Which kind of function best models the data in the table? Use differences or ratios.
x | y
0 | 0.6
1 | 4.2
2 | 29.4
3 | 205.8
4 | 1440.6
A. exponential
B. quadratic
C. linear
D. none of the above
I can't decide which one, but I think it's b.
You can only "decide" after you have done some analysis, you appear to
be just guessing.
Is there a common difference between consecutive y values ?
if so , you have a linear.
is there a common ration between consecutive y values?
if so, you have an exponential.
Did you check for those properties???
y(n+1) = 7 y(n)
y = a b^ x
if x = 0 y = 0.6 = a b^0 = a
so
y = 0.6 b^x
if x = 1, y = 4.2
4.2 = 0.6 b^1 = 0.6 b
so
b = 7
so
y = 0.6 * 7^x
==================
check
if x = 3
y = 0.6 * 7^3 = 205.8 sure enough
To determine which kind of function best models the data in the table, we can analyze the differences or ratios between the values of y. Let's consider the first differences and second differences to determine the pattern.
First differences:
1st diff between 0.6 and 4.2 = 4.2 - 0.6 = 3.6
1st diff between 4.2 and 29.4 = 29.4 - 4.2 = 25.2
1st diff between 29.4 and 205.8 = 205.8 - 29.4 = 176.4
1st diff between 205.8 and 1440.6 = 1440.6 - 205.8 = 1234.8
Second differences:
2nd diff between 3.6 and 25.2 = 25.2 - 3.6 = 21.6
2nd diff between 25.2 and 176.4 = 176.4 - 25.2 = 151.2
2nd diff between 176.4 and 1234.8 = 1234.8 - 176.4 = 1058.4
Looking at the first differences, we can see that the values are increasing at an exponential rate. However, when we look at the second differences, the values are not constant, which suggests that the data does not follow a quadratic pattern.
Since the first differences are increasing exponentially, we can conclude that the data is best modeled by an exponential function. Therefore, option A. "exponential" is the correct answer.
Your initial choice of option B. "quadratic" is incorrect, as the data does not follow a quadratic pattern.