Question

Write a rule for the function X -1 0 1 2 3 Y 1/8 1/4 1/2 1 2

Answer 1

An exponential function can be represented as:

$f(x)=a\cdot b^x$

To find the constans "a" and "b" we can choose 2 given points and substitute it in the general function.

First, lets choose the point (0,1/4):

$\begin{gathered} f(x)=a\cdot b^x \\ \frac{1}{4}=a\cdot b^0 \\ \frac{1}{4}=a\cdot1 \\ a=\frac{1}{4} \end{gathered}$

Now, we can write the general function again:

$f(x)=\frac{1}{4}b^x$

To find "b", we can choose another point. Lets choose (3,2)

$\begin{gathered} f(x)=\frac{1}{4}b^x \\ 2=\frac{1}{4}b^3 \\ 2\cdot4=b^3 \\ 8=b^3 \\ \text{If we factor 8, we will find that 8=2}^3 \\ \text{So,} \\ 2^3=b^3 \\ \text{If we have the same expoent, we have to have the same base. } \\ b=2 \end{gathered}$

Now, we write the rule of the fuction:

$\begin{gathered} f(x)=a\cdot b^x \\ f(x)=\frac{1}{4}\cdot2^x \\ or \\ f(x)=2^{-2}\cdot2^x \\ Since\text{ we have the same base, we can sum the expoents:} \\ f(x)=2^{x-2} \end{gathered}$

The exponential function can be expressed as:

$\begin{gathered} f(x)=\frac{1}{4}\cdot2^x \\ or \\ f(x)=2^{x-2} \end{gathered}$