Question

Write an exponential function y = abx for a graph that includes (–3, 16) and (–1, 4) f(x) = 2(0.5)x f(x) = 0.5(2)x f(x) = 4(0.3)x f(x) = 3(4)x

Answer 1

Answer:
`f(x)=2(0.5)^x`

Explanation:

Given, the exponential function
`y = ab^x` for a graph that includes (–3, 16) and (–1, 4).

On putting these points, we will have

`f(-3)=16=ab^(-3)\\\\ f(-1)=4=ab^(-1)`

Now,

`(f(-3))/(f(-1))=(16)/(4)=(ab^(-3))/(ab^(-1))\\\\\Rightarrow\ (4)/(1)=(b^(-3))/(b^(-1))\\\\\Rightarrow\ (4)/(1)=(b)/(b^3)=(1)/(b^2)\\\\\Rightarrow\ b^2=(1)/(4)\\\\\Rightarrow\ b=\pm(1)/(2)=\pm0.5`

since the multiplicative factor cannot be negative, so b= 0.5.

At b= 0.5

`4=a(0.5)^(-1)\\\\\Rightarrow\ 4=a((1)/(2))^(-1)\\\\\Rightarrow\ 4=a(2)\\\\\Rightarrow \ a=2`

So, the required function is
`f(x)=2(0.5)^x`.