Answer:
y = 0.5(3^x)
Explanation:
When matching a function to a table of values, the functions we usually try are linear, quadratic, and exponential. We can determine which of these might be appropriate by looking at differences in table values.
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first differences
Subtracting each height (y) value from the next, we get successive first differences of ...
1, 3, 9
These are not constant, and they do not have a constant difference of their own. However, we do notice they are related by a multiplicative factor of 3.
interpreting first differences
First differences are constant at a level equal to the degree of polynomial required to match the table values. If first differences are constant, then the table can be represented by a first-degree (linear) polynomial. Similarly, if second differences are constant, a second-degree (quadratic) polynomial will model the table.
If an exponential model is appropriate, differences will have the same constant ratio at every level. Here, first differences have ratios of 3/1 = 9/3 = 3. The second differences of 2 and 6 likewise have a ratio of 6/2 = 3. That ratio is the base of the exponential function.
exponential function
We have determined that the base of the exponential function is 3. The multiplier is the y-intercept (the value when x=0). The table tells us that is 0.5. Then we have ...
y = a·b^x . . . . . . 'a' = y-intercept; b = common ratio
y = 0.5·(3^x)
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Additional comment
The above discussion of differences applies to tables where the x-differences are constant. If they are other than 1, then the resulting function will need to be horizontally scaled. If the x-differences are not constant, other methods of regression analysis and interpolation are appropriate.