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: $f(x)=2(0.5)^x$

Step-by-step 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,

$\dfrac{f(-3)}{f(-1)}=\dfrac{16}{4}=\dfrac{ab^{-3}}{ab^{-1}}\\\\\Rightarrow\ \dfrac{4}{1}=\dfrac{b^{-3}}{b^{-1}}\\\\\Rightarrow\ \dfrac{4}{1}=\dfrac{b}{b^3}=\dfrac{1}{b^2}\\\\\Rightarrow\ b^2=\dfrac{1}{4}\\\\\Rightarrow\ b=\pm\dfrac{1}{2}=\pm0.5$

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

At b= 0.5

$4=a(0.5)^{-1}\\\\\Rightarrow\ 4=a(\dfrac{1}{2})^{-1}\\\\\Rightarrow\ 4=a(2)\\\\\Rightarrow \ a=2$

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