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Graphing an exponential function like \( f(x) = 2(3)^x \) involves several important steps. First, start with a clear coordinate plane. Once you have identified the function properties, begin by finding the y-intercept, which is the starting point of your graph. For \( f(x) = 2(3)^x \), the y-intercept is at the point (0, 2).

Next, use a series of strategic points to graph the curve of the exponential function. Exponential functions typically have a rapid increase or decrease, so ensure that your points capture this characteristic well. Use values both less than and greater than zero to see how the function behaves around the y-axis. Make a rough sketch by plotting these points: as \( x \) increases, \( f(x) \) increases rapidly, creating a characteristic upward curve.

Finally, draw a smooth curve through all the plotted points. Your graph should show that the function is asymptotic to the x-axis to the left, and it rises sharply to the right. Practicing these techniques will help you master graphing exponential functions.

Understanding the properties of an exponential function is crucial for accurate graphing. Exponential functions have a few distinct characteristics:

• **Rapid Change**: The base of the exponent dictates how quickly the function grows or decays. A base greater than 1, as in our function \( 2(3)^x \), indicates rapid growth.
• **Y-Intercept**: The y-intercept occurs at \( f(x) = 2 \) when \( x = 0 \), giving an initial value for sketching the graph.
• **Asymptote**: For \( x < 0 \), the function approaches but never touches the x-axis. This horizontal asymptote is a common feature of exponential functions.
• **Domain and Range**: The domain of exponential functions is all real numbers, while the range is strictly positive real numbers, owing to the increasing nature of the function.

Knowing these properties allows you to predict the behavior of any exponential function and aids in crafting a precise graph.

Creating a table of values is a straightforward method to understand the curve of an exponential function. Start by selecting a set of convenient \( x \) values, both negative and positive, and calculate the corresponding \( f(x) \) values.

For example, in \( f(x) = 2(3)^x \), if you choose \( x = -1, 0, 1, \) and \( 2 \), compute each \( f(x) \) as follows:

• At \( x = -1 \): \( f(x) = 2 \cdot 3^{-1} = \frac{2}{3} \)
• At \( x = 0 \): \( f(x) = 2 \cdot 3^{0} = 2 \)
• At \( x = 1 \): \( f(x) = 2 \cdot 3^{1} = 6 \)
• At \( x = 2 \): \( f(x) = 2 \cdot 3^{2} = 18 \)

With these calculated points, plot them on your graph. Observing how `f(x)` changes as `x` moves across the selected range gives insight into the overall curve of the function. This method not only helps in graphing but also deepens your understanding of the function's behavior.