–
4
–
18
–
3
–
13
–
2
–
8
–
1
–
3
0 2
Write a linear function (y=mx+b) or an exponential function (y=a(b)x) that models the data.
To determine whether the data can be modeled by a linear or exponential function, we can plot the points on a graph.
Plotting the points:
(4, 18)
(3, 13)
(2, 8)
(1, 3)
(0, 2)
From the graph, it is clear that the points do not lie on a straight line, so we can conclude that a linear function does not accurately model the data.
To determine if an exponential function will fit the data, we can calculate the ratios between the y-values:
18/4 = 4.5
13/3 = 4.33...
8/2 = 4
3/1 = 3
2/0
The ratios are approximately the same, indicating that an exponential function might be suitable to model the data.
Now, let's analyze the ratios to determine the constants a and b in the exponential function y = a(b)^x:
4.5 = a(b)^1
4.33... = a(b)^1
4 = a(b)^1
3 = a(b)^1
2 = a(b)^0
From the last equation, we can see that a(b)^0 = 2. Since any non-zero number raised to the power of 0 is 1, we can simplify this equation to a = 2.
Now, we have:
4.5 = 2(b)^1
4.33... = 2(b)^1
4 = 2(b)^1
3 = 2(b)^1
The common ratio is approximately 1.08. To determine b, we can use any of the equations above. Let's use the equation 4 = 2(b)^1:
4 = 2(b)^1
2 = b
So, the exponential function that models the data is:
y = 2(1.08)^x