x 3 6 9 15 21
f(x) 7.79 9.426 11.405 16.699 24.448
If the function is linear or exponential, write a formula for it below. If the function is neither, enter NONE.
since x grows by 3 each step, f(x) is linear if y does as well. The differences in y are
1.636, 1.979, ...
clearly not linear.
If f(x) is exponential, then the y terms have a constant ratio. Those are
1.21, 1.21, 1.464, 1.464
so, not exponential either
To determine if the given function is linear or exponential, we can look at the pattern in the data points.
First, let's calculate the difference between consecutive x-values:
3 - x = 3 - 0 = 3
6 - 3 = 3
9 - 6 = 3
15 - 9 = 6
21 - 15 = 6
Since the difference between consecutive x-values is not the same, this suggests that the function is not linear.
Next, let's calculate the ratio of consecutive y-values:
f(3) / f(0) = 7.79 / 0 ≈ undefined
f(6) / f(3) ≈ 9.426 / 7.79 ≈ 1.21
f(9) / f(6) ≈ 11.405 / 9.426 ≈ 1.21
f(15) / f(9) ≈ 16.699 / 11.405 ≈ 1.46
f(21) / f(15) ≈ 24.448 / 16.699 ≈ 1.46
The ratio between consecutive y-values is approximately the same, indicating that the function is exponential.
From the given data points, we can see that f(x) is increasing exponentially. To determine an exponential formula (y = ab^x), we can find the common ratio (b) by taking the ratio of any two consecutive y-values:
b = f(6) / f(3) ≈ 1.21
Now, we can solve for the base (b) by finding the square root of the common ratio:
b ≈ √1.21 ≈ 1.1
To find the initial value (a), we can substitute x = 0 and f(0) = a into the exponential formula:
a = f(0) = 7.79
Therefore, the formula for the function is approximately:
f(x) = 7.79 * (1.1)^x
Note: The formulas provided are approximations since we rounded the values during the calculations.