Look at this table:
x y
1 4.14
2 12.42
3 37.26
4 111.78
5 335.34
Write a linear function (y=mx+b) or an exponential function (y=a(b)x) that models the data.
y=
To determine whether the function that models the data is linear or exponential, we can compare the ratio of y values for consecutive x values.
Ratio of y values for consecutive x values:
4.14/12.42 = 0.333
12.42/37.26 = 0.333
37.26/111.78 = 0.333
111.78/335.34 = 0.333
Since the ratio of y values is constant (0.333), this indicates that the data follows an exponential function.
To write the exponential function (y = a(b)^x) that models the data, we need to find the values of a and b.
To find the value of b, we can divide any y value by the corresponding x value:
b = 4.14/1 = 12.42/2 = 37.26/3 = 111.78/4 = 335.34/5 = 3.14
The value of b is approximately 3.14.
Now we can substitute the value of b into the equation to find the value of a:
4.14 = a(3.14)^1
a = 4.14/3.14 = 1.316
The equation that models the data is:
y = 1.316(3.14)^x