Average Rate of Change
Key Questions
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The average rate of change is constant for a linear function.
Another way to state this is that the average rate of change remains the same for the entire domain of a linear function.
If the linear function is
#y=7x+12# then the average rate of change is 7 over any interval selected.Slope intercept form
#y=mx+b# , where#m# is the slope. -
Answer:
The rate of change is the slope of the graph.
Explanation:
It really doesn't make much sense to try to apply this to nonlinear functions, and you certainly cannot apply an "average" value to a non-linear function unless you first linearize it. Even then, the interpretation of what that "average" means must be carefully understood.
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Average rate of change is just another way of saying "slope".
For a given function, you can take the x-values and use them to calculate the y-values, then use the slope formula:#m=frac{y_2-y_1}{x_2-x_1}# Example: Given the function f(x) = 3x - 8, find the average rate of change between 1 and 4.
f(1) = 3(1) - 8 = -5 and f(4) = 3(4) - 8 = 4
m =
#frac{4-(-5)}{4-1}# =#frac{9}{3}# = 3 Surprised? No, because that is the slope between ANY two points on that line!Example: f(x) =
#x^2-3x# , find the average rate of change between 0 and 2.f(0) = 0 and f(2) = 4 - 6 = -2
m =
#frac{-2-0}{2-0}# =#frac{-2}{2}# = -1
Since this function is a curve, the average rate of change between any two points will be different.You would repeat the above procedure in order to find each different slope!
If you are interested in a more advanced look at "average rate of change" for curves and non linear functions, ask about the Difference Quotient.
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Answer:
The average rate of change of a function
#y=f(x)# , for example, tells you of how much the value of the function changes when#x# changes.Explanation:
Consider the following diagram:
when#x# changes from#x1# to#x2# the value of the function changes from#y1# to#y2# . The average rate of change will be:
#(y2-y1)/(x2-x1)# and it is, basically the slope of the blue line.For example:
if#x1=1# and#x2=5#
and:
#y1=2# and#y2=10#
you get that:
Average rate of change#=(10-2)/(5-1)=8/4=2# This means that for your function:
#color(red)("every time "x" increases of 1 then "y" increases of 2"#
Obviously your function is not a perfect straight line and it will change differently inside that interval but the average rate can only evaluate the change between the two given points not at each individual point.