91Ó°ÊÓ

To graph an exponential function like \( f(x) = 3^{-x} \), we first need to find specific points on the graph. This process is known as substituting points.
Start by choosing a set of x-values. Common choices are -2, -1, 0, 1, and 2, but you can pick any values that make the calculations easier and give a good range of the function's behavior.
Next, substitute each chosen x-value into the function to calculate the corresponding y-values. For instance, for x = -2, we substitute into the function to get \( f(-2) = 3^{2} = 9 \).
Follow this process for all chosen x-values to get their respective y-values. Record these pairs as (x, y) points. In this example, we would get:
\(-2, 9\)
\(-1, 3\)
\(0, 1\)
\(1, \frac{1}{3}\)
\(2, \frac{1}{9}\)
These points are vital as they give precise locations to plot the function on a graph.

After finding the (x, y) points by substituting values into the function, the next step is to plot these points on a coordinate plane.
Start by drawing your coordinate axes (x-axis and y-axis). Ensure you have a good range that includes all your points.
Mark each calculated point on the plane. For example, plot the point (-2, 9) by finding -2 on the x-axis and moving vertically to 9 on the y-axis. Repeat this for all other points.
Carefully ensure each point is accurately placed according to both its x and y values.
Once all points are plotted, connect them smoothly. An exponential function like \( f(x) = 3^{-x} \) will form a curve that decreases rapidly as x increases. Ensure the curve smoothly passes through all the plotted points to represent the function correctly.

To verify the accuracy of your manually plotted graph, you can use a graphing calculator. These devices help visualize mathematical functions easily and accurately.
First, turn on your graphing calculator and access the function graphing mode.
Enter the function \( f(x) = 3^{-x} \) into the calculator. Ensure you input the negative exponent correctly.
Press the graph button to display the function. The calculator will create a smooth curve representing the exponential function.
Compare this graph with the one you plotted manually. There should be a close match between both graphs. Any discrepancies can help you identify and correct potential errors in your manual plot.
Using a graphing calculator is a reliable way to check your work and confirm understanding of how exponential functions behave visually.