Given:
Required:
We need to find the first differences, second differences, and first difference ratios.
Explanation:
Recall that the first difference is the difference between values of the dependent variable by subtracting the previous value from each.
[tex]\frac{1}{9}-\frac{1}{27}=\frac{1\times3}{9\times3}-\frac{1}{27}=\frac{3}{27}-\frac{1}{27}=\frac{2}{27}[/tex][tex]\frac{1}{3}-\frac{1}{9}=\frac{1\times3}{3\times3}-\frac{1}{9}=\frac{3}{9}-\frac{1}{9}=\frac{2}{9}[/tex][tex]1-\frac{1}{3}=\frac{1\times3}{1\times3}-\frac{1}{3}=\frac{3}{3}-\frac{1}{3}=\frac{2}{3}[/tex][tex]3-1=2[/tex][tex]9-3=6[/tex]
Recall that the second difference is the difference of the first difference.
[tex]\frac{2}{9}-\frac{2}{27}=\frac{2\times3}{9\times3}-\frac{2}{27}=\frac{6}{27}-\frac{2}{27}=\frac{4}{27}[/tex][tex]\frac{2}{3}-\frac{2}{9}=\frac{2\times3}{3\times3}-\frac{2}{9}=\frac{6}{9}-\frac{2}{9}=\frac{4}{9}[/tex][tex]2-\frac{2}{3}=\frac{2\times3}{1\times3}-\frac{2}{3}=\frac{6}{3}-\frac{2}{3}=\frac{4}{3}[/tex][tex]6-2=4[/tex]
Divide the second number by the first number to find the ratio of the two numbers in the first difference.
[tex]\frac{2\/9}{2\/27}=\frac{2}{9}\times\frac{27}{2}=3[/tex][tex]\frac{2\/3}{2\/9}=\frac{2}{3}\times\frac{9}{2}=3[/tex][tex]\frac{2}{2\/3}=2\times\frac{3}{2}=3[/tex][tex]\frac{6}{2}=3[/tex]
We get the ratio of the first difference is constant so the given function is an exponential function.
Final answer:
[tex]An\text{ exponential function}[/tex]
