The standard exponential equation is expressed as:
[tex]y=a(b)^x[/tex]where:
• a is the ,intercept
,• b is the ,rate, (whether growth or decline)
Since we have several coordinate points from the table, we can make use of the coordinate points (3, 2/49) and (4, 2/343)
Set up a simultaneous equation using these coordinates as shown:
[tex]\begin{gathered} \frac{2}{49}=ab^3 \\ \frac{2}{343}=ab^4 \end{gathered}[/tex]Divide both equations to have:
[tex]\begin{gathered} \frac{(\frac{2}{49})}{(\frac{2}{343})}=\frac{ab^3}{ab^4} \\ \frac{\cancel{2}}{49}\times\frac{343}{\cancel{2}}=\frac{b^3}{b^4} \\ \frac{343}{49}=b^{3-4} \\ 7=b^{-1} \\ b=\frac{1}{7} \end{gathered}[/tex]Substitute the value of x, y, and b into any of the equations;
[tex]\begin{gathered} \frac{2}{49}=a(\frac{1}{7})^3 \\ \frac{2}{49}=(\frac{1}{343})a \\ \frac{2}{49}=\frac{a}{343} \\ 49a=2\times343 \\ 49a=686 \\ a=\frac{686}{49} \\ a=14 \end{gathered}[/tex]Get the required exponential function
[tex]\begin{gathered} y=a(b)^x \\ y=14(\frac{1}{7})^x \end{gathered}[/tex]Hence the required exponential equation is f(x) = 14(1/7)^x
