Look at this table:
x y
–
1 8.2
0 6.6
1 5
2 3.4
3 1.8
Write a linear function (y=mx+b) or an exponential function (y=a(b)x) that models the data.
y=
To determine whether a linear or exponential function would best model the data, we can look for any patterns or trends in the y-values corresponding to the x-values.
Examining the given data, it appears that as the x-values increase, the corresponding y-values decrease. This suggests an exponential relationship between x and y.
To write the exponential function that models the data, we need to find values for a and b.
Using the values provided in the table, we can find the values for a and b by substituting the x and y values into the exponential equation and solving for both variables. Since there are multiple data points, we can choose any two points to form two equations to solve for a and b.
Let's choose the points (1, 5) and (2, 3.4):
For the point (1, 5):
5 = a(b)^1
For the point (2, 3.4):
3.4 = a(b)^2
Now, we have a system of two equations:
5 = ab
3.4 = ab^2
We can divide these two equations to eliminate 'a':
(5/3.4) = b/b^2
Now, let's solve for 'b':
(5/3.4) = 1/b
b = 3.4/5
b = 0.68
Now, substitute this value of 'b' back into one of the original equations to solve for 'a'. Let's use 5 = ab:
5 = a(0.68)
a = 5/0.68 ≈ 7.35
Therefore, the exponential equation that models the data is:
y = 7.35(0.68)^x