Look at this table:
x y
–
1 2.2
0 4.4
1 8.8
2 17.6
3 35.2
Write a linear function (y=mx+b) or an exponential function (y=a(b)x) that models the data.
y=
To determine whether the data follows a linear or exponential pattern, we can check the ratios between consecutive values of y.
Let's calculate the ratios between each consecutive pair of y-values:
2.2/4.4 ≈ 0.5
4.4/8.8 ≈ 0.5
8.8/17.6 ≈ 0.5
17.6/35.2 ≈ 0.5
The fact that the ratios between consecutive y-values (0.5) are constant suggests that the data follows an exponential pattern.
To find the exponential function that models the data, let's look at the ratios between consecutive pairs of x-values:
1/0 ≈ undefined
0/1 = 0
1/0 ≈ undefined
2/1 = 2
3/2 ≈ 1.5
The ratios between the x-values are not constant, which means that the base (b) of the exponential function is not a constant. The exponential function that best models the data is therefore of the form y = a(b)^x.
Now, let's calculate the base (b) by dividing consecutive y-values:
4.4/2.2 = 2
8.8/4.4 = 2
17.6/8.8 = 2
35.2/17.6 = 2
The base (b) of the exponential function is 2.
Finally, let's calculate the coefficient (a) by dividing consecutive y-values by the base (b):
2.2/2 ≈ 1.1
4.4/2 ≈ 2.2
8.8/2 ≈ 4.4
17.6/2 ≈ 8.8
The coefficient (a) of the exponential function is roughly 1.1.
Therefore, the exponential function that models the data is:
y = 1.1(2)^x