Rate of change is one of the most deciding factors when you are constructing a function. Rate of change usually modifies the x variable in an f(x)=x style function.
The interesting part of the a rate of change is that adding a leading number to an equation is in some senses a "transformation" of the parent function y=x, which looks like this.
In this circumstance, the rate of change is '1', meaning if x changes 1, y also changes 1.
If we change our rate of change, it will transform the graph be changing it's slope.
an f(x) with a rate of change of 2 could look like this red function.
How can we make sense out of this very visually obvious concept in a numerical sense?
Rate of change is two dimensional, so both x and y must be taken into account.
The formula to find rate of change is Δy / Δx, y value change divided by x value change.
This can be expressed more simply as y2 - y1 / x2 - x1.
So in the parent function it comes out like this:
y2-y1 / x2 - x1 | 1-0 / 1-0 = rate of change 1 - using the two points (0,0) and (1,1)
We can find this based on any two points on the line.
It also works for y=2x
2-0 / 1-0 = 2 rate of change from the y=2x (0,0) and (2,1)
Rate of change is useful useful for finding equations of linear functions, as well as identifying linear and non linear functions. In a linear function, rate of change will be constant, other functions, will not have a constant rate of change.
Rate of change is unaffected by transformations like moving left right and up and down. y=2x +3 has the same rate of change as y=2x-4 (They're parallel)
Michael, thank you for your explanation on rate of change. You took the reader step by step which helped me get a clear understanding of the concept. The pictures also helped because they provided a visual on how rate of change is portrayed on a graph. Thank you!
ReplyDeletemichael,
ReplyDeletenice job of making the connection of these graphs to the idea of transformations. well done.
professor little