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

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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}

User Vfabre
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