Questions: Graph the given function by making a table of coordinates. f(x) = (2/3)^x Complete the table of coordinates. x -2 -1 0 1 2 y (Type integers or fractions. Simplify your answers.)

$Graph the given function by making a table of coordinates. f(x) = (2/3)^x Complete the table of coordinates. x -2 -1 0 1 2 y (Type integers or fractions. Simplify your answers.)$

Transcript text: Graph the given function by making a table of coordinates. \[ f(x)=\left(\frac{2}{3}\right)^{x} \] Complete the table of coordinates. \begin{tabular}{|c|c|c|c|c|c|} \hline $\mathbf{x}$ & -2 & -1 & 0 & 1 & 2 \\ \hline $\mathbf{y}$ & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ \\ \hline \end{tabular} (Type integers or fractions. Simplify your answers.)

Solution

Solution Steps

Step 1: Calculate $ f(x) $ for each $ x $ value

To complete the table, we need to calculate $ f(x) = \left(\frac{2}{3}\right)^{x} $ for each given $ x $ value.

Step 2: Calculate $ f(-2) $

\[ f(-2) = \left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^{2} = \frac{9}{4} \]

Step 3: Calculate $ f(-1) $

\[ f(-1) = \left(\frac{2}{3}\right)^{-1} = \frac{3}{2} \]

Step 4: Calculate $ f(0) $

\[ f(0) = \left(\frac{2}{3}\right)^{0} = 1 \]

Step 5: Calculate $ f(1) $

\[ f(1) = \left(\frac{2}{3}\right)^{1} = \frac{2}{3} \]

Step 6: Calculate $ f(2) $

\[ f(2) = \left(\frac{2}{3}\right)^{2} = \frac{4}{9} \]

Final Answer

The completed table of coordinates is: \[ \begin{tabular}{|c|c|c|c|c|c|} \hline \mathbf{x} & -2 & -1 & 0 & 1 & 2 \\ \hline \mathbf{y} & \frac{9}{4} & \frac{3}{2} & 1 & \frac{2}{3} & \frac{4}{9} \\ \hline \end{tabular} \]

{"axisType": 3, "coordSystem": {"xmin": -3, "xmax": 3, "ymin": -1, "ymax": 3}, "commands": ["y = (2/3)**x"], "latex_expressions": ["$y = \\left(\\frac{2}{3}\\right)^{x}$"]}

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