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To graph any function, it is helpful to start by plotting points. In this case with the exponential function \( f(x) = e^x - 2 \), we will use a table of values to determine precise points to plot. Select various values for \( x \), calculate the corresponding \( y \)-values, and then plot these points on a coordinate plane. For example, if we choose \( x = -2, -1, 0, 1, 2 \), we calculate as follows: ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀- For \( x = -2 \), \( y = e^{-2} - 2 \). ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀- For \( x = -1 \), \( y = e^{-1} - 2 \). ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀- For \( x = 0 \), \( y = e^{0} - 2 = -1 \). ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀- For \( x = 1 \), \( y = e^{1} - 2 \). ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀- For \( x = 2 \), \( y = e^2 - 2 \). Once these points are plotted, connect them smoothly to form the graph.

An important aspect of the given function \( f(x) = e^x - 2 \) is the vertical shift. Compare this with the basic exponential function \( e^x \), which has no shifts. The formula here indicates a downward shift by 2 units. This vertical shift means every point on the graph of \( e^x \) is moved down by 2 units. Visualizing this helps in understanding how the graph is modified from the base function. Note: The larger \( x \) gets, the graph gets closer to the horizontal asymptote at \( y = -2 \), but it never actually reaches it.

Graphing calculators can be a valuable tool for verifying your manually plotted graphs. Enter the function \( f(x) = e^x - 2 \) into the calculator, and check that the plotted graph matches your manual one. This step provides a visual confirmation and helps ensure accuracy. Besides verification, graphing calculators can also perform complex calculations and plot points quickly. Thus, they are useful for both learning and error-checking in plotting exponential functions. Remember to familiarize yourself with your calculator's functions to make the verification process smooth and efficient.