How do you find the average rate of change of a function ?

Question

HIX Tutor · Accepted Answer

undefined To find the average rate of change of a function, you can follow these steps:

1. Choose two points on the graph of the function. Let's call these points (x₁, y₁) and (x₂, y₂).
2. Calculate the change in y-values by subtracting y₁ from y₂: change in y = y₂ - y₁.
3. Calculate the change in x-values by subtracting x₁ from x₂: change in x = x₂ - x₁.
4. Divide the change in y-values by the change in x-values: average rate of change = change in y / change in x.

And that's how you find the average rate of change of a function!

Kennedy Adams · Answer

Example:

#f(x) = x^2-6x+3#

The average rate of change of a function over an interval is: #(Delta y)/(Delta x)# where #Delta y# is the total change in #y# and #Delta x# is the total change in #x#. In our example, we know: #Delta x = 3 - 1 = 2# We want: #-2 = (Delta y)/(Delta x) = (Delta y)/2# Hence: #Delta y = -4# That is, we want to find #f(x)# with #f(3) = f(1) - 4# A basic linear function would be something like: #g(x) = -2x# noting that the slope of the line is #-2#, which is the average rate of change too. To this we can add a quadratic function that takes the same value at #x=1# and #x=3#. Let's make that value #0# by defining: #h(x) = (x-1)(x-3) = x^2-4x+3# So we can put: #f(x) = g(x)+h(x) = -2x+(x^2-4x+3) = x^2-6x+3# graph{(y-(x^2-6x+3))((x-1)^2+(y+2)^2-0.014)((x-3)^2+(y+6)^2-0.014) = 0 [-8.88, 11.12, -7.52, 2.48]}

Eli Assouline · Answer

undefined To find the average rate of change of a function over a given interval, you use the formula:

Average Rate of Change = (f(b) - f(a)) / (b - a)

Where:
- f(b) represents the value of the function at the end of the interval (point b).
- f(a) represents the value of the function at the beginning of the interval (point a).
- (b - a) represents the change in the input variable (x) over the interval.

You evaluate the function at the endpoints of the interval and then subtract the initial value of the function from the final value. Finally, divide this difference by the change in the input variable (the length of the interval) to obtain the average rate of change of the function over that interval.