Look at this table:
x y
5 17.72
6 21.72
7 25.72
8 29.72
9 33.72
Write a linear function (y=mx+b) or an exponential function (y=a(b)x) that models the data.
y=
To determine whether the given data can be modeled by a linear function or an exponential function, we can analyze the relationship between the x-values and the corresponding y-values.
By observing the table, we can see that as the x-values increase, the y-values also increase, but not at a steady rate. This indicates that the data cannot be best modeled by a linear function since a linear function would have a constant slope.
Instead, let's analyze if the data can be modeled by an exponential function. We'll compare the ratios of the y-values for consecutive x-values:
Ratio between y-values for consecutive x-values:
21.72 / 17.72 ≈ 1.2287
25.72 / 21.72 ≈ 1.1866
29.72 / 25.72 ≈ 1.1536
33.72 / 29.72 ≈ 1.1343
The fact that these ratios are roughly constant suggests that the data can be modeled by an exponential function.
We can now use the formula for an exponential function, y = a(b)^x, to model the data:
y = a(b)^x
We need to find the values of a and b that best fit the data. Let's choose the first data point (5, 17.72) and substitute it into the equation:
17.72 = a(b)^5
Now let's choose another data point, (6, 21.72), and substitute it into the equation:
21.72 = a(b)^6
We now have a system of two equations with two variables: a and b. We can solve this system of equations to find the values of a and b that best fit the data.
If you provide additional data points or any other guidelines for finding a and b, I can provide a more accurate and specific response.