graph the exponential function.\n\n$f(x)=(\\frac{2}{3})^x$\n\nplot five points on the graph of the function…

Explanation:

Step1: Choose x - values

Let (x=- 2,-1,0,1,2).

Step2: Calculate y - values for (x = - 2)

Substitute (x=-2) into (y = (\frac{2}{3})^x), we get (y=(\frac{2}{3})^{-2}=\frac{1}{(\frac{2}{3})^2}=\frac{9}{4} = 2.25).

Step3: Calculate y - values for (x=-1)

Substitute (x = - 1) into (y=(\frac{2}{3})^x), we get (y=(\frac{2}{3})^{-1}=\frac{3}{2}=1.5).

Step4: Calculate y - values for (x = 0)

Substitute (x = 0) into (y=(\frac{2}{3})^x), we get (y=(\frac{2}{3})^0 = 1).

Step5: Calculate y - values for (x = 1)

Substitute (x = 1) into (y=(\frac{2}{3})^x), we get (y=\frac{2}{3}\approx0.67).

Step6: Calculate y - values for (x = 2)

Substitute (x = 2) into (y=(\frac{2}{3})^x), we get (y=(\frac{2}{3})^2=\frac{4}{9}\approx0.44).

Step7: Determine the asymptote

For an exponential function of the form (y = a^x) where (0\lt a\lt1) (here (a=\frac{2}{3})), the horizontal asymptote is (y = 0). The five points are ((-2,2.25),(-1,1.5),(0,1),(1,\frac{2}{3}),(2,\frac{4}{9})) and the horizontal asymptote is (y = 0). You can plot these points on the graph and draw the asymptote (y = 0) (the x - axis).

Answer:

The five points are ((-2,2.25),(-1,1.5),(0,1),(1,\frac{2}{3}),(2,\frac{4}{9})) and the horizontal asymptote is (y = 0).