Question 1189588
the equation is f(x) = -3 * x^2


the value of x is from -3 to -1.


when x = -3, f(x) = -3 * (-3)^2 = -3 * 9 = -27
when x = -1, f(x) = -3 * (-1)^2 = -3 * 1 = -3


you can use the geometric series formula to solve this.


the geometric series formula is An = A1 * r^(n-1).


A1 = the value of f(x) when x = -3.
that becomes -3 * (-3)^2 = -3 * 9 = -27


the formula becomes An = -27 * r^(n-1)


A3 is equal to f(x) when x = -1 because x = -1 is 2 units more positive than x = -3.


the formula becomes A3 = -27 * r^(3-1) which becomes A3 = -27 * r^2.
when An = A3, the value of n is equal to 3.
the value of A3 is the value of f(x), when x = -1.
that becomes -3 * (-1)^2 = -3.


the formula becomes -3 = -27 * r^(3-1) which becomes -3 = -27 * r^2.


solve for r to get:


r = (-3/-27)^(1/2) = .333333333.....


what this says is that, each successive value in the geometric series is 1/3 the value of the preceding value.


you start at A1 = -27
then A2 = -9
then A3 = -3.


your common ratio is .3333333..... = 1/3.


that's your average rate of change.
each succeeding term in the sequence is 1/3 * the value of the the preceding term.