use a table of coordinates to graph the exponential function. begin by selecting - 2, - 1, 0, 1, and 2 for…

use a table of coordinates to graph the exponential function. begin by selecting - 2, - 1, 0, 1, and 2 for x. based on your graph, describe the shape of a scatter plot that can be modeled by ( f(x)=b^{x},0 < b < 1 )\n( f(x)=left(\frac{1}{5}\right)^{x} )\ndescribe the shape of a scatter plot that can be modeled by ( f(x)=b^{x},0 < b < 1 ). choose the correct description below.\na. increasing and then decreasing\nb. decreasing, with the rate of decrease slowing down\nc. decreasing and then increasing\nd. decreasing more and more rapidly\nf lncreasinn with the rate of increase slnwinn down

use a table of coordinates to graph the exponential function. begin by selecting - 2, - 1, 0, 1, and 2 for x. based on your graph, describe the shape of a scatter plot that can be modeled by ( f(x)=b^{x},0 < b < 1 )\n( f(x)=left(\frac{1}{5}\right)^{x} )\ndescribe the shape of a scatter plot that can be modeled by ( f(x)=b^{x},0 < b < 1 ). choose the correct description below.\na. increasing and then decreasing\nb. decreasing, with the rate of decrease slowing down\nc. decreasing and then increasing\nd. decreasing more and more rapidly\nf lncreasinn with the rate of increase slnwinn down

Answer

Explanation:

Step1: Analyze the function ( y = b^x) with (0 < b<1)

For an exponential function (y = b^x) where (0 < b<1), when (x) increases, (y) decreases. Let's take (b=\frac{1}{5}) (as in (f(x)=\left(\frac{1}{5}\right)^x)) and calculate values for (x=- 2,-1,0,1,2)

  • When (x = - 2), (y=\left(\frac{1}{5}\right)^{-2}=5^{2}=25)
  • When (x=-1), (y=\left(\frac{1}{5}\right)^{-1}=5)
  • When (x = 0), (y=\left(\frac{1}{5}\right)^{0}=1)
  • When (x = 1), (y=\frac{1}{5}=0.2)
  • When (x = 2), (y=\left(\frac{1}{5}\right)^{2}=\frac{1}{25}=0.04)

Step2: Study the rate of change

The function (y = b^x) with (0 < b<1) is a decreasing function. The derivative of (y = b^x) (using the formula ((a^x)^\prime=a^x\ln a)) is (y^\prime=b^x\ln b). Since (0 < b<1), (\ln b<0) and (y^\prime<0) (function is decreasing). Also, the second - derivative (y^{\prime\prime}=b^x(\ln b)^2>0) (the function is concave up). A concave - up decreasing function has a rate of decrease that is slowing down.

Answer:

B. Decreasing, with the rate of decrease slowing down