Average rate of change between two points

Question

Consider the function f(x)=x^2 +2

Find the average rate of change between the points:

1) (1, f(-1)) and 4, f(4))

I know the answer for this one is 3 because 18-3 divided by 4-(-1) =15/5=3

But for the following two I can seem to get them right

2) (a, f(a)) and (b, f(b))

 

 

3) (x, f(x)) and (x+h0, f(x+h))

 

 

Answer 1

2).  Average rate of change = (f(b) - f(a))/(b - a)

 

                                  = [(b² + 2) - (a² + 2)]/(b - a)

 

                                  = (b² - a²)/(b - a)

 

                                  = (b + a)(b - a)/(b - a)

 

                                  = b + a

 

3).  Average rate of change

 

                   = [f(x + h) - f(x)]/[x + h - x]

 

                   = [(x + h)² + 2 - (x² + 2)]/h

 

                   = [x² +2xh + h² - x²]/h

 

                   = [2xh + h²]/h

 

                   = [h(2x + h)]/h = 2x + h