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An exponential function graph represents the visual depiction of an exponential equation, which typically takes the form of \( f(x) = a^x \), where \( a \) is a constant base and \( x \) is the exponent. When sketching these graphs, there are several characteristics to keep in mind:

• The graph always passes through the point \( (0,1) \), because any non-zero number to the power of 0 equals 1.
• The rate of increase or decrease of the function is dependent on whether the base \( a \) is greater than 1 or between 0 and 1, respectively.
• If the base \( a \) is greater than 1, the graph will rise rapidly, indicating growth. Conversely, if the base \( a \) is less than 1, the graph will descend rapidly, indicating decay.
• As the value of \( x \) increases, the value of \( f(x) \) moves farther from the x-axis in case of a growth function, or approaches the x-axis in case of a decay function – while never touching or crossing the x-axis.

In our exercise, we are dealing with \( f(x) = \left(\frac{1}{3}\right)^x \), which is an exponential decay function due to the base \( \frac{1}{3} \) being less than 1. The graph will therefore show a decreasing pattern as \( x \) increases, reflecting the nature of the function.

To create an accurate graph of a function, it's essential to plot several key points of the function. These points serve as a guide to drawing the curve. Here are steps to follow:

• Begin with easy-to-calculate points such as when \( x = 0 \), which often yields \( (0,1) \) in exponential functions.
• Choose some integer values of \( x \) and calculate the corresponding \( y \) values to get a sense of how rapidly the function increases or decreases.
• Plot each calculated point on the coordinate plane, using a dot to represent the location of the point with its exact \( x \) and \( y \) values.
• Once enough points are plotted, draw a smooth curve through them to represent the continuous nature of the function.

For example, in our function \( f(x) = \left(\frac{1}{3}\right)^x \), after plotting the point (0,1), we could also plot (1, \( \frac{1}{3} \)) to illustrate the decrease. It's beneficial to plot points where \( x \) has negative values as well, to show the behavior when \( x \) is less than 0, which will demonstrate the curve moving upwards in those cases.

A horizontal asymptote of a graph is a horizontal line that the graph approaches but never actually touches or crosses. It represents the behavior of a graph as the independent variable \( x \) approaches positive or negative infinity. For an exponential function, such as our example \( f(x) = \left(\frac{1}{3}\right)^x \), the horizontal asymptote typically occurs at \( y = 0 \). This is because as the exponent \( x \) becomes very large (positively or negatively), the value of the exponential function approaches 0, but never quite reaches it.
Recognizing the horizontal asymptote is crucial when sketching exponential graphs as it ensures the curve is approaching the correct value at the extremes and helps in maintaining the integrity of the graph's shape. Remember, for an exponential decay function where the base is between 0 and 1, the x-axis itself acts as the horizontal asymptote.