure that the curve drawn gets closer to the line \( y = -2 \) without crossing it, reflecting the concept of an asymptote accurately.

An asymptote provides guidance on how the function behaves at extremely large or small values of \( x \). It's a roadmap for how to position the curve of a graph correctly.

Transformations of Exponential Functions

Transformations are changes made to the basic exponential function to alter its graph. In \( f(x) = e^x - 2 \), a downward vertical translation is the key transformation under consideration. This influences the shape and position of the graph significantly.

Here's how transformations can be handled:

• Vertical Shifts: The \(-2\) in the expression \( e^x - 2 \) represents a move 2 units down from the function \( e^x \). All points on the graph drop 2 spaces vertically.
• Effect on Y-intercept: Originally at \( (0, 1) \) for \( e^x \), it changes to \( (0, -1) \) after the vertical shift.
• Impact on Asymptotes: Similarly, the horizontal asymptote at \( y = 0 \) descends to \( y = -2 \), following the shift.

Transformations make graphs versatile, allowing them to fit diverse real-world contexts by adjusting parameters like vertical shifts, stretching, compressing, or reflecting across axes. This flexibility makes exponential functions useful in modeling various growth and decay scenarios.