The rate in which a function increases or decreases between its points are called the slope, or the rate of change. For any function, the rate of change is calculated by the slope formula. Slope formula: where m = slope (a, f(a)) and (b, f(b)) are two points on the function. Here is an example: For the function, f(x) = 2x - 1, calculate the rate of change between the points, (-1, f(-1)) and (4, f(4)). Let(-1,f(-1))=(a,f(a)) and (4,f(4))=(b,f(b)). Use the given functin, f(x)=2x-1, to complete the points. For (-1, f(-1): For (4,f(4)) f(x)=2x-1 f(x)=2x-1 f(-1)=2(-1)-1 f(4)=2(4)-1 f(-1)=-3 f(4)=7 (-1,f(-1))=(-1,-3) (4,f(4))=(4,7) Next, calculate the slope for the function, using the points (-1, -3) and (4, 7). For the formula, let(-1,-3) =(a,f(a)) and (4,7) = (b,f(b)): The slope of the function, f(x)=2x-1, between the points (-1, -3) and (4,7) is 2. For each function, use the slope formula to calculate the rate of change between the points. In your final answer, include all of your calculations. f(x): (a, f(a) and (b,f(b) 1.) f(x)=x - 3 (0,f(0)) and (6,f(6)) 2.) f(x) = -x (-4,f(-4)) and(2,f(2)) 3.) f(x)=x2 (-2,f(-2)) and (0,f(0)) 4.) f(x)=x3 (-1,f(-1)) and (1,f(1)) 5.) f(x)=2x (0,f(0)) and (4,f(4))

Accepted Solution

I'll focus on one example:    f(x)=x^2:    (-2,f(-2)) and (0,f(0)) 4)

Find the average rate of change of f(x) = x^2 from x = -2 to x = 0:
           f(0) - f(-2)           (0)^2 - (-2)^2)         0-4
arc = ------------------ = ----------------------  = ---------- = -2 (answer)
              0-(-2)                         2                    2