Look at this table:
x y
1 4.77
2 14.31
3 42.93
4 128.79
5 386.37
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 accurately modeled by a linear function or an exponential function, let's analyze the relationship between the x and y values:
When we look at the y-values, we can see that they increase by a factor of approximately 3 each time the x-value increases by 1:
14.31 ÷ 4.77 ≈ 3
42.93 ÷ 14.31 ≈ 3
128.79 ÷ 42.93 ≈ 3
386.37 ÷ 128.79 ≈ 3
Since the y-values increase exponentially by a factor of approximately 3 each time the x-value increases by 1, the data can be accurately modeled by an exponential function.
Now, let's write the exponential function that models the data:
Let y = a(b)^x
To find the values of a and b, we can use any of the given data points. Let's use the point (1, 4.77):
4.77 = a(b)^1
4.77 = ab
Now, let's use a different data point to create a system of equations. Let's use the point (2, 14.31):
14.31 = a(b)^2
We already know that ab = 4.77, so we can substitute that value into the second equation:
14.31 = (4.77)b^2
14.31 ÷ 4.77 = b^2
3 = b^2
b = √3
Substituting the value of b back into ab = 4.77:
a(√3) = 4.77
a = 4.77 ÷ √3
Therefore, the exponential function that models the given data is:
y = (4.77 ÷ √3)(√3)^x