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Magazine Chapter 11: Problem 28
Graph each function. $$ f(x)=e^{x}-2 $$
Short Answer
Expert verified
Graph the exponential curve of \( f(x) = e^x - 2 \) with a horizontal asymptote at \( y = -2 \).
Step by step solution
01
Understanding the Function
The function given is \( f(x) = e^{x} - 2 \). This is an exponential function that represents a vertical shift of the basic function \( e^x \) by 2 units downwards.
02
Table of Values
To graph the function, we start by calculating a few values. Choose some x-values and evaluate the function at these points. For example: - \( x = -1 \) : \( f(x) = e^{-1} - 2 \approx 0.37 - 2 = -1.63 \) - \( x = 0 \) : \( f(x) = e^{0} - 2 = 1 - 2 = -1 \) - \( x = 1 \) : \( f(x) = e^{1} - 2 \approx 2.72 - 2 = 0.72 \)
03
Draw the Axes
On graph paper or using a graphing tool, draw the horizontal x-axis and the vertical y-axis. Make sure they intersect at the origin (0,0) and the scales are equal.
04
Plot the Points
Using the table of values, plot the points on the graph: - Point \((-1, -1.63)\)- Point \((0, -1)\)- Point \((1, 0.72)\). Make sure the points are plotted accurately according to the scales on your axes.
05
Draw the Exponential Curve
Draw a smooth curve through the points plotted. The curve should approach the line \( y = -2 \) as \( x \) goes to negative infinity (the horizontal asymptote), and rise sharply as \( x \) increases. Ensure that the curve is smooth and appropriately represents exponential growth.
06
Identify Key Features
Identify the asymptote, which is the line \( y = -2 \). Note that the y-intercept of the graph occurs at \((0, -1)\) and the graph passes through this key point. The curve never touches the asymptote.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Techniques
Graphing exponential functions such as \( f(x) = e^x - 2 \) involves several systematic steps. Start by understanding the general shape of the exponential function, which grows rapidly as \( x \) increases. This function appears as a rising curve when graphed. Drawing a graph begins with noting the axes: the horizontal x-axis and the vertical y-axis. Both axes intersect at the origin, point \( (0, 0) \).
- Use a consistent scale for both axes to ensure accuracy.
- Sketch the axes on graph paper or with digital tools for a more precise graph.
Vertical Shifts
Vertical shifts occur when a constant is added or subtracted from the exponential function. For \( f(x) = e^x - 2 \), subtracting 2 indicates a vertical shift downward by 2 units from the basic exponential function \( e^x \). In the graph:
- This shift affects the entire curve, moving it down parallel to its original path.
- The basic shape and growth pattern of the curve remain unchanged, but every point on the graph is now 2 units lower.
Asymptotes
Asymptotes are lines that the graph of a function approaches but never actually touches. In the case of \( f(x) = e^x - 2 \), the horizontal asymptote is \( y = -2 \). This is where the vertical shift comes into play, as it lowers the asymptote by 2 units from the typical \( y = 0 \) of the \( e^x \) function.
- The graph gets indefinitely close to this line as \( x \) moves toward negative infinity.
- The asymptote serves as a boundary that guides the path of the function.
Table of Values
Creating a table of values is a helpful tool for graphing any function, especially exponential ones. For \( f(x) = e^x - 2 \), it's important to choose a range of \( x \)-values and calculate the corresponding \( y \)-values. For example:
- At \( x = -1 \), \( f(x) = e^{-1} - 2 \approx -1.63 \)
- At \( x = 0 \), \( f(x) = e^{0} - 2 = -1 \)
- At \( x = 1 \), \( f(x) = e^{1} - 2 \approx 0.72 \)
