graph the exponential function.\n\n$f(x)=(\\frac{4}{3})^x$\n\nplot five points on the graph of the function…

graph the exponential function.\n\n$f(x)=(\\frac{4}{3})^x$\n\nplot five points on the graph of the function, and also draw the asymptote. then click on the graph - a - function button.

graph the exponential function.\n\n$f(x)=(\\frac{4}{3})^x$\n\nplot five points on the graph of the function, and also draw the asymptote. then click on the graph - a - function button.

Answer

Answer:

  1. Find five points:
    • When (x = - 2), (f(-2)=\left(\frac{4}{3}\right)^{-2}=\left(\frac{3}{4}\right)^{2}=\frac{9}{16}=0.5625). So the point is ((-2,0.5625)).
    • When (x=-1), (f(-1)=\left(\frac{4}{3}\right)^{-1}=\frac{3}{4} = 0.75). So the point is ((-1,0.75)).
    • When (x = 0), (f(0)=\left(\frac{4}{3}\right)^{0}=1). So the point is ((0,1)).
    • When (x = 1), (f(1)=\frac{4}{3}\approx1.33). So the point is ((1,\frac{4}{3})).
    • When (x = 2), (f(2)=\left(\frac{4}{3}\right)^{2}=\frac{16}{9}\approx1.78). So the point is ((2,\frac{16}{9})).
  2. Find the asymptote:
    • For an exponential - function of the form (y = a\cdot b^{x}) (in this case (a = 1) and (b=\frac{4}{3}>1)), the horizontal asymptote is (y = 0).

Explanation:

Step1: Calculate points

We substitute different (x) - values into the function (y=\left(\frac{4}{3}\right)^{x}) to get the corresponding (y) - values.

Step2: Determine asymptote

For an exponential function (y = b^{x}) with (b>0,b\neq1), when (b > 1) (as (\frac{4}{3}>1) here), the horizontal asymptote is (y = 0) because as (x\to-\infty), (y=\left(\frac{4}{3}\right)^{x}\to0).