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 \\ (1)/(4)=a\cdot b^0 \\ (1)/(4)=a\cdot1 \\ a=(1)/(4) \end{gathered}`

Now, we can write the general function again:

`f(x)=(1)/(4)b^x`

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

`\begin{gathered} f(x)=(1)/(4)b^x \\ 2=(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)=(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)=(1)/(4)\cdot2^x \\ or \\ f(x)=2^(x-2) \end{gathered}`