Average Rate of Change

Question

Find the average rate of change of f(x) = 4x^2-9 between the points (x, f(x)) and (x+h, f(x+h))

Answer 1

Average rate of change is the slope of the line between points

(f(x+h)-f(x))/(x+h-x)<-(denominator simplifies to just h)

plug "x+h" and "x" into f(x)

(4(x+h)²-9-(4x²-9))/(h)

Expand:

(4(x²+2xh+h²)-9-(4x²-9))/(h)

Distribute

(4x²+8xh+4h²-9-4x²+9))/(h)

Combine like terms

(8xh+4h²))/(h)

Factor an h our of the numerator

h(8x+4h)/h

Cancel h

8x+4h