Look at this table:
x y
1 3.87
2 11.61
3 34.83
4 104.49
5 313.47
Write a linear function (y=mx+b) or an exponential function (y=a(b)x) that models the data.
y=
To determine if the function that models the data is linear or exponential, we can examine the ratios between consecutive y-values.
For the given data:
3.87 / 1 = 3.87
11.61 / 2 ≈ 5.80
34.83 / 3 ≈ 11.61
104.49 / 4 ≈ 26.12
313.47 / 5 ≈ 62.69
The ratios between consecutive y-values are not consistent, indicating that the data does not follow a linear pattern. However, the ratios are roughly consistent when rounded to two decimal places, suggesting that the data may follow an exponential pattern.
To find the exponential function that models the data, we need to determine the values of a and b in the equation y = a(b)^x.
Choosing an initial point (1, 3.87) as a reference, we have the equation:
3.87 = a(b)^1
Simplifying this equation, we have:
3.87 = ab
To find the value of a, we can select another point from the table. Using the point (2, 11.61), we have the equation:
11.61 = ab^2
Dividing this equation by the previous equation (3.87 = ab), we have:
11.61 / 3.87 = b^2
Simplifying this equation, we find:
3 = b^2
Taking the square root of both sides, we get:
b = √3 or b ≈ 1.732
Substituting the value of b back into the previous equation (3.87 = ab), we can solve for a:
3.87 = a * 1.732
3.87 / 1.732 = a
a ≈ 2.235
Therefore, the exponential function that models the data is:
y ≈ 2.235(1.732)^x