Look at this table:
x y
1 4.59
2 13.77
3 41.31
4 123.93
5 371.79
Write a linear function (y=mx+b) or an exponential function (y=a(b)x) that models the data.
y=
This table shows an exponential relationship between x and y. To find the exponential function that models the data, we need to find the values of a and b in the equation y = a(b)^x.
To do this, let's calculate the common ratio, b, by dividing each y value by the previous y value:
\(b = \frac{y_{n}}{y_{n-1}}\)
Using the given data:
\(b = \frac{4.59}{13.77} = 0.333333333333\)
\(b = \frac{13.77}{41.31} = 0.333333333333\)
\(b = \frac{41.31}{123.93} = 0.333333333333\)
\(b = \frac{123.93}{371.79} = 0.333333333333\)
From these calculations, we can see that the common ratio is \(b = 0.333333333333\).
Now, let's substitute one of the points (x, y) into the equation y = a(b)^x to solve for a:
Using the point (1, 4.59):
\(4.59 = a(0.333333333333)^1\)
Solving for a:
\(a = \frac{4.59}{0.333333333333}\)
\(a \approx 13.77\)
Now that we have the values of a and b, we can write the exponential function that models the data:
\(y \approx 13.77(0.333333333333)^x\)