Chapter 9.9, Problem 34HP

To determine

The relationship between the coefficients of $x^2$ and the constant second difference

Answer to Problem 34HP

For any interval of length 1, we see that the factor of growth for the exponential function is a

Explanation of Solution

Given:

The data modeled by a quadratic function $y=a{x}^2+c$

Concept Used:

If first difference of y-values is equal, then model be linear

If second difference be equal, then the model be quadratic

If first and second differences are not equal then the model will be exponential

Calculation:

In order to check relationship between the coefficients of $x^2$ and the constant second difference Consider a generic linear function $y=ax$ the difference for the first intervals is $a_1-a_0=a$ and the difference for the second intervals is

  $\begin{array}{c}{a}_2-a_1=2a-a\\ =a\end{array}$

So we see that for any interval of length 1, the difference in y is a

Consider a generic exponential function $y=a^x$ the difference for the first intervals is $\frac{a_1}{a_0}=a$ and the difference for the second intervals is $\frac{a_2}{a_1}=a$

Thus, for any interval of length 1, we see that the factor of growth for the exponential function is a