Question

the table shows values for the function f at selected values of x.
which of the following function types best model these data?
a linear, because these data demonstrate roughly constant rates of change.
b exponential, because there is a common ratio between consecutive terms.
c quadratic, because these data demonstrate constant nonzero 2nd differences.
d polynomial greater than degree 2, because these data demonstrate non - linear rates of change.

Explanation:

Step1: Calculate the first - differences

The first - difference formula is \(\Delta f(x)=f(x + 1)-f(x)\).
For \(x = 1\) to \(x = 2\): \(\Delta f(1)=38 - 15=23\)
For \(x = 2\) to \(x = 5\): \(\Delta f(2)=70 - 38 = 32\)
For \(x = 5\) to \(x = 6\): \(\Delta f(5)=100 - 70=30\)
For \(x = 6\) to \(x = 7\): \(\Delta f(6)=123 - 100 = 23\)
For \(x = 7\) to \(x = 8\): \(\Delta f(7)=148 - 123=25\)
Since the first - differences (\(23,32,30,23,25\)) are not constant, the function is not linear (so option A is incorrect).

Step2: Calculate the ratios of consecutive terms

The ratio formula is \(r=\frac{f(x + 1)}{f(x)}\).
For \(x = 1\) to \(x = 2\): \(r_1=\frac{38}{15}\approx2.53\)
For \(x = 2\) to \(x = 5\): \(r_2=\frac{70}{38}\approx1.84\)
Since the ratios are not constant, the function is not exponential (so option B is incorrect).

Step3: Calculate the second - differences (if needed for quadratic)

But we can already see from the non - constant first - differences and non - constant ratios that the function has non - linear rates of change.
A polynomial of degree greater than \(2\) (or other non - linear functions) can have non - linear rates of change. Since we have ruled out linear (\(y=mx + b\), constant first - differences) and exponential (\(y = a\cdot b^x\), constant ratios), and the data does not show constant non - zero second - differences (for a quadratic \(y=ax^{2}+bx + c\), second - differences are constant), the function has non - linear rates of change.

Answer:

D. non - linear rates of change